inventory routing problem under uncertainty · inventory routing problem under uncertainty zheng...

Submitted tomanuscript (Please provide the mansucript number)

Inventory Routing Problem under Uncertainty

Zheng CUI Daniel Zhuoyu LONGDepartment of Systems Engineering and Engineering Management Chinese University of Hong Kong

New Territories Hong Kong zcuisecuhkeduhk zylongsecuhkeduhk

Jin QIDepartment of Industrial Engineering and Decision Analytics Hong Kong University of Science and Technology

Kowloon Hong Kong jinqiusthk

Lianmin ZHANGDepartment of Management Science and Engineering Nanjing University Nanjing zhanglmnjueducn

We study a stochastic inventory routing problem with a finite horizon The supplier acts as a central planner

who determines replenishment quantities and also the times and routes for delivery to all retailers We

allow ambiguity in the probability distribution of each retailerrsquos uncertain demand Adopting a service-level

viewpoint we minimize the risk of uncertain inventory levels violating a pre-specified acceptable range We

quantify that risk using a novel decision criterion the Service Violation Index that accounts for how often

and how severely the inventory requirement is violated The solutions proposed here are adaptive in that

they vary with the realization of uncertain demand We provide algorithms to solve the problem exactly and

then demonstrate the superiority of our solutions by comparing them with several benchmarks

Key words inventory routing distributionally robust optimization risk management adjustable robust

optimization decision rule

1 Introduction

A key concept in supply chain management is the coordination of different stakeholders to minimize

the overall cost In that regard the vendor-managed inventory (VMI) system as a relatively new

business model has attracted extensive attention in both academia and industry In the VMI

system the supplier (vendor) behaves like a central decision maker determining not only the timing

and quantities of the replenishment for all retailers but also the route on which they are visited

Under this arrangement retailers benefit from saving efforts to control inventory Meanwhile the

supplier can improve the service level and reduce its costs by using transportation capacity more

1

Z Cui et al Inventory Routing Problem under Uncertainty2 Article submitted to manuscript no (Please provide the mansucript number)

efficiently For a more detailed discussion of the VMI conceptrsquos advantages see Waller et al (1999)

Cetinkaya and Lee (2000) Cheung and Lee (2002) and Dong and Xu (2002)

This joint management of inventory and vehicle routing gives rise to the inventory routing

problem (IRP) The term IRP was first used by Golden et al (1984) when defining a routing

problem with an explicit inventory feature Federgruen and Zipkin (1984) study a single-period

IRP and describe the economic benefits of coordinating the decisions related to distribution and

inventory Since then many variants of the IRP have been proposed over the past three decades

(Coelho et al 2014a) planning horizons can be finite or infinite excess demand can be either lost

or back-ordered and there can be either just one or more than one vehicle Scholars have used

these formulations on many real-world applications of the IRP which include maritime logistics

(Christiansen et al 2011 Papageorgiou et al 2014) the transportation of gas and oil (Bell et al

1983 Campbell and Savelsbergh 2004 Groslashnhaug et al 2010) groceries (Gaur and Fisher 2004

Custodio and Oliveira 2006) perishable products (Federgruen et al 1986) and bicycle sharing

(Brinkmann et al 2016) We refer interested readers to Andersson et al (2010) for more on IRP

applications

Since the basic IRP incorporates a classical vehicle routing problem that is already NP-hard

it follows that the stochastic IRP must be even more complex We next focus our review on

the stochastic version of the IRP especially in the case of uncertain demand A classical way

to formulate the stochastic IRP is using Markov decision process which assumes that the joint

probability distribution of retailersrsquo demands is known Campbell et al (1998) introduce a dynamic

programming model in which the state is the current inventory level at each retailer and the Markov

transition matrix is obtained from the known probability distribution of demand in their model

the objective is to minimize the expected total discounted cost over an infinite horizon This work is

extended by Kleywegt et al (2002 2004) who solve the problem by constructing an approximation

to the optimal value function Adelman (2004) decomposes the optimal value function as the sum

of single-customer inventory value functions which are approximated by the optimal dual prices

Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 3

derived under a linear program Hvattum and Loslashkketangen (2009) and Hvattum et al (2009) solve

the same problem using heuristics based on finite scenario trees

Another way to solve the stochastic IRP is via stochastic programming Federgruen and Zipkin

(1984) accommodate inventory and shortage cost in the vehicle routing problem and offer a heuristic

for that problem in a scenario-based random demand environment Coelho et al (2014b) address

the dynamic and stochastic inventory routing problem in the case where the distribution of demand

changes over time They give the algorithms for four solution policies whose use depends on whether

demand forecasts are used and whether emergency trans-shipments are allowed Adulyasak et al

(2015) consider the stochastic production routing problem with demand uncertainty under both

two-stage and multi-stage decision processes They minimize average total cost by enumerating all

possible scenarios and solve the problem by deriving some valid inequalities and taking a Benders

decomposition approach

The stochastic IRP literature is notable however for not adequately addressing two impor-

tant issues The first of these involves demand uncertainty Both the Markov decision process

and stochastic programming methods assume a known probability distribution for the uncertain

demand but full distributional information is seldom available in practice Owing to various rea-

sons such as estimation error and the lack of historical data we may have only partial informa-

tion on the probability distribution To cope with this shortcoming classical robust optimization

methodology has been applied to the IRP (see eg Aghezzaf 2008 Solyalı et al 2012 Bertsimas et

al 2016) These authors use a polyhedron to characterize the realization of uncertain demand and

then minimize the cost for the worst-case scenario In particular Aghezzaf (2008) assumes nor-

mally distributed demands and minimizes the worst-case cost for possible realizations of demands

within certain confidence level Solyalı et al (2012) consider a robust IRP by using the ldquobudget

of uncertaintyrdquo approach proposed by Bertsimas and Sim (2003 2004) and solve the problem via

a branch-and-cut algorithm Bertsimas et al (2016) consider the robust IRP from the perspective

of scalability they propose a robust and adaptive formulation that can be solved for instances of


around 6000 customers However traditional robust optimization does not capture any frequency

information and focuses only on the worst-case realization mdash an approach that some consider to be

unnecessarily conservative We therefore propose a distributionally robust optimization framework

that does exploit the frequency information The resulting model allows us to make robust inventory

replenishment and routing decisions that can immune against the effect of data uncertainty

Furthermore since this is a multi-period problem with uncertain demand each periodrsquos opera-

tional decisions should be adjusted in response to previous demand realization in order to achieve

appealing performance For such multi-stage problems Ben-Tal et al (2004) propose an adjustable

decision rule approach that models decisions as functions of the uncertain parametersrsquo realizations

Yet because finding the optimal decision rule is computationally intractable various decision rules

have been proposed to solve the problem approximately by restricting the set of feasible adaptive

decisions to some particular and simple functional forms (see eg Ben-Tal et al 2004 Chen et al

2008 Goh and Sim 2010 Bertsimas et al 2011 Kuhn et al 2011 Georghiou et al 2015) readers

are referred to the tutorial by Delage and Iancu (2015) To reduce computational complexity we

will use a linear decision rule that restricts the decisions in each period to be affine functions of

previous periodsrsquo uncertain demands The effectiveness of this decision rule has been established

both computationally (eg Ben-Tal et al 2005 Bertsimas et al 2018) and analytically (eg Kuhn

et al 2011 Bertsimas and Goyal 2012) To the best of our knowledge Bertsimas et al (2016) is

the only paper that incorporates the linear decision rule in IRP and hence is able to solve large-

scale instances However our paper differs from theirs in three ways we account for the frequency

information of demands we incorporate an enhanced decision rule and we propose a new decision

criterion that is consistent with risk aversion

The second issue that is largely ignored in the literature arises from the stochastic IRPrsquos objective

function Most research on this topic minimizes the expectation of operational cost which is the sum

of holding cost shortage cost and transportation cost However the calibration of cost parameters

especially the per-unit shortage cost is challenging in many practical cases (Brandimarte and


Zotteri 2007) And even if cost parameters could be precisely estimated the inventory cost is a

piecewise linear function and so calculating its expectation involves the sort of multi-dimensional

integration that increases the computational complexity (Ardestani-Jaafari and Delage 2016) Last

but not least expected cost is a risk-neutral criterion and thus fails to account for decision makersrsquo

attitudes toward risk Yet risk aversion is well documented empirically (as in the St Petersburg

paradox) and applied in inventory literatures (eg Chen et al 2007) So instead of optimizing the

expected cost we present an alternative approach that resolves the issues previously described

Our objective function is related to the concept of service level which is a prevalent requirement

in supply chain management (Miranda and Garrido 2004 Bernstein and Federgruen 2007 Bertsi-

mas et al 2016) From a practical standpoint it is extremely important for retailers to maintain

inventory levels within a certain range On the one hand the upper limit corresponds to a retailerrsquos

capacity eg storage capacity On the other hand running out of inventory will damage brand

image and also impose substantial cost due to emergent replenishments It follows that establishing

a lower limit will also help control inventory levels The rationale of such a service-level require-

ment also relates to the target-driven decision making It can be traced back to Simon (1955)

who suggests that the main goal for most decision making is not to maximize returns or minimize

costs but rather to achieve certain targets Since then a rich body of descriptive literatures has

confirmed that the decision making is often driven by targets (Mao 1970) Indeed with the absence

of inventory control Jaillet et al (2016) and Zhang et al (2018) have investigated the target-driven

decision making in the vehicle routing problem It is therefore natural to assume that in stochastic

IRP the objective is to maintain the inventory within certain target levels defined by an upper

and a lower bound

Given such a range imposed on inventory there are two classical ways to model the service-level

requirement The first approach is to ensure that the inventory level remains within the desired

range for all possible demand realizations however this approach is often viewed as being too

conservative The second way of modeling the service-level requirement is to minimize the violation


probability via chance constrained optimization Yet chance constraints are well known to be

nonconvex and computationally intractable in most cases (eg Ben-Tal et al 2009) that approach

has also been criticized for being unable to capture a violationrsquos magnitude (Diecidue and van

de Ven 2008) Hence we introduce the Service Violation Index (SVI) a general decision criterion

that we use to evaluate the risk of violating the service-level requirement and to address the issues

discussed so far Our SVI decision criterion generalizes the performance measure proposed by Jaillet

et al (2016) when studying the stochastic vehicle routing problem It can be represented from

classical expected utility theory and can therefore be constructed using any utility function In

the case of single period the SVI is associated with the satisficing measure axiomatized by Brown

and Sim (2009)

We summarize our main contributions as follows

bull Instead of minimizing the expectation of inventory and transportation cost we study robust

IRP from the service-level perspective and aim to keep the inventory level within a certain range

The existence of uncertainty in demand motivates us to propose a multi-period decision criterion

that allows decision makers to account for the risk that the uncertain inventory level of each retailer

in each period will violate the stipulated service requirement window

bull We use a distributionally robust optimization framework to calibrate uncertain demand

by way of descriptive statistics to derive solutions that protect against demand variation This

approach incorporates more distributional information and will yield less conservative results than

does the classical robust optimization approach

bull We derive adaptive inventory replenishment solutions that can be updated with observed

demand by implementing a decision rule approach Then the robust IRP can be formulated as a

large-scale mixed-integer linear optimization problem

bull We provide algorithms that can solve our model efficiently by way of the Benders decomposi-

tion approach

This rest of our paper proceeds as follows Section 2 defines the problem and discusses how

we model and evaluate uncertainty in the multi-period case In Section 3 we formulate adaptive


decisions using our decision rule approach and Section 4 proposes a solution procedure under

various information sets Section 5 reports results from several computational studies We conclude

in Section 6 with a brief summary

Notations A boldface lowercase letter such as x represents a column vector row vectors are

given within parentheses and with a comma separating each element xprime = (x1 xn) We use

(yxminusi) to denote the vector with all elements equal to those in x except the ith element which

is equal to y that is (yxminusi) = (x1 ximinus1 y xi+1 xT ) Boldface uppercase letters such as

Y represent matrices thus Y i is the ith column vector of Y and yprimei is its ith row vector For

two matrices XY isin Rmtimesn the standard inner product is given by 〈XY 〉 =m983123i=1

n983123j=1

XijYij We

use ldquolerdquo to represent an element-wise comparison for example X le Y means Xij le Yij for all

i= 1 m j = 1 n We denote uncertain quantities by characters with ldquo˜rdquo sign such as d and

the corresponding characters themselves such as d as the realization We denote by V the space

of random variables and let the probability space be (ΩF P) We assume that full information

about the probability distribution P is not known Instead we know only that P belongs to an

uncertainty set P which is a set of probability distributions characterized by particular descriptive

statistics An inequality between two uncertain parameters t ge v implies statewise dominance

that is it implies t(ω) ge v(ω) for all ω isin Ω In addition the strict inequality t gt v implies that

there exists an 983171gt 0 for which tge v+ 983171 We let em isinRT be the mth standard basis vector We use

[middot] to represent a set of running indices eg [T ] = 1 T and denote the cardinality of the set

S as |S|

2 Model

Section 21 defines the inventory routing problem In Section 22 we introduce a new service

measure to evaluate the uncertain outcome from any inventory routing decision

21 Problem statement

We consider a basic stochastic inventory routing problem in which there is one supplier andN retail-

ers over T periods The problem is defined formally on a directed network G = ([N ]cup0A) where


node 0 is the supplier node and A is the set of arcs Given any subset of nodes S sube ([N ]cup 0) we

define A(S) as the set of all arcs that connect two nodes in S That is A(S) = (i j)isinA i j isin S

We let D isin VNtimesT denote the matrix of uncertain demand where Dnt is retailer nrsquos uncertain

demand in period t Also Dt=

983059D1 Dt

983060isin VNtimest is the matrix of uncertain demand from

period 1 to period t hence we have DT= D We assume that D has an upper and a lower bound

which are denoted (respectively) D and D The support of D we denote by W = D|DleDleD

Any excess demand is assumed to be backlogged For the sake of simplicity we assume that the

transportation cost from node i to node j is deterministic and denoted by cij for (i j)isinA

We assume that the supplier uses only a single vehicle but one of unconstrained capacity to

deliver products In each period the supplier observes the inventory levels of all retailers and

then makes three decisions jointly which subset of retailers to visit which route to use and what

quantities of product to replenish

The decisions made in period tisin [T ] are affected by the demand realization in previous periods

we therefore represent those decisions as functions of previous uncertain demands (ie Dtminus1

) The

routing decisions are denoted by ytn z

tij isinBt where Bt is the space of all measurable functions that

map from RNtimes(tminus1) to 01 More specifically in period t we have ytn

983059D

tminus1983060= 1 if retailer n is

visited and yt0

983059D

tminus1983060= 1 if the supplierrsquos vehicle is used to replenish a subset of retailers

Moreover for (i j) isinA we have ztij

983059D

tminus1983060= 1 if that vehicle travels directly from node i to

node j Retailer nrsquos replenishment decision is denoted by qtn

983059D

tminus1983060isinRt where Rt is the space of

measurable functions that map from RNtimes(tminus1) to R For retailer n in period t we use xtn

983059D

t983060isinRt+1

to represent the ending inventory Without loss of generality (WLOG) we assume that the system

has no inventory at the start of the planning horizon x0n = 0 Furthermore we put D

0= empty so that

any function on D0is essentially a constant The inventory dynamics can therefore be calculated

as

xtn

983059D

t983060= xtminus1

n

983059D

tminus1983060+ qtn

983059D

tminus1983060minus Dnt =

t983131

m=1

983059qmn

983059D

mminus1983060minus Dnm

983060 foralltisin [T ]

We summarize the decision variables and parameters in Table 1


Parameters Description

cij (i j)isinA Transportation cost from node i to node j

Dnt nisin [N ] tisin [T ] Uncertain demand of retailer n in period t

Decision variables Description

yt0 isinBt tisin [T ] yt

0

983059D

tminus1983060= 1 if the vehicle is used in period t (= 0 otherwise)

ytn isinBt nisin [N ] tisin [T ] yt

n

983059D

tminus1983060= 1 if retailer n is replenished in period t (= 0 otherwise)

ztij isinBt (i j)isinA ztij

983059D

tminus1983060= 1 if the arc (i j) is traversed in period t (= 0 otherwise)

qtn isinRt nisin [N ] tisin [T ] qtn

983059D

tminus1983060is the replenishment quantity of retailer n in period t

xtn isinRt+1 nisin [N ] tisin [T ] xt

n

983059D

t983060is the inventory level of retailer n at the end of period t

Table 1 Parameters and decisions

We now present the constraints on the routing and replenishment decisions with a feasible set

Z as

Z =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

yt0 y

tn isinBt

ztij isinBt

qtn isinRt

forallnisin [N ] tisin [T ]

(i j)isinA

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

0le qtn

983059D

tminus1983060leMyt

n

983059D

tminus1983060 forallnisin [N ] tisin [T ] (a)

ytn

983059D

tminus1983060le yt

0

983059D

tminus1983060 forallnisin [N ] tisin [T ] (b)

983131

j(nj)isinA

ztnj

983059D

tminus1983060= yt

n

983059D

tminus1983060 forallnisin [N ]cup 0 tisin [T ] (c)

983131

j(jn)isinA

ztjn

983059D

tminus1983060= yt

n

983059D

tminus1983060 forallnisin [N ]cup 0 tisin [T ] (d)

983131

(ij)isinA(S)

ztij

983059D

tminus1983060le983131

nisinS

ytn

983059D

tminus1983060minus yt

k

983059D

tminus1983060

forallS sube [N ] |S|ge 2 k isin S tisin [T ] (e)

983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102

(1)

With M being a large number constraint (1a) uses the Big-M method to infer that the ordering

decision can be made only when retailer n is served Constraint (1b) ensures that if any retailer

n is served in period t then the route in that period must ldquovisitrdquo the supplier at node 0 The

constraints (1c) and (1d) guaranteen that each visited node n has only one arc entering n and one

arc exiting n Constraint (1e) uses a two-index vehicle flow formulation which has been shown to

be computationally attractive to eliminate subtours (see eg Solyalı et al 2012)


We require the period-t inventory level xtn

983059D

t983060

to be within a pre-specified interval known

as the requirement window [τ tn τ

tn] The requirement window can be determined by practical

considerations For example τ tn = 0 indicates a strong preference for avoiding stockouts and an

upper bound τ tn can be specified in accordance with the storage capacity which if exceeded would

incur a high holding cost Similarly we set a budget Bt for the transportation cost in each period

tisin [T ]

Therefore our major concern now is to evaluate the risk that the uncertain inventory level fails to

satisfy the requirement or that the transportation cost violates the budget constraint We propose

a new performance measure to evaluate this risk and then present a comprehensive model

22 Service Violation Index

We now introduce a service quality measure to evaluate the risk of service violation in terms of

inventory level and transportation cost Inspired by Chen et al (2015) and Jaillet et al (2016)

we propose a new and more general index to evaluate this risk Our construction of this index is

based on classical expected utility theory and risk measures

In particular given an I-dimensional random vector x isin VI we must evaluate the risk that x

realizes to be outside of the required window [τ τ ]subeRI Toward that end we propose the following

notion of Service Violation Index We first define a function vτ τ (middot) as vτ τ (x) = maxxminus τ τ minus

x= (maxxi minus τ i τ i minusxi)iisin[I] which is the violation of x with regard to the requirement window

[τ τ ] Specifically a negative value of violation eg v03(2) =minus1 implies that a change can still

be made in any arbitrary direction without violating the requirement window

Definition 1 A class of functions ρτ τ (middot) VI rarr [0infin] is a Service Violation Index (SVI) if for

all x y isin VI it satisfies the following properties

1 Monotonicity ρτ τ (x)ge ρτ τ (y) if vτ τ (x)ge vτ τ (y)

2 Satisficing

(a) Attainment content ρτ τ (x) = 0 if vτ τ (x)le 0

(b) Starvation aversion ρτ τ (x) =infin if there exists an iisin [I] such that vτ iτ i(xi)gt 0


3 Convexity ρτ τ (λx+(1minusλ)y)le λρτ τ (x)+ (1minusλ)ρτ τ (y) for any λisin [01]

4 Positive homogeneity ρλτ λτ (λx) = λρτ τ (x) for any λge 0

5 Dimension-wise additivity ρτ τ (x) =983123I

i=1 ρτ τ ((xiwminusi)) for any w isin [τ τ ]

6 Order invariance ρτ τ (x) = ρPτ Pτ (P x) for any permutation matrix P

7 Left continuity limadarr0

ρminusinfin0(xminus a1) = ρminusinfin0(x) where (minusinfin) is the vector all of whose com-

ponents are minusinfin

We define the SVI such that a low value indicates a low risk of violating the requirement window

Monotonicity states that if a given violation is always greater than another then the former will

be associated with higher risk and hence will be less preferred Satisficing specifies the riskiness

in two extreme cases Under attainment content if any realization of the uncertain attributes lies

within the requirement window then there is no risk of violation In contrast starvation aversion

implies that risk is highest when there is at least one attribute that always violates the requirement

window Our convexity property mimics risk managementrsquos preference for diversification Positive

homogeneity enables the cardinal nature the motivation of which can be found in Artzner et al

(1999) Dimension-wise additivity and order invariance imply that overall risk is the aggregation

of individual risk and the insensitivity to the sequence of dimensions respectively Both properties

are justified in a multi-stage setting by Chen et al (2015) Finally for solutions to be tractable

we need left continuity together with other properties it also allows us to represent the SVI in

terms of a convex risk measure as follows

Theorem 1 A function ρτ τ (middot) VI rarr [0infin] is an SVI if and only if it has the representation as

ρτ τ (x) = inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτ iτ i(xi)

αi

983062le 0αi gt 0 foralliisin [I]

983070 (2)

where we define inf empty=infin by convention and micro(middot) V rarrR is a normalized convex risk measure ie

micro(middot) satisfies the following properties for all x y isin V

1 Monotonicity micro(x)ge micro(y) if xge y

2 Cash invariance micro(x+w) = micro(x)+w for any w isinR


3 Convexity micro(λx+(1minusλ)y)le λmicro(x)+ (1minusλ)micro(y) for any λisin [01]

4 Normalization micro(0) = 0

Conversely given an SVI ρτ τ (middot) the underlying convex risk measure is given by

micro(x) =mina |ρminusinfin0((xminus a)e1)le 1 (3)

For the sake of brevity we present all proofs in the appendix

To calculate the SVI value efficiently and to illustrate some insights into its connection with the

classical expected utility criterion we focus on the SVI whose underlying convex risk measure is the

shortfall risk measure defined by Follmer and Schied (2002) In order to incorporate distributional

ambiguity we redefine it as follows

Definition 2 A shortfall risk measure is any function microu(middot) V rarrR defined by

microu(x) = infη

983069η

983055983055983055983055supPisinP

EP [u(xminus η)]le 0

983070 (4)

where u is an increasing convex normalized utility function such that u(0) = 0

The operation supPisinP allows us to take the robustness into account Hence the shortfall risk can

be interpreted as the minimum amount that needs to be subtracted from the uncertain attribute

so that this attributersquos expected utility is less than 0 ie the risk becomes acceptable We now

use the dual representation described in Theorem 1 to show that when shortfall risk measure is

the underlying risk measure the corresponding SVI has the following representation

Proposition 1 The shortfall risk measure is a normalized convex risk measure Its corresponding

SVI which we refer to as the utility-based SVI can be represented as follows

ρτ τ (x) = inf

983099983103

983101983131

iisin[I]

αi

983055983055983055983055983055983055supPisinP

EP

983061u

983061vτ iτ i(xi)

αi

983062983062le 0αi gt 0foralliisin [I]

983100983104

983102 (5)

According to the equation (5) the utility-based SVI is the aggregation of the smallest positive

scalar factors αi over all indexes i isin [I] such that vτ iτ i(xi)αi is within the acceptable set or


equivalently vτ iτ i(xi)αi is with nonpositive expected utility This particular index allows the

utility function to take different forms For instance if the utility function is exponential as u(x) =

exp(x)minus 1 then the index can be reformulated as

ρτ τ (x) = inf

983099983103

983101983131

iisin[I]

αi

983055983055983055983055983055983055Cαi

(xi)le 0αi ge 0foralliisin [I]

983100983104

983102

where

Cαi(xi) = αi ln

983061supPisinP

EP

983061exp

983061vτ iτ i(xi)

αi

983062983062983062

Function Cαi(xi) is the worst-case certainty equivalent of the uncertain violation of xi with respect

to the requirement window [τ i τ i] when the parameter of absolute risk aversion is 1αi (Gilboa

and Schmeidler 1989) More details can be found in Hall et al (2015) and Jaillet et al (2016)

For the sake of computational tractability we focus on the SVI whose construction is based on

the piecewise linear utility function u In particular we let u(x) =maxkisin[K]akx+ bk with ak ge 0

for all k isin [K] Adopting this piecewise linear utility u preserves the optimization problemrsquos linear

structure which eases the computational burden even while accounting for risk We remark that

practically speaking a piecewise linear utility can be used to approximate any type of increasing

convex utility Because αi gt 0 for the piecewise linear function considered here we multiply both

the left-hand side (LHS) and the right-hand side (RHS) of the constraint by αi Then the SVI can

be simplified as

ρτ τ (x) = inf

983099983105983105983103

983105983105983101

983131

iisin[I]

αi

983055983055983055983055983055983055983055983055

supPisinP

EP

983061maxkisin[K]

983051akvτ iτ i(xi)+ bkαi

983052983062le 0

αi gt 0foralliisin [I]

983100983105983105983104

983105983105983102

Having specified the objective function we can now present our stochastic IRP model as follows


inf983131

tisin[T ]

983131

nisin[N ]

αtn +

983131

tisin[T ]

βt

st supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minus xt

n

983059D

t983060 xt

n

983059D

t983060minus τ t

n

983154+ bkα

tn

983154983062le 0 forallnisin [N ] tisin [T ]

supPisinP

EP

983091

983107maxkisin[K]

983099983103

983101ak

983091

983107983131

(ij)isinA

cijztij

983059D

tminus1983060minusBt

983092

983108+ bkβt

983100983104

983102

983092

983108le 0 foralltisin [T ]

xtn

983059D

t983060=

t983131

m=1

(qmn

983059D

mminus1983060minus Dnm) forallnisin [N ] tisin [T ]

αtn gt 0 forallnisin [N ] tisin [T ]

(qtn yt0 y

tn z

tij nisin [N ] tisin [T ] (i j)isinA)isinZ

(6)

3 Adaptive Decisions

We facilitate the optimization by adopting a decision rule approach to derive adaptive decisions

In particular each periodrsquos decision is assumed to be a specific mapping from previous demand

realizations At each stage the decisions involve both binary ones (ie routing) and continuous

ones (ie replenishment) Although decision rules with respect to continuous decisions have been

investigated by a rich body of literatures the rules related to binary decisions have attracted

much less attention and is still computationally challenging (eg Bertsimas and Caramanis 2007

Bertsimas and Georghiou 2014 2015) Since this study is the first to consider such a complex

stochastic IRP we simplify the model by assuming that the binary decisions are nonadaptable

That is we consider the problem in two steps In the first step the visiting decisions and routing

decisions for all periods are determined at the beginning of planning horizon Thus all routing

decisions are made a priori and irrespective of demand realizations This approach approximates the

optimal solution and indeed is a reasonable one in practice Visiting plans must be communicated

to retailers in advance so that they can adequately prepare besides scheduling the vehicle and

informing the staff on short notice is not always practically feasible (Bertsimas et al 2016) In the

second step the supplier makes replenishment decisions at the start of each period after demand

realizations in previous periods In this sense then inventory decisions are made adaptively Figure

1 summarizes the timeline applicable to these decisions


Period 1 Period 2 Period Thellip

Routing decisions

Order quantities Order quantities

Actual demand Actual demand

Decisionmaking

Uncertaintyrealization

Order quantities

Actual demand

Figure 1 Sequence of routing decisions and inventory replenishment decisions

For the sake of tractability we now introduce a special case of decision rules mdash namely a

linear decision rule mdash whereby each decision is assumed to be an affine function of uncertainty

realizations Following this linear decision rule we can write the order quantity as

qtn

983059D

tminus1983060= qtn0 + 〈Qt

nD〉 nisin [N ] tisin [T ]

where qtn0 isin R and Qtn isin RNtimesT are decision variables In particular we let the lth column of Qt

n

denoted by (Qtn)l be the zero vector 0 when lge t If t= 1 then Q1

n = 0 That captures the ldquonon-

anticipativerdquo property since there is no way for the order quantity to depend on future demand

realization

So at the end of each period the inventory level is also an affine function of demand realization

This follows because for all tisin [T ]

xtn

983059D

t983060=

t983131

m=1

983059qmn

983059D


983060=

t983131

m=1

983059qmn0 + 〈Qm

n D〉minus Dnm

983060= xt

n0 + 〈XtnD〉 (7)

here xtn0 =

t983123m=1

qmn0 and Xtn =

t983123m=1

(Qmn minus ene

primem)

Since now the routing decisions ztij for (i j) isinA and t isin [T ] are nonadaptable the transporta-

tion cost983123

(ij)isinA cijztij is independent of the demand realizations Hence the constraint in model

(6) supPisinP EP

983059u983059983059983123

(ij)isinA cijztij minusBt

983060βt

983060983060le 0 reduces to the deterministic budget constraint

983123(ij)isinA cijz

tij leBt because the utility function u is non-decreasing and u(0) = 0 Under this deci-

sion rule approach our model can be formulated as follows


inf983131

tisin[T ]

983131

nisin[N ]

αtn

st supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt

n0 minus〈XtnD〉 xt

n0 + 〈XtnD〉minus τ t

n

983154+ bkα

tn

983154983062le 0forallnisin [N ] tisin [T ] (a)

qtn0 + 〈QtnD〉 ge 0 forallD isinW nisin [N ] tisin [T ] (b)

qtn0 + 〈QtnD〉 leMyt

n forallD isinW nisin [N ] tisin [T ] (c)

xtn0 =

t983131

m=1

qmn0 forallnisin [N ] tisin [T ]

Xtn =

t983131

m=1

(Qmn minus ene

primem) forallnisin [N ] tisin [T ]

983043Qt

n

983044l= 0 forallnisin [N ] lge t l tisin [T ]

αtn ge 983171 forallnisin [N ] tisin [T ]

(yt0 y

tn z

tij nisin [N ] tisin [T ] (i j)isinA)isinZR

(8)

Here ZR is the feasible set for static routing decisions

ZR =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

yt0 y

tn z

tij isin 01


(i j)isinA

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

983131

(ij)isinA

cijztij leBt foralltisin [T ]

ytn le yt

0 forallnisin [N ] tisin [T ]

983131

j(nj)isinA

ztnj = ytn forallnisin [N ]cup 0 tisin [T ]

983131

j(jn)isinA

ztjn = ytn forallnisin [N ]cup 0 tisin [T ]

983131

(ij)isinA(S)

ztij le983131

nisinS

ytn minus yt

k forallS sube [N ] |S|ge 2 k isin S tisin [T ]

983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102

(9)

Instead of directly using the constraint αtn gt 0 as in the original problem (6) here we use αt

n ge 983171

so that the feasible set will be closed Indeed we can always choose a positive 983171 small enough that

optimality is not compromised (Chen et al 2015)

4 Solution Procedure

The main problem (8) involves an inner optimization over the set of possible distributions ie

the operation of supPisinP in the first constraint Therefore the optimization procedure inevitably


depends on the information available about uncertain demand In this section we describe different

types of information sets P and then give the corresponding reformulations and solution techniques

41 Stochastic approach

We first consider the case when uncertain demand follows a known distribution ie P is a singleton

Thus we assume that D takes the value ofD(m) with probability pmmisin [Ms] where983123

misin[Ms]pm =

1 In this case

P =

983099983103

983101P

983055983055983055983055983055983055P983059D=D(m)

983060= pmmisin [Ms]

983131

misin[Ms]

pm = 1

983100983104

983102 (10)

and the LHS of (8a) becomes

supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

=983131

misin[Ms]

pm

983061maxkisin[K]

983153akmax

983153τ tn minusxt

n0 minus〈XtnD

(m)〉 xtn0 + 〈Xt

nD(m)〉minus τ t

n

983154+ bkα

tn

983154983062

Hence we can now reformulate the overall problem as a mixed-integer linear programming (MILP)

Proposition 2 Given P as defined by the equation (10) the constraints (8a)ndash(8c) are equivalent

to these constraints

983131

misin[Ms]

pmνmnt le 0 forallnisin [N ] tisin [T ]

νmnt ge ak

983059τ tn minusxt

n0 minus〈XtnD

(m)〉983060+ bkα

tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]

νmnt ge ak

983059xtn0 + 〈Xt

nD(m)〉minus τ t

n

983060+ bkα


qtn0 + 〈QtnD

(m)〉 ge 0 forallmisin [Ms] nisin [N ] tisin [T ]

qtn0 + 〈QtnD

(m)〉 leMytn forallmisin [Ms] nisin [N ] tisin [T ]

It follows that problem (8) can be reformulated as an MILP

42 Robust approach with mean absolute deviation

When knowledge about distribution is incomplete we can also characterize uncertain demand using

a distributionally robust optimization framework (see eg Delage and Ye 2010 Goh and Sim 2010

Wiesemann et al 2014) Because the deterministic version of IRP is already difficult to solve we

need to choose the ambiguity set P so as not to significantly increase computational complexity


We accommodate that consideration by introducing a P that allows us to capture the correlation

effect among uncertain demands Formally

P =

983099983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059Dle DleD

983060= 1 (a)

EP

983059D983060=Ξ (b)

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ (c)

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094

983096le 983171lhforalll isin [L] hisin [H] (d)

983100983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983102

(11)

In this set P the equation (11a) specifies the uncertain demandrsquos bound support while the

equation (11b) specifies its mean The inequality (11c) specifies the parameter Σnt as the bound of

uncertain demandrsquos mean absolute deviation Similar to the standard deviation this mean absolute

deviation provides a direct measure of uncertain demandrsquos dispersion about its mean in addition

it can be obtained in closed form for most common distributions (see eg Pham-Gia and Hung

2001) For example if Dnt is distributed uniformly on [01] then the mean absolute deviation

takes the value 14 The mean absolute deviation of a normal distribution amounts to the standard

deviation multiplied by9831552π

In constraint (11d) fl(middot) is a piecewise linear convex function that can be represented byKl pieces

as fl(x) = maxkisin[Kl] olkx+ plk This information set is intended to capture the correlation of

uncertain demand among different retailers over several periods In practice the respective demand

of these retailers may be correlated since retailers in general are geographically dispersed within a

nearby region and serve common customers As an illustrative example we consider a special case of

(11d) in which f1(x) =maxxminusx= |x| If S1 = (i τ)iisin[N ] then we have EP

983075983055983055983055983055983055983123

iisin[N ]

DiτminusΞiτΣiτ

983055983055983055983055983055

983076le

98317111 in essence this is a bound on the variation of total normalized demand over all retailers in

period τ Similarly for S2 = (i τ)τisin[T ] we have EP

983075983055983055983055983055983055983123

τisin[T ]

DiτminusΞiτΣiτ

983055983055983055983055983055

983076le 98317112 which bounds the

variation of the total normalized demand over all periods at node i We remark that with typical

characterization of correlation effect such as the covariance matrix the inventory management

problem is computationally challenging even when there is just one retailer Nonetheless we shall


later show that our problem remains tractable with the joint dispersion information described by

constraint (11d)

We now demonstrate that for the distributional uncertainty set P defined by the equation (11)

the constraints in problem (8) can be transformed into linear ones

Proposition 3 Let P be as defined in the equation (11) Then for any n isin [N ] and t isin [T ] the

constraint (8a) can be reformulated as the following set of linear constraints

stn0 + 〈StnΞ〉+ 〈T t

nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh le 0

〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091

983107plkprime minus olkprime983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (xtn0 minus τ t

n)+ bkαtn minus stn0 forallk isin [K]

〈DU tnk〉minus 〈DV t

nk〉+ 〈ΞGtnk minusF t

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt

n0)+ bkαtn minus stn0 forallk isin [K]

983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131

hisinhisin[H](iτ)isinSh

983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ

foralliisin [N ] τ isin [T ] k isin [K]

983043U t

nk minusV tnk minusF t

nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


Ft

nk +Gt

nk = T tn forallk isin [K]

F tnk +Gt


983131

kprimeisin[Kl]

wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]

983131

kprimeisin[Kl]


rtnlh ge 0 foralll isin [L] hisin [H]

T tn ge 0

Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG

tnk ge 0 forallk isin [K]

wtnklhkprime w

tnklhkprime ge 0 foralll isin [L] hisin [H] k isin [K] kprime isin [Kl]


A similar technique can be used to transform constraint (8b) as follows

Proposition 4 Let P be defined as in the equation (11) Then for any n isin [N ] and t isin [T ] the

constraint (8b) can be written as

minusqtn0 + 〈DLtn〉minus 〈DH t

n〉 le 0

Qtn +Lt

n minusH tn = 0

LtnH

tn ge 0

We remark that much as in Proposition 4 the constraint (8c) can also be represented equivalently

by a set of linear constraints Now we can achieve the following reformulation of the overall problem

Corollary 1 Given P as defined in the equation (11) the problem (8) can be formulated as an

MILP

A full description of the reformulation is given in the Appendix F

The solution quality can be further improved by incorporating a more sophisticated decision

rule For instance Bertsimas et al (2018) define the lifted ambiguity set by introducing certain

auxiliary random variables and then incorporating them into an approximation of a linear decision

rule To differentiate that approach from the traditional one we call it the enhanced linear decision

rule (ELDR) We can use this ELDR in the case of the ambiguity set P considered previously For

that purpose we introduce the auxiliary random variables U and V and then define the following

lifted ambiguity set Q

Q=

983099983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983101

Q

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

Q983059983059

D U V983060isin 983142W

983060= 1

EQ

983059D983060=Ξ

EQ

983059U983060leΣ

EQ [vlh]le 983171lhforalll isin [L] hisin [H]

983100983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983102

(12)


here 983142W is the lifted support defined formally as

983142W =

983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101

983059D U V

983060isinRNtimesT timesRNtimesT timesRLtimesH

983055983055983055983055983055983055983055983055983055983055983055983055983055

Dle DleD983055983055983055DminusΞ

983055983055983055le U

fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108le vlhforalll isin [L] hisin [H]

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102

where vlh denotes the element located at the lth row and hth column of matrix V Wiesemann

et al (2014) show that the ambiguity set P is equivalent to the set of marginal distributions of

D under Q for all Q isin Q Under ELDR the ordering quantity qtn n isin [N ] t isin [T ] is restricted

to depend linearly not only on the primary random variable D but also on the auxiliary random

variables U and V Then the order quantity can be written as

qtn

983059D U V

983060= qtn0 + 〈Qt

0nD〉+ 〈Qt1n U〉+ 〈Qt

2n V 〉 nisin [N ] tisin [T ]

here qtn0 isinR Qt0n isinRNtimesT Qt

1n isinRNtimesT and Qt2n isinRLtimesH are decision variables Since the adaptive

decisions cannot be anticipated both983043Qt

0n

983044land

983043Qt

1n

983044lare set to 0 for all l ge t we also put

983043Qt

2n

983044h= 0 when there exists some τ ge t such that (i τ)isin Sh Now constraint (8a) can be written

as

supQisinQ

EQ

983061maxkisin[K]

983153akmax

983153τ tn minus qtn0 minus〈Qt

0nD〉minus 〈Qt1n U〉minus 〈Qt

2n V 〉

qtn0 + 〈Qt0nD〉+ 〈Qt

1n U〉+ 〈Qt2n V 〉minus τ t

n

983154+ bkα

tn

983154983060le 0

(13)

Here problem (8) can be solved by the corresponding reformulation method This claim is formal-

ized in our next proposition

Proposition 5 Let the distribution ambiguity set Q be as defined in the equation (12) and let

constraint (8a) be replaced by (13) Then problem (8) can be equivalently formulated as an MILP

43 Robust approach with Wasserstein distance

Considerable interest has recently been shown in ambiguity sets whose construction is based on the

notion of statistical distance In particular by measuring the proximity of probability distributions

with statistical distance measures the ambiguity sets are defined as the set of probability distri-

butions which are ldquocloserdquo to the reference distribution The examples of such statistical distance


are φ-divergence (Ben-Tal et al 2013 Wang et al 2016 Jiang and Guan 2016) and Wasserstein

distance (Esfahani and Kuhn 2017 Gao and Kleywegt 2016) Among others Wasserstein distancersquos

increasing popularity is due to its powerful out-of-sample performance Here we show that our

model also enables use of an ambiguity set based on Wasserstein distance when the matrix norm

is chosen as entrywise 1-norm

We denote by P0(W) the space of all probability distributions supported on W The reference

probability distribution Pdagger is given for the random variable Ddaggerwhich takes values D(1) D(Mr)

with equal likelihood that is

Pdagger983059D

dagger= D

(m)983060=

1

Mr

forallmisin [Mr]

Given any distribution PisinP0(W) on D the Wasserstein distance between P and Pdagger is defined as

dW (PPdagger) = inf EP

983059983056983056983056Dminus Ddagger983056983056983056983060

st (DDdagger)sim P

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

Here P is the joint distribution of D and Ddagger the terms ΠDP and Π

DdaggerP stand for the marginal

distribution of P on D and Ddagger respectively and 983042 middot 983042 represents an arbitrary norm Now the

ambiguity set can be constructed as the Wasserstein ball of radius θgt 0 centered at the reference

probability distribution Pdagger

P =983051PisinP0(W)|dW (PPdagger)le θ

983052 (14)

Esfahani and Kuhn (2017) and Gao and Kleywegt (2016) show that strong duality holds while

studying the distributionally robust optimization problem with Wasserstein distance We now

incorporate that result into our problem In this case reformulating constraints (8b)ndash(8c) to a finite

set of linear constraints can proceed as described in Proposition 4 hence we focus on constraints

(8a)


Proposition 6 Let P be as defined in the equation (14) and let the norm 983042 middot 983042 be chosen as the

entrywise 1-norm Then for any given nisin [N ] and tisin [T ] the constraints (8a) are equivalent to

rtnθ+983131

misin[Mr ]

stnm le 0

〈DV tnmk〉minus 〈DU t

nmk〉+ 〈D(m)

Rtnmk〉 leMrs

tnm minus ak (τ

tn minusxt

n0)minus bkαtnforallmisin [Mr] k isin [K]

〈DVt

nmk〉minus 〈DUt

nmk〉+ 〈D(m)

Rt

nmk〉 leMrstnm minus ak (x

tn0 minus τ t

n)minus bkαtnforallmisin [Mr] k isin [K]

minusU tnmk +V t

nmk =minusakXtn minusRt

nmk forallmisin [Mr] k isin [K]

minusUt

nmk +Vt

nmk = akXtn minusR

t


983043Rt

nmk

983044iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]

983043Rt

nmk

983044iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]

983059R

t

nmk

983060

iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]

983059R

t

nmk

983060

iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]

U tnmkV

tnmkU

t

nmkVt

nmk ge 0 forallmisin [Mr] k isin [K]

rtn ge 0stn isinRMs

Therefore problem (8) can be formulated as an MILP

Remark 1 After accounting for the risk described in Section 22 and for the uncertainty sets

introduced in Sections 41-43 we find that our problemrsquos final formulation is still an MILP Com-

pared with the deterministic inventory routing formulation only a set of linear constraints and

continuous random variables are added

Note that the computation can be expedited by applying in our stochastic framework meth-

ods designed for the deterministic IRP Most valid cuts developed for deterministic problem to

strengthen the routing decisions remain applicable in this context example include ztij le yti z

tij le yt

j

and983123

nisin[N ] qtn

983059D

tminus1983060le yt

0 (Archetti et al 2007) We can also facilitate computation by limiting

the feasible routes to a pre-determined subset such that instances of much larger scale can be solved

(Bertsimas et al 2016)

Finally there are many possible directions in which this framework can be extended From the

operational perspective one could forgo specifying the budget for each period and instead stipulate


a total budget that controls the transportation cost within a given planning horizon Furthermore

it would be straightforward to consider the case of multiple vehicles with capacity constraints The

model we propose also allows for incorporating the production decisions (Adulyasak et al 2013

2015)

44 Benders decomposition

In Sections 41ndash43 we showed that our IRP under uncertainty can be formulated as an MILP

Although the size of the resulting MILP is large we can solve it using the Benders (1962) decom-

position approach (for additional details see Rahmaniani et al 2016) For completeness we also

describe how our own model can incorporate the Benders decomposition and hence be solved more

efficiently

Our preceding reformulations of the inventory routing problem as an MILP differ since they use

different information set to characterize uncertainty Here we express all those formulations in the

following general form

min gprimeφ

st aprimeiφ+ bprimeiy0 + 〈Cy

i Y 〉+ 〈Czi Z〉 le si foralliisin [I]

(y0Y Z)isinZR

(15)

where φ is used to represent all continuous decision variables the terms y0 isin 01T Y isin 01NtimesT

and Z isin 01|A|timesT are the routing decisions all of which are binary So if we now define a function

f ZR rarrRcup infin as

f(y0Y Z) =min gprimeφ

st aprimeiφle si minus bprimeiy0 minus〈Cy

i Y 〉minus 〈Czi Z〉 foralliisin [I]

(16)

then the main problem (15) is equivalent to min(y0Y Z)isinZRf(y0Y Z) In defining f(y0Y Z) we

consider the minimization problem to be a primal problem its dual problem can be easily obtained

as

fd(y0Y Z) =max983131

iisin[I]

(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz

i Z〉)ψi

st983131

iisin[I]

aiψi = g

(17)


We note that problem (15) is bounded from below by NT 983171 Hence the primal problem (16)

must be either finite or infeasible from which it follows that the dual problem (17) must be either

finite or unbounded In the former case by strong duality we have f(y0Y Z) = fd(y0Y Z)

The latter case occurs if and only if983123

iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz

i Z〉)ψi gt 0 for some ψ with

983123iisin[I]aiψi = 0 Define an optimization problem as follows

min η

st983131

iisin[I]


i Z〉)ψi le η forallψ isin G1

983131

iisin[I]


i Z〉)ψi le 0 forallψ isin G2

(y0Y Z)isinZR

(18)

If G1 = ψ|983123

iisin[I]aiψi = g and G2 = ψ|983123

iisin[I]aiψi = 0 then the main problem (15) is equivalent

to (18)

We incorporate the Benders decomposition as described in the following algorithm

Algorithm Benders Decomposition

1 Initialize G1 as a singleton containing an arbitrary extreme point of the polyhedron

ψ|983123

iisin[I]aiψi = g and initialize G2 = empty

2 Solve the problem (18) and denote the optimal solution by (ηlowastylowast0Y

lowastZlowast)

3 Solve problem (17) with (y0Y Z) = (ylowast0Y

lowastZlowast) If fd(ylowast0Y

lowastZlowast) is finite then let ψlowast

denote an optimal extreme-point solution and update G1 = G1 cup ψlowast Otherwise let ψlowast signify

the direction with983123


i Z〉)ψi gt 0 and983123

iisin[I]aiψi = 0 and update

G2 = G2 cup ψlowast

4 If ηlowast = fd(ylowast0Y

lowastZlowast) then terminate the algorithm and return (xlowastylowast0Y

lowastZlowast) as the optimal

solution otherwise return to Step 2

It is easy to show that in theory the algorithm can be terminated within a finite number of

iterations since in Step 3 there is only a finite number of extreme pointsrays On a practical

level it has been well recognized that this approach can help solve MILP with greater efficiency

(Rahmaniani et al 2016)


5 Computational Study

In this section we carry out computational studies to show that our proposed model is practically

solvable and can yield an appealing solution under uncertainty The program is coded in C++ and

run on an Intel Core i5 PC with a 270-GHz CPU and 8 GB of RAM using CPLEX 125 Unless

otherwise stated the maximum CPU time is set to two hours

To assess the solvability of our model and the quality of its solutions we test instances with

N = 8 retailers on a randomly generated graph with T = 4 planning periods The transportation

cost between any two nodes is proportional to the distance between them

Taking the network as given we calculate the minimum cost to visit all nodes based on the

traditional traveling salesman problem We denote this minimum cost as TSP Since the initial

inventory level of each retailer is set as zero we let the budget for the first period be the TSP

such that all retailers can be served For subsequent periods we let the budget be κtimes TSP in

this setup κlt 1 indicates that the supplierrsquos vehicle cannot visit all retailers in a single period

Our experiment considers two cases κ = 07 and κ = 08 We put Dnt = 10 + 30σnt where σnt

follows the distribution of Beta(24) for all n isin [N ] and t isin [T ] In all periods all retailers share

a common inventory level requirement window [0 τ ] we vary τ to capture the effects of different

requirement windows For simplicity we choose u(x) =maxminus1 x as the utility function We also

conduct these computational studies for other distributions The results are quite consistent across

distributions so we only report the case with a Beta distribution

51 Comparison between our approaches and benchmark approach

As a means of further evaluating and understanding the benefit of our approach we compare the

quality of solutions from three approaches Two of them are variants of our SVI method and one

is a benchmark approach based on minimizing the probability of violation

511 Robust approach with mean absolute deviation (SVI-R) The first approach

which we dub the SVI-R model is from our robust optimization using information of mean absolute


deviation (Section 42) When solving for the optimal solution we use the information characterized

as follows

P =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059dle Dnt le d

983060= 1

EP

983059Dnt

983060= micro

EP

983059983055983055983055Dnt minusmicro983055983055983055983060le σ

EP

983091

983107

983055983055983055983055983055983055

983131

tisin[T ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092

983108le 983171oforallnisin [N ]

EP

983091

983107

983055983055983055983055983055983055

983131

nisin[N ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092

983108le 983171lowastforalltisin [T ]

983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102

Here d= 10+30times 0 = 10 d= 10+30times 1 = 40 and the mean value is micro= 10+30times 22+4

= 20 The

values of σ 983171o and 983171lowast can similarly be obtained from the underlying distribution of Dnt Observe

that the penultimate constraint is to incorporate the correlation effect among uncertain demand

of all periods for each retailer the last constraint is for the correlation among uncertain demand

of all retailers for each period We lift the ambiguity set P to Q as described in the equation (12)

after which we solve the problem using ELDR as described in Proposition 5

512 Stochastic approach (SVI-S) In the second approach called the SVI-S model we

have exact information on the distribution of D (Section 41) Following the underlying distribu-

tion on D we generate Ms samples of D and use the sampling distribution as P in Section 41 As

expected performance and computation time depend strongly on the number Ms of training sam-

ples A large number of samples will describe the distribution better and lead to a good solution

but computation time will increase significantly because there are more constraints involved We

test the performance of our SVI-S model in two cases Ms = 100 and Ms = 500

Note that as the feature of distributional robustness has been reflected by the SVI-R model

here we do not incorporate the model based on Wasserstein distance (Section 43) to avoid the

specification of θ which might be abstract in the computational study Indeed the SVI-S is its

special case with θ= 0


513 Minimizing violation probability (MVP) Since we focus on the service level a

natural benchmark is to minimize the probability of an inventory requirement violation that

is to minimize 1NT

983123nisin[N ]

983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly

intractable we can use the sampling average approximation to derive only static decisions not

adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)

misin [Mc] Then the problem of MVP can be approximated as

min1

McNT

983131

misin[Mc]

983131

tisin[T ]

983131

nisin[N ]

I(m)nt

st I(m)nt ge x

(m)nt minus τ

Mforallmisin [Mc] nisin [N ] tisin [T ]

I(m)nt ge minusx

(m)nt

M forallmisin [Mc] nisin [N ] tisin [T ]

qtn leMytn forallnisin [N ] tisin [T ]

qtn ge 0 forallnisin [N ] tisin [T ]

I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]

983043yt0 y

tn z

tij nisin [N ] tisin [T ] (i j)isinA

983044isinZR

Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M

method If the sample size Mc is large enough then this formulation closely approximates the MVP

model however a large sample size will also result in long computation times In this experiment

we set Mc = 100

514 Comparison For each approach we first derive the optimal solution and then generate

Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-

mance analysis by aggregating the uncertain violation at all nodes and all periods into a single

uncertain violation v In particular v is the violation at node n in period t with equal likelihood

for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution

we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as

measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected

value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th


percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to

better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU

timerdquo

The performance of our SVI-R model is presented in Table 2 As the transportation budget

increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of

violation If the transportation budget is held constant then when τ increases the ending inventory

requirement is less restrictive once again violations are thus less likely The SVI-R model entails

reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network

over four periods

κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56

30 911 042 185 265 1044 9235 553 027 148 064 849 86

08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62

Table 2 Performance of the SVI-R model

We summarize the performance of other models in Table 3 For purposes of comparison we

normalize the performance of these models by that of our SVI-R model Hence a value greater

than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that

for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar

Yet performance is poor as reflected by most values exceeding unity When the sample size Ms

increases to 500 computation time increases significantly but performance is only slightly better

than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall

that performance of the MVP model is also affected by the training sample size But even for 100

samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP

performance is the worst among all these models in terms not only of the violation probability but

also of the violation magnitude We therefore conclude that our proposed SVI is a good measure

for capturing the risk of a requirement violation and we know that it can be computed efficiently


For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)

in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive

Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159

(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105

08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155

SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243

35 124 070 068 145 062 112808 27 114 107 099 110 096 752

30 127 092 084 129 078 53735 125 074 071 na 064 395

MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372

08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613

Table 3 Comparison of performance for different benchmarks

52 Robustness

We shall now explain the need to incorporate distributional ambiguity From a practical standpoint

given a set of historical data we can extract certain information and then derive the corresponding

optimal solution For the SVI-R model we use information of mean absolute deviation for both the

SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate

from the true distribution We test the performance of different models under this scenario

In particular we assume that the true distribution of demand is as given in Section 51 Dnt =

10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are

drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw

nt where σwnt follows the Beta(48) distribution

Note that the true and wrong distributions do not deviate too much in the sense that Dnt and

Dwnt have the same mean and bound We derive the optimal solution for all models based on the

samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn

from the true distribution Dnt Because the mean absolute deviation information we extract from


both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51

And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance

is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances

which are again normalized by that of the SVI-R model

Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143

(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110

08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141

SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083

35 162 100 086 273 08008 27 137 124 104 125 098

30 161 117 093 168 08535 177 113 090 NA 085

MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140

08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139

Table 4 Performance of different models when given the wrong distribution information

Since the performance of models SVI-S and MVP are highly dependent on the training samples

the solutions these models derive from the wrong distribution can easily lead to performance far

inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported

in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach

will better immunize the decision maker against the effects of data ambiguity

6 Conclusions

We take a distributionally robust optimization approach to exploring the stochastic inventory

routing problem After proposing a new decision criterion for evaluating the associated risk we

develop methods geared to solve for the optimal affine policies Our model can easily be extended to

incorporate various operational features (eg production) and its computational efficiency can be


improved by employing some well-developed heuristics (eg restricting to only a subset of routes)

We undertake computational studies to illustrate the strength of our method


Appendix A Proof of Theorem 1

We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the

equation (2) is an SVI by showing its satisfaction of the properties in Definition 1

1 Monotonicity If vτiτi(xi)ge vτiτi

(yi)foralliisin [I] then by the monotonicity of micro we have

micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi

983062 forallαi gt 0

Hence ρτ τ (x)ge ρτ τ (y)

2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0

micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0

Hence ρτ τ (x) = 0

2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi

(xi)αi gt

0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and

micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0

Therefore ρτ τ (x) =infin

3 Convexity Given any xisin VI we define a set of α as

S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070

We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the

definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence

micro

983061vτiτi

(λxi +(1minusλ)yi)

αλi

983062le micro

983061λvτiτi

(xi)+ (1minusλ)vτiτi(yi)

αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi

983062+(1minus o)micro

983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α

λi isin [01] The first inequality holds because of the convexity of vτiτi

(middot) and the monotonicity

of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition

of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get

λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf

983069I983123

i=1

αxi α

x isin S(x)

983070+(1minusλ) inf

983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1

(λαxi +(1minusλ)αy

i ) αx isin S(x)αy isin S(y)

983070

ge inf

983069I983123

i=1

αiαisin S(λx+(1minusλ)y)

983070

= ρτ τ (λx+(1minusλ)y)

4 Positive homogeneity For all λgt 0

ρλτ λτ (λx) = inf

983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi

983062le 0βi gt 0foralliisin [I]

983084

= λρτ τ (x)

5 Dimension-wise additivity It is obvious due to representation (2)

6 Order invariance It is obvious due to representation (2)

7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we

have αlowast =983123

iisin[I]αlowasti We next prove the left continuity in two cases

Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin

9830430 δ

983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we

have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The

case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast

i = 0 Then

we have 983171lt αlowast =983123

iisinI αlowasti Hence we can find a vector 983171iiisinI such that

983123iisinI 983171i = 983171 and 983171i isin (0αlowast

i ) foralliisin I

Choose δ =miniisinI(αlowasti minus 983171i)micro(

vminusinfin0(xi)

αlowastiminus983171i

) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ

is strictly positive Considering forallδ isin (0 δ) and iisin I we have

micro

983061vminusinfin0(xi minus δ)

αlowasti minus 983171i

983062= micro

983061vminusinfin0(xi)minus δ


983062= micro

983061vminusinfin0(xi)


983062minus δ


gtmicro



983062minus δ


ge 0

where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second

equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma


1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously

we have ρminusinfin0((xi minus δ)ei) = 0 Therefore

ρminusinfin0(xminus δ1) =983131

iisinI

ρminusinfin0((xi minus δ)ei)ge983131

iisinI

(αlowasti minus 983171i) = αlowast minus 983171

Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim

adarr0ρminusinfin0(xminusa1) is not infin ie lim

adarr0ρminusinfin0(xminus

a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ

983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]

Since αlowasti = ρminusinfin0(xiei) =infin we have micro(

vminusinfin0(xi)

α)gt 0 forallαgt 0 Hence we are able to choose

δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)

where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ

983044 we still have ρminusinfin0((xi minus

δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171

We can also get that

micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α

983062minus983061micro


α

983062minus η

983062= ηgt 0

which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0

ρminusinfin0(xminus a1) =infin

We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a

convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x

1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the

monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)

we get micro(x)ge micro(y)

2 Cash invariance Notice that forallw isinR

micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1

= mina+w983055983055ρminusinfin0((xminus a)e1)le 1

= micro(x)+w

3 Convexity

ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)

le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)

le λ+(1minusλ)

= 1


where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on

the definition of micro Hence we have

micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1

983052le λmicro(x)+ (1minusλ)micro(y)

4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore

micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1

983052= 0

Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI

the minimum in (3) is achievable Therefore

min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1

983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro

983061vminusinfin0(xminus a)

α1

983062le 0 micro

983061vminusinfin0(0)

αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070

= mina |micro (vminusinfin0(xminus a))le 0

= mina |micro (xminus a)le 0

= mina |micro (x)le a

= micro(x)

where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition

of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due

to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which

we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for

any such αisin (01]

micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α

983062+(1minusα)micro(0) = αmicro

983061w

α

983062le 0

where inequalities are due to the convexity of micro and the condition B) respectively QED

Appendix B Proof of Proposition 1

The shortfall risk measure without supPisinP

has been shown to be a convex risk measure (Follmer and Schield

2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be

convex risk measure as well


By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as

ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi

983055983055983055983055existηi le 0 supPisinP

EP

983147u983059

vτiτi(xi)

αiminus ηi

983060983148le 0 αi gt 0 foralliisin [I]

983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070

The last equation holds due to the equivalence between condition A) existη le 0 such that

supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 and condition B) sup

PisinPEP

983147u983059

vτiτi(xi)

αi

983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0

For the direction ldquoArArrBrdquo we see that u983059

vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η

983060for η le 0 because u is an increasing

function So supPisinP

EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED

Appendix C Proof of Proposition 2

It is straightforward QED

Appendix D Proof of Proposition 3

For any node nisin [N ] and period tisin [T ] consider

supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming

problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962


Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then

supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094

983096le 983171lh foralll isin [L] hisin [H]

= min stn0 + 〈StnΞ〉+ 〈T t

nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh

st stn0 + 〈StnD〉+ 〈T t

n |DminusΞ|〉+983131

lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα

tn forallDleDleD k isin [K] (a)

stn0 + 〈StnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα

tn forallDleDleD k isin [K] (b)


T tn ge 0

The last optimization problem is a semi-infinite linear programming problem with infinite number of con-

straints For forallk isinK the constraint (a) is equivalent to

minDleDleD

983099983103

983101〈Stn minus akX

tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104

983102ge ak (xtn0 minus τ t

n)

+bkαtn minus stn0


Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left

hand side of the above inequality can be written as a linear optimization problem

min 〈Stn minus akX

tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlh maxkprimeisin[Kl]

983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD

Since strong duality holds it is equivalent to

max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091

983107plkprime minus olkprime

983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ

foralliisin [N ] τ isin [T ]

Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]

wtnklhkprime = rtnlh foralll isin [L] hisin [H]

Ut

nkVt

nkFt

nkGt

nk ge 0

wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]

Consequently constraint (a) is equivalent to the following constraints

〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime


n)+ bkαtn minus stn0

983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



Following the same analysis the constraint (b) is equivalent to the following constraints



nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt

n0)+ bkαtn minus stn0

983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0


Combining all of these analysis parts together we can derive the constraints stated in the proposition 3

QED

Appendix E Proof of Proposition 4

When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD

983052 for any

nisin [N ] tisin [T ] we can transform the constraint (8b) as

qn0 + 〈DQtn〉 ge 0 forallD isinW

hArr qn0 + minDisinW

〈DQtn〉 ge 0

hArr qn0 + minDleDleD

〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

minusqtn0 + 〈DLtn〉minus 〈DHt

n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0

The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can

be formulated as


qtn0 + 〈DQtn〉 leMyt

n forallD isinW

hArr qtn0 + maxDisinW

〈DQtn〉 leMyt

n

hArr qtn0 + maxDleDleD

〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED


Appendix F Proof of Corollary 1

inf983131

tisin[T ]

983131

nisin[N]

αtn

st stn0 + 〈StnΞ〉+ 〈T t

nΣ〉+983123

lisin[L]

983123hisin[H]

983171lhrtnlh le 0 foralltisin [T ] nisin [N ]

stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ

tn foralltisin [T ] nisin [N ] k isin [K]

stn0 + 〈DU tnk〉minus 〈DV t


nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ

foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]

983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt

nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]

F tnk +Gt


983131

kprimeisin[Kl]

wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]

983131

kprimeisin[Kl]



n〉 le 0 foralltisin [T ] nisin [N ]

qtn0 + 〈DLt

n〉minus 〈DHt

n〉minusMytn le 0 foralltisin [T ] nisin [N ]

Qtn +Lt

n minusHtn = 0 foralltisin [T ] nisin [N ]

Qtn minusL

t

n +Ht

n = 0 foralltisin [T ] nisin [N ]

983043Qt

n


αtn ge 983171 foralltisin [T ] nisin [N ]

rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]

T tnL

tnH

tnL

t

nHt

n ge 0 foralltisin [T ] nisin [N ]

Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG

tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]

wtnklhkprime wt

nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]

(yt0 y

tn z



QED

Appendix G Proof of Proposition 5

It is similar to that for Proposition 4 QED

Appendix H Proof of Proposition 6

For completeness here we also present a full proof customized to our problem For any node n and period t

we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n

and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side

(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)

Then it can be formulated as follow

sup EP

983061maxkisin[K]

983153ak max

983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ

983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)

Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D

(m) m isin

[Mr] In another word Pm

983059D=D

983060= P

983059D=D|Ddagger

= D(m)

983060 With the law of total probability we have

P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D

983060 Hence the problem (19)

can be reformulated as follows

sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]


The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its

dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]

ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062

forallD isinWmisin [Mr]

rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D

(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα

forallD isinWmisin [Mr] k isin [K]

rge 0sisinRMr

For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following

two inequalities

r983056983056983056Dminus D

(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW

r983056983056983056Dminus D

(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW

Then can be further equivalently reformulated as

supDisinW

983153minusak〈XD〉minus r

983056983056983056Dminus D(m)

9830569830569830561

983154le Mrsm minus ak (τ minusx0)minus bkα (a)

supDisinW

983153ak〈XD〉minus r

983056983056983056Dminus D(m)

9830569830569830561

983154le Mrsm minus ak (x0 minus τ)minus bkα (b)

(20)

By the definition of the dual norm the 1-norm in (20) can be written as

r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112


with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-

straint (20a) is equivalent to

supDisinW

983083minusak〈XD〉minus max

983042Rmk983042infinler

983111983059Dminus D

(m)983060Rmk

983112983084leMrsm minus ak (τ minusx0)minus bkα

lArrrArr supDisinW

983083min


983153minusak〈XD〉minus

983111983059Dminus D

(m)983060Rmk


lArrrArr min983042Rmk983042infin

ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk


983042Rmk983042infin le r

lArrrArr

983099983105983105983103

983105983105983101

supDisinW

〈minusakX minusRmkD〉+983111D

(m)Rmk



Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization

over D and minimization over Rmk are interchangable By strong duality we have

supDisinW

〈minusakX minusRmkD〉

= sup 〈minusakX minusRmkD〉

st DgeD

DleD

= inf 〈DV mk〉minus 〈DUmk〉

st minusUmk +V mk =minusakX minusRmk

UmkV mk ge 0

where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]

|(Rmk)iτ | Therefore the constraint

983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101

〈DV mk〉minus 〈DUmk〉+ 〈D(m)

Rmk〉 leMrsm minus ak (τ minusx0)minus bkα

minusUmk +V mk =minusakX minusRmk

(Rmk)iτ le r foralliisin [N ] τ isin [T ]

(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]

UmkV mk ge 0


By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101


Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα

minusUmk +V mk = akX minusRmk

983043Rmk

983044iτle r foralliisin [N ] τ isin [T ]

983043Eτ

i Rmk

983044iτgeminusr foralliisin [N ] τ isin [T ]

UmkV mk ge 0

By combining the analysis above we get the results of proposition 6 QED


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Chen X Sim M Simchi-Levi D Sun P (2007) Risk aversion in inventory management Operations

Research 55(5) 828-842

Chen X Sim M Sun P Zhang J (2008) A linear decision-based approximation approach to stochas-

tic programming Operations Research 56(2) 344-357

Cheung K L Lee H L (2002) The inventory benefit of shipment coordination and stock rebalancing

in a supply chain Management Science 48(2) 300-306

Christiansen M Fagerholt K Flatberg T Haugen Oslash Kloster O Lund E H (2011) Maritime inven-

tory routing with multiple products A case study from the cement industry European Journal

of Operational Research 208(1) 86-94

Coelho L C Cordeau J F Laporte G (2014a) Thirty years of inventory routing Transportation

Science 48(1) 1-19

Coelho L C Cordeau J F Laporte G (2014b) Heuristics for dynamic and stochastic inventory-

routing Computers amp Operations Research 52 55-67

Custodio A L Oliveira R C (2006) Redesigning distribution operations a case study on integrating

inventory management and vehicle routes design International Journal of Logistics Research

and Applications 9(2) 169-187


Delage E Iancu D A (2015) Robust multistage decision making INFORMS Tutorials in Operations

Research 20-46

Delage E Ye Y (2010) Distributionally robust optimization under moment uncertainty with appli-

cation to data-driven problems Operations Research 58(3) 595-612

Diecidue E van de Ven J (2008) Aspiration level probability of success and failure and expected

utility International Economic Review 49 683-700

Dong Y Xu K (2002) A supply chain model of vendor managed inventory Transportation Research

Part E Logistics and Transportation Review 38(2) 75-95

Esfahani P M Kuhn D (2017) Data-driven distributionally robust optimization using the Wasser-

stein metric Performance guarantees and tractable reformulations Mathematical Programming

Federgruen A Prastacos G Zipkin P H (1986) An allocation and distribution model for perishable

products Operations Research 34(1) 75-82

Federgruen A Zipkin P (1984) A combined vehicle routing and inventory allocation problem


Follmer H Schied A (2002) Convex measures of risk and trading constraints Finance and Stochas-

tics 6(4) 429-447

Gao R Kleywegt A J (2016) Distributionally robust stochastic optimization with Wasserstein

distance arXiv preprint arXiv160402199

Gaur V Fisher M L (2004) A periodic inventory routing problem at a supermarket chain Opera-

tions Research 52(6) 813-822

Georghiou A Wiesemann W Kuhn D (2015) Generalized decision rule approximations for stochas-

tic programming via liftings Mathematical Programming 152(1-2) 301-338

Gilboa I Schmeidler D (1989) Maxmin expected utility with non-unique prior Journal of Mathe-

matical Economics 18(2) 141-153

Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-

ations Research 58(4-part-1) 902-917


Golden B Assad A Dahl R (1984) Analysis of a large scale vehicle routing problem with an

inventory component Large Scale Systems 7(2-3) 181-190

Groslashnhaug R Christiansen M Desaulniers G Desrosiers J (2010) A branch-and-price method for

a liquefied natural gas inventory routing problem Transportation Science 44(3) 400-415

Hall N G Long D Z Qi J Sim M (2015) Managing underperformance risk in project portfolio

selectionOperations Research 63(3) 660-675

Hvattum L M Loslashkketangen A (2009) Using scenario trees and progressive hedging for stochastic

inventory routing problems Journal of Heuristics 15(6) 527-557

Hvattum L M Loslashkketangen A Laporte G (2009) Scenario tree-based heuristics for stochastic

inventory-routing problems INFORMS Journal on Computing 21(2) 268-285

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Mathematics 14(1) 185-197

Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)

186-200

Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-

gramming 158(1-2) 291-327

Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem

with direct deliveries Transportation Science 36(1) 94-118

Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a

stochastic inventory routing problem Transportation Science 38(1) 42-70

Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic

and robust optimization Mathematical Programming 130(1) 177-209

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349-360

Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-

bution network design model with stochastic demand Transportation Research Part E Logistics

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for a class of long-horizon maritime inventory routing problems Transportation Science 49(4)

870-885

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Computer Modelling 34(7-8) 921-936

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efficiently For a more detailed discussion of the VMI conceptrsquos advantages see Waller et al (1999)

Cetinkaya and Lee (2000) Cheung and Lee (2002) and Dong and Xu (2002)

This joint management of inventory and vehicle routing gives rise to the inventory routing

problem (IRP) The term IRP was first used by Golden et al (1984) when defining a routing

problem with an explicit inventory feature Federgruen and Zipkin (1984) study a single-period

IRP and describe the economic benefits of coordinating the decisions related to distribution and

inventory Since then many variants of the IRP have been proposed over the past three decades

(Coelho et al 2014a) planning horizons can be finite or infinite excess demand can be either lost

or back-ordered and there can be either just one or more than one vehicle Scholars have used

these formulations on many real-world applications of the IRP which include maritime logistics

(Christiansen et al 2011 Papageorgiou et al 2014) the transportation of gas and oil (Bell et al

1983 Campbell and Savelsbergh 2004 Groslashnhaug et al 2010) groceries (Gaur and Fisher 2004

Custodio and Oliveira 2006) perishable products (Federgruen et al 1986) and bicycle sharing

(Brinkmann et al 2016) We refer interested readers to Andersson et al (2010) for more on IRP

applications

Since the basic IRP incorporates a classical vehicle routing problem that is already NP-hard

it follows that the stochastic IRP must be even more complex We next focus our review on

the stochastic version of the IRP especially in the case of uncertain demand A classical way

to formulate the stochastic IRP is using Markov decision process which assumes that the joint

probability distribution of retailersrsquo demands is known Campbell et al (1998) introduce a dynamic

programming model in which the state is the current inventory level at each retailer and the Markov

transition matrix is obtained from the known probability distribution of demand in their model

the objective is to minimize the expected total discounted cost over an infinite horizon This work is

extended by Kleywegt et al (2002 2004) who solve the problem by constructing an approximation

to the optimal value function Adelman (2004) decomposes the optimal value function as the sum

of single-customer inventory value functions which are approximated by the optimal dual prices













































































and a lower bound















and Sim (2009)















tion approach













n983123j=1

XijYij We











S as |S|

2 Model











983059D1 Dt












) The




983059D


visited and yt0

983059D



983059D



983059D



983059D

t983060isinRt+1



0= empty so that


as

xtn

983059D

t983060= xtminus1

n

983059D

tminus1983060+ qtn

983059D


t983131

m=1

983059qmn

983059D










0

983059D



n

983059D



983059D



983059D



n

983059D




Z as

Z =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

yt0 y

tn isinBt

ztij isinBt

qtn isinRt


(i j)isinA

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

0le qtn

983059D

tminus1983060leMyt

n

983059D


ytn

983059D

tminus1983060le yt

0

983059D


983131

j(nj)isinA

ztnj

983059D

tminus1983060= yt

n

983059D


983131

j(jn)isinA

ztjn

983059D

tminus1983060= yt

n

983059D


983131

(ij)isinA(S)

ztij

983059D

tminus1983060le983131

nisinS

ytn

983059D


k

983059D

tminus1983060


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102

(1)









983059D

t983060







tisin [T ]


















2 Satisficing


























ρτ τ (x) = inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτ iτ i(xi)

αi


983070 (2)
















microu(x) = infη

983069η

983055983055983055983055supPisinP


983070 (4)









ρτ τ (x) = inf

983099983103

983101983131

iisin[I]

αi

983055983055983055983055983055983055supPisinP

EP

983061u

983061vτ iτ i(xi)

αi


983100983104

983102 (5)







ρτ τ (x) = inf

983099983103

983101983131

iisin[I]

αi

983055983055983055983055983055983055Cαi


983100983104

983102

where

Cαi(xi) = αi ln

983061supPisinP

EP

983061exp

983061vτ iτ i(xi)

αi

983062983062983062











be simplified as

ρτ τ (x) = inf

983099983105983105983103

983105983105983101

983131

iisin[I]

αi

983055983055983055983055983055983055983055983055

supPisinP

EP

983061maxkisin[K]


983052983062le 0


983100983105983105983104

983105983105983102



inf983131

tisin[T ]

983131

nisin[N ]

αtn +

983131

tisin[T ]

βt

st supPisinP

EP

983061maxkisin[K]

983153akmax


n

983059D

t983060 xt

n

983059D

t983060minus τ t

n

983154+ bkα

tn


supPisinP

EP

983091

983107maxkisin[K]

983099983103

983101ak

983091

983107983131

(ij)isinA

cijztij

983059D


983092

983108+ bkβt

983100983104

983102

983092


xtn

983059D

t983060=

t983131

m=1

(qmn

983059D



(qtn yt0 y

tn z


(6)





















Routing decisions



Decisionmaking


Order quantities

Actual demand





qtn

983059D




n




realization



xtn

983059D

t983060=

t983131

m=1

983059qmn

983059D


983060=

t983131

m=1

983059qmn0 + 〈Qm

n D〉minus Dnm

983060= xt

n0 + 〈XtnD〉 (7)

here xtn0 =

t983123m=1

qmn0 and Xtn =

t983123m=1

(Qmn minus ene

primem)


tion cost983123


(6) supPisinP EP

983059u983059983059983123


983060βt






inf983131

tisin[T ]

983131

nisin[N ]

αtn

st supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt



n

983154+ bkα

tn





xtn0 =

t983131

m=1


Xtn =

t983131

m=1

(Qmn minus ene


983043Qt

n



(yt0 y

tn z


(8)


ZR =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

yt0 y

tn z

tij isin 01


(i j)isinA

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

983131

(ij)isinA


ytn le yt


983131

j(nj)isinA


983131

j(jn)isinA


983131

(ij)isinA(S)

ztij le983131

nisinS

ytn minus yt


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102

(9)


n ge 983171












misin[Ms]pm =

1 In this case

P =

983099983103

983101P

983055983055983055983055983055983055P983059D=D(m)


983131

misin[Ms]

pm = 1

983100983104

983102 (10)


supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

=983131

misin[Ms]

pm

983061maxkisin[K]

983153akmax

983153τ tn minusxt

n0 minus〈XtnD

(m)〉 xtn0 + 〈Xt

nD(m)〉minus τ t

n

983154+ bkα

tn

983154983062




983131

misin[Ms]


νmnt ge ak

983059τ tn minusxt

n0 minus〈XtnD

(m)〉983060+ bkα


νmnt ge ak

983059xtn0 + 〈Xt

nD(m)〉minus τ t

n

983060+ bkα


qtn0 + 〈QtnD


qtn0 + 〈QtnD











P =

983099983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059Dle DleD

983060= 1 (a)

EP

983059D983060=Ξ (b)

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ (c)

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094


983100983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983102

(11)















983075983055983055983055983055983055983123

iisin[N ]

DiτminusΞiτΣiτ

983055983055983055983055983055

983076le



983075983055983055983055983055983055983123

τisin[T ]

DiτminusΞiτΣiτ

983055983055983055983055983055







constraint (11d)






nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh le 0

〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime





nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]



T tn ge 0

Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime w







n〉 le 0

Qtn +Lt

n minusH tn = 0

LtnH

tn ge 0




MILP









Q=

983099983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983101

Q

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

Q983059983059

D U V983060isin 983142W

983060= 1

EQ

983059D983060=Ξ

EQ

983059U983060leΣ


983100983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983102

(12)



983142W =

983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101

983059D U V


983055983055983055983055983055983055983055983055983055983055983055983055983055


983055983055983055le U

fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092


983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102






qtn

983059D U V

983060= qtn0 + 〈Qt






0n

983044land

983043Qt

1n


983043Qt

2n


as

supQisinQ

EQ

983061maxkisin[K]

983153akmax



2n V 〉



n

983154+ bkα

tn

983154983060le 0

(13)



















Pdagger983059D

dagger= D

(m)983060=

1

Mr

forallmisin [Mr]



983059983056983056983056Dminus Ddagger983056983056983056983060

st (DDdagger)sim P

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1







983052 (14)





(8a)




rtnθ+983131

misin[Mr ]

stnm le 0


nmk〉+ 〈D(m)

Rtnmk〉 leMrs

tnm minus ak (τ

tn minusxt


〈DVt

nmk〉minus 〈DUt

nmk〉+ 〈D(m)

Rt


tn0 minus τ t


minusU tnmk +V t



minusUt

nmk +Vt

nmk = akXtn minusR

t


983043Rt

nmk


983043Rt

nmk


983059R

t

nmk

983060


983059R

t

nmk

983060


U tnmkV

tnmkU

t

nmkVt


rtn ge 0stn isinRMs









tij le yt

j

and983123

nisin[N ] qtn

983059D

tminus1983060le yt










2015)






efficiently




min gprimeφ



(y0Y Z)isinZR

(15)







(16)



as


iisin[I]


i Z〉)ψi

st983131

iisin[I]

aiψi = g

(17)









min η

st983131

iisin[I]



983131

iisin[I]



(y0Y Z)isinZR

(18)

If G1 = ψ|983123



to (18)




ψ|983123



lowastZlowast)














































as follows

P =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059dle Dnt le d

983060= 1

EP

983059Dnt

983060= micro

EP


EP

983091

983107

983055983055983055983055983055983055

983131

tisin[T ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


EP

983091

983107

983055983055983055983055983055983055

983131

nisin[N ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102


= 20 The




















is to minimize 1NT

983123nisin[N ]





min1

McNT

983131

misin[Mc]

983131

tisin[T ]

983131

nisin[N ]

I(m)nt

st I(m)nt ge x

(m)nt minus τ


I(m)nt ge minusx

(m)nt





983043yt0 y

tn z


983044isinZR




we set Mc = 100












timerdquo






over four periods


30 911 042 185 265 1044 9235 553 027 148 064 849 86

08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62


















(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105

08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155

SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243

35 124 070 068 145 062 112808 27 114 107 099 110 096 752

30 127 092 084 129 078 53735 125 074 071 na 064 395

MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372

08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613


52 Robustness




















(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110

08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141

SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083

35 162 100 086 273 08008 27 137 124 104 125 098

30 161 117 093 168 08535 177 113 090 NA 085

MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140

08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139







6 Conclusions














micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi




micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0



(xi)αi gt


micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0



S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070



micro

983061vτiτi


αλi

983062le micro

983061λvτiτi


αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi


983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376















































































and a lower bound















and Sim (2009)















tion approach













n983123j=1

XijYij We











S as |S|

2 Model











983059D1 Dt












) The




983059D


visited and yt0

983059D



983059D



983059D



983059D

t983060isinRt+1



0= empty so that


as

xtn

983059D

t983060= xtminus1

n

983059D

tminus1983060+ qtn

983059D


t983131

m=1

983059qmn

983059D










0

983059D



n

983059D



983059D



983059D



n

983059D




Z as

Z =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

yt0 y

tn isinBt

ztij isinBt

qtn isinRt


(i j)isinA

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

0le qtn

983059D

tminus1983060leMyt

n

983059D


ytn

983059D

tminus1983060le yt

0

983059D


983131

j(nj)isinA

ztnj

983059D

tminus1983060= yt

n

983059D


983131

j(jn)isinA

ztjn

983059D

tminus1983060= yt

n

983059D


983131

(ij)isinA(S)

ztij

983059D

tminus1983060le983131

nisinS

ytn

983059D


k

983059D

tminus1983060


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102

(1)









983059D

t983060







tisin [T ]


















2 Satisficing


























ρτ τ (x) = inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτ iτ i(xi)

αi


983070 (2)
















microu(x) = infη

983069η

983055983055983055983055supPisinP


983070 (4)









ρτ τ (x) = inf

983099983103

983101983131

iisin[I]

αi

983055983055983055983055983055983055supPisinP

EP

983061u

983061vτ iτ i(xi)

αi


983100983104

983102 (5)







ρτ τ (x) = inf

983099983103

983101983131

iisin[I]

αi

983055983055983055983055983055983055Cαi


983100983104

983102

where

Cαi(xi) = αi ln

983061supPisinP

EP

983061exp

983061vτ iτ i(xi)

αi

983062983062983062











be simplified as

ρτ τ (x) = inf

983099983105983105983103

983105983105983101

983131

iisin[I]

αi

983055983055983055983055983055983055983055983055

supPisinP

EP

983061maxkisin[K]


983052983062le 0


983100983105983105983104

983105983105983102



inf983131

tisin[T ]

983131

nisin[N ]

αtn +

983131

tisin[T ]

βt

st supPisinP

EP

983061maxkisin[K]

983153akmax


n

983059D

t983060 xt

n

983059D

t983060minus τ t

n

983154+ bkα

tn


supPisinP

EP

983091

983107maxkisin[K]

983099983103

983101ak

983091

983107983131

(ij)isinA

cijztij

983059D


983092

983108+ bkβt

983100983104

983102

983092


xtn

983059D

t983060=

t983131

m=1

(qmn

983059D



(qtn yt0 y

tn z


(6)





















Routing decisions



Decisionmaking


Order quantities

Actual demand





qtn

983059D




n




realization



xtn

983059D

t983060=

t983131

m=1

983059qmn

983059D


983060=

t983131

m=1

983059qmn0 + 〈Qm

n D〉minus Dnm

983060= xt

n0 + 〈XtnD〉 (7)

here xtn0 =

t983123m=1

qmn0 and Xtn =

t983123m=1

(Qmn minus ene

primem)


tion cost983123


(6) supPisinP EP

983059u983059983059983123


983060βt






inf983131

tisin[T ]

983131

nisin[N ]

αtn

st supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt



n

983154+ bkα

tn





xtn0 =

t983131

m=1


Xtn =

t983131

m=1

(Qmn minus ene


983043Qt

n



(yt0 y

tn z


(8)


ZR =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

yt0 y

tn z

tij isin 01


(i j)isinA

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

983131

(ij)isinA


ytn le yt


983131

j(nj)isinA


983131

j(jn)isinA


983131

(ij)isinA(S)

ztij le983131

nisinS

ytn minus yt


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102

(9)


n ge 983171












misin[Ms]pm =

1 In this case

P =

983099983103

983101P

983055983055983055983055983055983055P983059D=D(m)


983131

misin[Ms]

pm = 1

983100983104

983102 (10)


supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

=983131

misin[Ms]

pm

983061maxkisin[K]

983153akmax

983153τ tn minusxt

n0 minus〈XtnD

(m)〉 xtn0 + 〈Xt

nD(m)〉minus τ t

n

983154+ bkα

tn

983154983062




983131

misin[Ms]


νmnt ge ak

983059τ tn minusxt

n0 minus〈XtnD

(m)〉983060+ bkα


νmnt ge ak

983059xtn0 + 〈Xt

nD(m)〉minus τ t

n

983060+ bkα


qtn0 + 〈QtnD


qtn0 + 〈QtnD











P =

983099983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059Dle DleD

983060= 1 (a)

EP

983059D983060=Ξ (b)

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ (c)

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094


983100983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983102

(11)















983075983055983055983055983055983055983123

iisin[N ]

DiτminusΞiτΣiτ

983055983055983055983055983055

983076le



983075983055983055983055983055983055983123

τisin[T ]

DiτminusΞiτΣiτ

983055983055983055983055983055







constraint (11d)






nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh le 0

〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime





nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]



T tn ge 0

Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime w







n〉 le 0

Qtn +Lt

n minusH tn = 0

LtnH

tn ge 0




MILP









Q=

983099983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983101

Q

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

Q983059983059

D U V983060isin 983142W

983060= 1

EQ

983059D983060=Ξ

EQ

983059U983060leΣ


983100983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983102

(12)



983142W =

983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101

983059D U V


983055983055983055983055983055983055983055983055983055983055983055983055983055


983055983055983055le U

fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092


983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102






qtn

983059D U V

983060= qtn0 + 〈Qt






0n

983044land

983043Qt

1n


983043Qt

2n


as

supQisinQ

EQ

983061maxkisin[K]

983153akmax



2n V 〉



n

983154+ bkα

tn

983154983060le 0

(13)



















Pdagger983059D

dagger= D

(m)983060=

1

Mr

forallmisin [Mr]



983059983056983056983056Dminus Ddagger983056983056983056983060

st (DDdagger)sim P

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1







983052 (14)





(8a)




rtnθ+983131

misin[Mr ]

stnm le 0


nmk〉+ 〈D(m)

Rtnmk〉 leMrs

tnm minus ak (τ

tn minusxt


〈DVt

nmk〉minus 〈DUt

nmk〉+ 〈D(m)

Rt


tn0 minus τ t


minusU tnmk +V t



minusUt

nmk +Vt

nmk = akXtn minusR

t


983043Rt

nmk


983043Rt

nmk


983059R

t

nmk

983060


983059R

t

nmk

983060


U tnmkV

tnmkU

t

nmkVt


rtn ge 0stn isinRMs









tij le yt

j

and983123

nisin[N ] qtn

983059D

tminus1983060le yt










2015)






efficiently




min gprimeφ



(y0Y Z)isinZR

(15)







(16)



as


iisin[I]


i Z〉)ψi

st983131

iisin[I]

aiψi = g

(17)









min η

st983131

iisin[I]



983131

iisin[I]



(y0Y Z)isinZR

(18)

If G1 = ψ|983123



to (18)




ψ|983123



lowastZlowast)














































as follows

P =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059dle Dnt le d

983060= 1

EP

983059Dnt

983060= micro

EP


EP

983091

983107

983055983055983055983055983055983055

983131

tisin[T ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


EP

983091

983107

983055983055983055983055983055983055

983131

nisin[N ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102


= 20 The




















is to minimize 1NT

983123nisin[N ]





min1

McNT

983131

misin[Mc]

983131

tisin[T ]

983131

nisin[N ]

I(m)nt

st I(m)nt ge x

(m)nt minus τ


I(m)nt ge minusx

(m)nt





983043yt0 y

tn z


983044isinZR




we set Mc = 100












timerdquo






over four periods


30 911 042 185 265 1044 9235 553 027 148 064 849 86

08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62


















(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105

08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155

SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243

35 124 070 068 145 062 112808 27 114 107 099 110 096 752

30 127 092 084 129 078 53735 125 074 071 na 064 395

MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372

08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613


52 Robustness




















(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110

08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141

SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083

35 162 100 086 273 08008 27 137 124 104 125 098

30 161 117 093 168 08535 177 113 090 NA 085

MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140

08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139







6 Conclusions














micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi




micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0



(xi)αi gt


micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0



S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070



micro

983061vτiτi


αλi

983062le micro

983061λvτiτi


αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi


983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376













and Sim (2009)















tion approach













n983123j=1

XijYij We











S as |S|

2 Model











983059D1 Dt












) The




983059D


visited and yt0

983059D



983059D



983059D



983059D

t983060isinRt+1



0= empty so that


as

xtn

983059D

t983060= xtminus1

n

983059D

tminus1983060+ qtn

983059D


t983131

m=1

983059qmn

983059D










0

983059D



n

983059D



983059D



983059D



n

983059D




Z as

Z =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

yt0 y

tn isinBt

ztij isinBt

qtn isinRt


(i j)isinA

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

0le qtn

983059D

tminus1983060leMyt

n

983059D


ytn

983059D

tminus1983060le yt

0

983059D


983131

j(nj)isinA

ztnj

983059D

tminus1983060= yt

n

983059D


983131

j(jn)isinA

ztjn

983059D

tminus1983060= yt

n

983059D


983131

(ij)isinA(S)

ztij

983059D

tminus1983060le983131

nisinS

ytn

983059D


k

983059D

tminus1983060


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102

(1)









983059D

t983060







tisin [T ]


















2 Satisficing


























ρτ τ (x) = inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτ iτ i(xi)

αi


983070 (2)
















microu(x) = infη

983069η

983055983055983055983055supPisinP


983070 (4)









ρτ τ (x) = inf

983099983103

983101983131

iisin[I]

αi

983055983055983055983055983055983055supPisinP

EP

983061u

983061vτ iτ i(xi)

αi


983100983104

983102 (5)







ρτ τ (x) = inf

983099983103

983101983131

iisin[I]

αi

983055983055983055983055983055983055Cαi


983100983104

983102

where

Cαi(xi) = αi ln

983061supPisinP

EP

983061exp

983061vτ iτ i(xi)

αi

983062983062983062











be simplified as

ρτ τ (x) = inf

983099983105983105983103

983105983105983101

983131

iisin[I]

αi

983055983055983055983055983055983055983055983055

supPisinP

EP

983061maxkisin[K]


983052983062le 0


983100983105983105983104

983105983105983102



inf983131

tisin[T ]

983131

nisin[N ]

αtn +

983131

tisin[T ]

βt

st supPisinP

EP

983061maxkisin[K]

983153akmax


n

983059D

t983060 xt

n

983059D

t983060minus τ t

n

983154+ bkα

tn


supPisinP

EP

983091

983107maxkisin[K]

983099983103

983101ak

983091

983107983131

(ij)isinA

cijztij

983059D


983092

983108+ bkβt

983100983104

983102

983092


xtn

983059D

t983060=

t983131

m=1

(qmn

983059D



(qtn yt0 y

tn z


(6)





















Routing decisions



Decisionmaking


Order quantities

Actual demand





qtn

983059D




n




realization



xtn

983059D

t983060=

t983131

m=1

983059qmn

983059D


983060=

t983131

m=1

983059qmn0 + 〈Qm

n D〉minus Dnm

983060= xt

n0 + 〈XtnD〉 (7)

here xtn0 =

t983123m=1

qmn0 and Xtn =

t983123m=1

(Qmn minus ene

primem)


tion cost983123


(6) supPisinP EP

983059u983059983059983123


983060βt






inf983131

tisin[T ]

983131

nisin[N ]

αtn

st supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt



n

983154+ bkα

tn





xtn0 =

t983131

m=1


Xtn =

t983131

m=1

(Qmn minus ene


983043Qt

n



(yt0 y

tn z


(8)


ZR =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

yt0 y

tn z

tij isin 01


(i j)isinA

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

983131

(ij)isinA


ytn le yt


983131

j(nj)isinA


983131

j(jn)isinA


983131

(ij)isinA(S)

ztij le983131

nisinS

ytn minus yt


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102

(9)


n ge 983171












misin[Ms]pm =

1 In this case

P =

983099983103

983101P

983055983055983055983055983055983055P983059D=D(m)


983131

misin[Ms]

pm = 1

983100983104

983102 (10)


supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

=983131

misin[Ms]

pm

983061maxkisin[K]

983153akmax

983153τ tn minusxt

n0 minus〈XtnD

(m)〉 xtn0 + 〈Xt

nD(m)〉minus τ t

n

983154+ bkα

tn

983154983062




983131

misin[Ms]


νmnt ge ak

983059τ tn minusxt

n0 minus〈XtnD

(m)〉983060+ bkα


νmnt ge ak

983059xtn0 + 〈Xt

nD(m)〉minus τ t

n

983060+ bkα


qtn0 + 〈QtnD


qtn0 + 〈QtnD











P =

983099983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059Dle DleD

983060= 1 (a)

EP

983059D983060=Ξ (b)

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ (c)

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094


983100983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983102

(11)















983075983055983055983055983055983055983123

iisin[N ]

DiτminusΞiτΣiτ

983055983055983055983055983055

983076le



983075983055983055983055983055983055983123

τisin[T ]

DiτminusΞiτΣiτ

983055983055983055983055983055







constraint (11d)






nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh le 0

〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime





nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]



T tn ge 0

Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime w







n〉 le 0

Qtn +Lt

n minusH tn = 0

LtnH

tn ge 0




MILP









Q=

983099983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983101

Q

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

Q983059983059

D U V983060isin 983142W

983060= 1

EQ

983059D983060=Ξ

EQ

983059U983060leΣ


983100983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983102

(12)



983142W =

983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101

983059D U V


983055983055983055983055983055983055983055983055983055983055983055983055983055


983055983055983055le U

fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092


983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102






qtn

983059D U V

983060= qtn0 + 〈Qt






0n

983044land

983043Qt

1n


983043Qt

2n


as

supQisinQ

EQ

983061maxkisin[K]

983153akmax



2n V 〉



n

983154+ bkα

tn

983154983060le 0

(13)



















Pdagger983059D

dagger= D

(m)983060=

1

Mr

forallmisin [Mr]



983059983056983056983056Dminus Ddagger983056983056983056983060

st (DDdagger)sim P

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1







983052 (14)





(8a)




rtnθ+983131

misin[Mr ]

stnm le 0


nmk〉+ 〈D(m)

Rtnmk〉 leMrs

tnm minus ak (τ

tn minusxt


〈DVt

nmk〉minus 〈DUt

nmk〉+ 〈D(m)

Rt


tn0 minus τ t


minusU tnmk +V t



minusUt

nmk +Vt

nmk = akXtn minusR

t


983043Rt

nmk


983043Rt

nmk


983059R

t

nmk

983060


983059R

t

nmk

983060


U tnmkV

tnmkU

t

nmkVt


rtn ge 0stn isinRMs









tij le yt

j

and983123

nisin[N ] qtn

983059D

tminus1983060le yt










2015)






efficiently




min gprimeφ



(y0Y Z)isinZR

(15)







(16)



as


iisin[I]


i Z〉)ψi

st983131

iisin[I]

aiψi = g

(17)









min η

st983131

iisin[I]



983131

iisin[I]



(y0Y Z)isinZR

(18)

If G1 = ψ|983123



to (18)




ψ|983123



lowastZlowast)














































as follows

P =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059dle Dnt le d

983060= 1

EP

983059Dnt

983060= micro

EP


EP

983091

983107

983055983055983055983055983055983055

983131

tisin[T ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


EP

983091

983107

983055983055983055983055983055983055

983131

nisin[N ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102


= 20 The




















is to minimize 1NT

983123nisin[N ]





min1

McNT

983131

misin[Mc]

983131

tisin[T ]

983131

nisin[N ]

I(m)nt

st I(m)nt ge x

(m)nt minus τ


I(m)nt ge minusx

(m)nt





983043yt0 y

tn z


983044isinZR




we set Mc = 100












timerdquo






over four periods


30 911 042 185 265 1044 9235 553 027 148 064 849 86

08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62


















(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105

08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155

SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243

35 124 070 068 145 062 112808 27 114 107 099 110 096 752

30 127 092 084 129 078 53735 125 074 071 na 064 395

MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372

08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613


52 Robustness




















(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110

08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141

SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083

35 162 100 086 273 08008 27 137 124 104 125 098

30 161 117 093 168 08535 177 113 090 NA 085

MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140

08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139







6 Conclusions














micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi




micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0



(xi)αi gt


micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0



S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070



micro

983061vτiτi


αλi

983062le micro

983061λvτiτi


αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi


983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376













n983123j=1

XijYij We











S as |S|

2 Model











983059D1 Dt












) The




983059D


visited and yt0

983059D



983059D



983059D



983059D

t983060isinRt+1



0= empty so that


as

xtn

983059D

t983060= xtminus1

n

983059D

tminus1983060+ qtn

983059D


t983131

m=1

983059qmn

983059D










0

983059D



n

983059D



983059D



983059D



n

983059D




Z as

Z =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

yt0 y

tn isinBt

ztij isinBt

qtn isinRt


(i j)isinA

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

0le qtn

983059D

tminus1983060leMyt

n

983059D


ytn

983059D

tminus1983060le yt

0

983059D


983131

j(nj)isinA

ztnj

983059D

tminus1983060= yt

n

983059D


983131

j(jn)isinA

ztjn

983059D

tminus1983060= yt

n

983059D


983131

(ij)isinA(S)

ztij

983059D

tminus1983060le983131

nisinS

ytn

983059D


k

983059D

tminus1983060


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102

(1)









983059D

t983060







tisin [T ]


















2 Satisficing


























ρτ τ (x) = inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτ iτ i(xi)

αi


983070 (2)
















microu(x) = infη

983069η

983055983055983055983055supPisinP


983070 (4)









ρτ τ (x) = inf

983099983103

983101983131

iisin[I]

αi

983055983055983055983055983055983055supPisinP

EP

983061u

983061vτ iτ i(xi)

αi


983100983104

983102 (5)







ρτ τ (x) = inf

983099983103

983101983131

iisin[I]

αi

983055983055983055983055983055983055Cαi


983100983104

983102

where

Cαi(xi) = αi ln

983061supPisinP

EP

983061exp

983061vτ iτ i(xi)

αi

983062983062983062











be simplified as

ρτ τ (x) = inf

983099983105983105983103

983105983105983101

983131

iisin[I]

αi

983055983055983055983055983055983055983055983055

supPisinP

EP

983061maxkisin[K]


983052983062le 0


983100983105983105983104

983105983105983102



inf983131

tisin[T ]

983131

nisin[N ]

αtn +

983131

tisin[T ]

βt

st supPisinP

EP

983061maxkisin[K]

983153akmax


n

983059D

t983060 xt

n

983059D

t983060minus τ t

n

983154+ bkα

tn


supPisinP

EP

983091

983107maxkisin[K]

983099983103

983101ak

983091

983107983131

(ij)isinA

cijztij

983059D


983092

983108+ bkβt

983100983104

983102

983092


xtn

983059D

t983060=

t983131

m=1

(qmn

983059D



(qtn yt0 y

tn z


(6)





















Routing decisions



Decisionmaking


Order quantities

Actual demand





qtn

983059D




n




realization



xtn

983059D

t983060=

t983131

m=1

983059qmn

983059D


983060=

t983131

m=1

983059qmn0 + 〈Qm

n D〉minus Dnm

983060= xt

n0 + 〈XtnD〉 (7)

here xtn0 =

t983123m=1

qmn0 and Xtn =

t983123m=1

(Qmn minus ene

primem)


tion cost983123


(6) supPisinP EP

983059u983059983059983123


983060βt






inf983131

tisin[T ]

983131

nisin[N ]

αtn

st supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt



n

983154+ bkα

tn





xtn0 =

t983131

m=1


Xtn =

t983131

m=1

(Qmn minus ene


983043Qt

n



(yt0 y

tn z


(8)


ZR =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

yt0 y

tn z

tij isin 01


(i j)isinA

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

983131

(ij)isinA


ytn le yt


983131

j(nj)isinA


983131

j(jn)isinA


983131

(ij)isinA(S)

ztij le983131

nisinS

ytn minus yt


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102

(9)


n ge 983171












misin[Ms]pm =

1 In this case

P =

983099983103

983101P

983055983055983055983055983055983055P983059D=D(m)


983131

misin[Ms]

pm = 1

983100983104

983102 (10)


supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

=983131

misin[Ms]

pm

983061maxkisin[K]

983153akmax

983153τ tn minusxt

n0 minus〈XtnD

(m)〉 xtn0 + 〈Xt

nD(m)〉minus τ t

n

983154+ bkα

tn

983154983062




983131

misin[Ms]


νmnt ge ak

983059τ tn minusxt

n0 minus〈XtnD

(m)〉983060+ bkα


νmnt ge ak

983059xtn0 + 〈Xt

nD(m)〉minus τ t

n

983060+ bkα


qtn0 + 〈QtnD


qtn0 + 〈QtnD











P =

983099983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059Dle DleD

983060= 1 (a)

EP

983059D983060=Ξ (b)

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ (c)

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094


983100983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983102

(11)















983075983055983055983055983055983055983123

iisin[N ]

DiτminusΞiτΣiτ

983055983055983055983055983055

983076le



983075983055983055983055983055983055983123

τisin[T ]

DiτminusΞiτΣiτ

983055983055983055983055983055







constraint (11d)






nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh le 0

〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime





nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]



T tn ge 0

Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime w







n〉 le 0

Qtn +Lt

n minusH tn = 0

LtnH

tn ge 0




MILP









Q=

983099983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983101

Q

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

Q983059983059

D U V983060isin 983142W

983060= 1

EQ

983059D983060=Ξ

EQ

983059U983060leΣ


983100983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983102

(12)



983142W =

983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101

983059D U V


983055983055983055983055983055983055983055983055983055983055983055983055983055


983055983055983055le U

fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092


983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102






qtn

983059D U V

983060= qtn0 + 〈Qt






0n

983044land

983043Qt

1n


983043Qt

2n


as

supQisinQ

EQ

983061maxkisin[K]

983153akmax



2n V 〉



n

983154+ bkα

tn

983154983060le 0

(13)



















Pdagger983059D

dagger= D

(m)983060=

1

Mr

forallmisin [Mr]



983059983056983056983056Dminus Ddagger983056983056983056983060

st (DDdagger)sim P

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1







983052 (14)





(8a)




rtnθ+983131

misin[Mr ]

stnm le 0


nmk〉+ 〈D(m)

Rtnmk〉 leMrs

tnm minus ak (τ

tn minusxt


〈DVt

nmk〉minus 〈DUt

nmk〉+ 〈D(m)

Rt


tn0 minus τ t


minusU tnmk +V t



minusUt

nmk +Vt

nmk = akXtn minusR

t


983043Rt

nmk


983043Rt

nmk


983059R

t

nmk

983060


983059R

t

nmk

983060


U tnmkV

tnmkU

t

nmkVt


rtn ge 0stn isinRMs









tij le yt

j

and983123

nisin[N ] qtn

983059D

tminus1983060le yt










2015)






efficiently




min gprimeφ



(y0Y Z)isinZR

(15)







(16)



as


iisin[I]


i Z〉)ψi

st983131

iisin[I]

aiψi = g

(17)









min η

st983131

iisin[I]



983131

iisin[I]



(y0Y Z)isinZR

(18)

If G1 = ψ|983123



to (18)




ψ|983123



lowastZlowast)














































as follows

P =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059dle Dnt le d

983060= 1

EP

983059Dnt

983060= micro

EP


EP

983091

983107

983055983055983055983055983055983055

983131

tisin[T ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


EP

983091

983107

983055983055983055983055983055983055

983131

nisin[N ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102


= 20 The




















is to minimize 1NT

983123nisin[N ]





min1

McNT

983131

misin[Mc]

983131

tisin[T ]

983131

nisin[N ]

I(m)nt

st I(m)nt ge x

(m)nt minus τ


I(m)nt ge minusx

(m)nt





983043yt0 y

tn z


983044isinZR




we set Mc = 100












timerdquo






over four periods


30 911 042 185 265 1044 9235 553 027 148 064 849 86

08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62


















(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105

08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155

SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243

35 124 070 068 145 062 112808 27 114 107 099 110 096 752

30 127 092 084 129 078 53735 125 074 071 na 064 395

MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372

08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613


52 Robustness




















(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110

08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141

SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083

35 162 100 086 273 08008 27 137 124 104 125 098

30 161 117 093 168 08535 177 113 090 NA 085

MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140

08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139







6 Conclusions














micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi




micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0



(xi)αi gt


micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0



S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070



micro

983061vτiτi


αλi

983062le micro

983061λvτiτi


αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi


983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376








983059D1 Dt












) The




983059D


visited and yt0

983059D



983059D



983059D



983059D

t983060isinRt+1



0= empty so that


as

xtn

983059D

t983060= xtminus1

n

983059D

tminus1983060+ qtn

983059D


t983131

m=1

983059qmn

983059D










0

983059D



n

983059D



983059D



983059D



n

983059D




Z as

Z =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

yt0 y

tn isinBt

ztij isinBt

qtn isinRt


(i j)isinA

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

0le qtn

983059D

tminus1983060leMyt

n

983059D


ytn

983059D

tminus1983060le yt

0

983059D


983131

j(nj)isinA

ztnj

983059D

tminus1983060= yt

n

983059D


983131

j(jn)isinA

ztjn

983059D

tminus1983060= yt

n

983059D


983131

(ij)isinA(S)

ztij

983059D

tminus1983060le983131

nisinS

ytn

983059D


k

983059D

tminus1983060


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102

(1)









983059D

t983060







tisin [T ]


















2 Satisficing


























ρτ τ (x) = inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτ iτ i(xi)

αi


983070 (2)
















microu(x) = infη

983069η

983055983055983055983055supPisinP


983070 (4)









ρτ τ (x) = inf

983099983103

983101983131

iisin[I]

αi

983055983055983055983055983055983055supPisinP

EP

983061u

983061vτ iτ i(xi)

αi


983100983104

983102 (5)







ρτ τ (x) = inf

983099983103

983101983131

iisin[I]

αi

983055983055983055983055983055983055Cαi


983100983104

983102

where

Cαi(xi) = αi ln

983061supPisinP

EP

983061exp

983061vτ iτ i(xi)

αi

983062983062983062











be simplified as

ρτ τ (x) = inf

983099983105983105983103

983105983105983101

983131

iisin[I]

αi

983055983055983055983055983055983055983055983055

supPisinP

EP

983061maxkisin[K]


983052983062le 0


983100983105983105983104

983105983105983102



inf983131

tisin[T ]

983131

nisin[N ]

αtn +

983131

tisin[T ]

βt

st supPisinP

EP

983061maxkisin[K]

983153akmax


n

983059D

t983060 xt

n

983059D

t983060minus τ t

n

983154+ bkα

tn


supPisinP

EP

983091

983107maxkisin[K]

983099983103

983101ak

983091

983107983131

(ij)isinA

cijztij

983059D


983092

983108+ bkβt

983100983104

983102

983092


xtn

983059D

t983060=

t983131

m=1

(qmn

983059D



(qtn yt0 y

tn z


(6)





















Routing decisions



Decisionmaking


Order quantities

Actual demand





qtn

983059D




n




realization



xtn

983059D

t983060=

t983131

m=1

983059qmn

983059D


983060=

t983131

m=1

983059qmn0 + 〈Qm

n D〉minus Dnm

983060= xt

n0 + 〈XtnD〉 (7)

here xtn0 =

t983123m=1

qmn0 and Xtn =

t983123m=1

(Qmn minus ene

primem)


tion cost983123


(6) supPisinP EP

983059u983059983059983123


983060βt






inf983131

tisin[T ]

983131

nisin[N ]

αtn

st supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt



n

983154+ bkα

tn





xtn0 =

t983131

m=1


Xtn =

t983131

m=1

(Qmn minus ene


983043Qt

n



(yt0 y

tn z


(8)


ZR =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

yt0 y

tn z

tij isin 01


(i j)isinA

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

983131

(ij)isinA


ytn le yt


983131

j(nj)isinA


983131

j(jn)isinA


983131

(ij)isinA(S)

ztij le983131

nisinS

ytn minus yt


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102

(9)


n ge 983171












misin[Ms]pm =

1 In this case

P =

983099983103

983101P

983055983055983055983055983055983055P983059D=D(m)


983131

misin[Ms]

pm = 1

983100983104

983102 (10)


supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

=983131

misin[Ms]

pm

983061maxkisin[K]

983153akmax

983153τ tn minusxt

n0 minus〈XtnD

(m)〉 xtn0 + 〈Xt

nD(m)〉minus τ t

n

983154+ bkα

tn

983154983062




983131

misin[Ms]


νmnt ge ak

983059τ tn minusxt

n0 minus〈XtnD

(m)〉983060+ bkα


νmnt ge ak

983059xtn0 + 〈Xt

nD(m)〉minus τ t

n

983060+ bkα


qtn0 + 〈QtnD


qtn0 + 〈QtnD











P =

983099983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059Dle DleD

983060= 1 (a)

EP

983059D983060=Ξ (b)

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ (c)

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094


983100983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983102

(11)















983075983055983055983055983055983055983123

iisin[N ]

DiτminusΞiτΣiτ

983055983055983055983055983055

983076le



983075983055983055983055983055983055983123

τisin[T ]

DiτminusΞiτΣiτ

983055983055983055983055983055







constraint (11d)






nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh le 0

〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime





nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]



T tn ge 0

Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime w







n〉 le 0

Qtn +Lt

n minusH tn = 0

LtnH

tn ge 0




MILP









Q=

983099983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983101

Q

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

Q983059983059

D U V983060isin 983142W

983060= 1

EQ

983059D983060=Ξ

EQ

983059U983060leΣ


983100983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983102

(12)



983142W =

983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101

983059D U V


983055983055983055983055983055983055983055983055983055983055983055983055983055


983055983055983055le U

fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092


983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102






qtn

983059D U V

983060= qtn0 + 〈Qt






0n

983044land

983043Qt

1n


983043Qt

2n


as

supQisinQ

EQ

983061maxkisin[K]

983153akmax



2n V 〉



n

983154+ bkα

tn

983154983060le 0

(13)



















Pdagger983059D

dagger= D

(m)983060=

1

Mr

forallmisin [Mr]



983059983056983056983056Dminus Ddagger983056983056983056983060

st (DDdagger)sim P

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1







983052 (14)





(8a)




rtnθ+983131

misin[Mr ]

stnm le 0


nmk〉+ 〈D(m)

Rtnmk〉 leMrs

tnm minus ak (τ

tn minusxt


〈DVt

nmk〉minus 〈DUt

nmk〉+ 〈D(m)

Rt


tn0 minus τ t


minusU tnmk +V t



minusUt

nmk +Vt

nmk = akXtn minusR

t


983043Rt

nmk


983043Rt

nmk


983059R

t

nmk

983060


983059R

t

nmk

983060


U tnmkV

tnmkU

t

nmkVt


rtn ge 0stn isinRMs









tij le yt

j

and983123

nisin[N ] qtn

983059D

tminus1983060le yt










2015)






efficiently




min gprimeφ



(y0Y Z)isinZR

(15)







(16)



as


iisin[I]


i Z〉)ψi

st983131

iisin[I]

aiψi = g

(17)









min η

st983131

iisin[I]



983131

iisin[I]



(y0Y Z)isinZR

(18)

If G1 = ψ|983123



to (18)




ψ|983123



lowastZlowast)














































as follows

P =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059dle Dnt le d

983060= 1

EP

983059Dnt

983060= micro

EP


EP

983091

983107

983055983055983055983055983055983055

983131

tisin[T ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


EP

983091

983107

983055983055983055983055983055983055

983131

nisin[N ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102


= 20 The




















is to minimize 1NT

983123nisin[N ]





min1

McNT

983131

misin[Mc]

983131

tisin[T ]

983131

nisin[N ]

I(m)nt

st I(m)nt ge x

(m)nt minus τ


I(m)nt ge minusx

(m)nt





983043yt0 y

tn z


983044isinZR




we set Mc = 100












timerdquo






over four periods


30 911 042 185 265 1044 9235 553 027 148 064 849 86

08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62


















(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105

08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155

SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243

35 124 070 068 145 062 112808 27 114 107 099 110 096 752

30 127 092 084 129 078 53735 125 074 071 na 064 395

MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372

08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613


52 Robustness




















(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110

08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141

SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083

35 162 100 086 273 08008 27 137 124 104 125 098

30 161 117 093 168 08535 177 113 090 NA 085

MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140

08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139







6 Conclusions














micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi




micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0



(xi)αi gt


micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0



S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070



micro

983061vτiτi


αλi

983062le micro

983061λvτiτi


αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi


983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376









0

983059D



n

983059D



983059D



983059D



n

983059D




Z as

Z =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

yt0 y

tn isinBt

ztij isinBt

qtn isinRt


(i j)isinA

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

0le qtn

983059D

tminus1983060leMyt

n

983059D


ytn

983059D

tminus1983060le yt

0

983059D


983131

j(nj)isinA

ztnj

983059D

tminus1983060= yt

n

983059D


983131

j(jn)isinA

ztjn

983059D

tminus1983060= yt

n

983059D


983131

(ij)isinA(S)

ztij

983059D

tminus1983060le983131

nisinS

ytn

983059D


k

983059D

tminus1983060


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102

(1)









983059D

t983060







tisin [T ]


















2 Satisficing


























ρτ τ (x) = inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτ iτ i(xi)

αi


983070 (2)
















microu(x) = infη

983069η

983055983055983055983055supPisinP


983070 (4)









ρτ τ (x) = inf

983099983103

983101983131

iisin[I]

αi

983055983055983055983055983055983055supPisinP

EP

983061u

983061vτ iτ i(xi)

αi


983100983104

983102 (5)







ρτ τ (x) = inf

983099983103

983101983131

iisin[I]

αi

983055983055983055983055983055983055Cαi


983100983104

983102

where

Cαi(xi) = αi ln

983061supPisinP

EP

983061exp

983061vτ iτ i(xi)

αi

983062983062983062











be simplified as

ρτ τ (x) = inf

983099983105983105983103

983105983105983101

983131

iisin[I]

αi

983055983055983055983055983055983055983055983055

supPisinP

EP

983061maxkisin[K]


983052983062le 0


983100983105983105983104

983105983105983102



inf983131

tisin[T ]

983131

nisin[N ]

αtn +

983131

tisin[T ]

βt

st supPisinP

EP

983061maxkisin[K]

983153akmax


n

983059D

t983060 xt

n

983059D

t983060minus τ t

n

983154+ bkα

tn


supPisinP

EP

983091

983107maxkisin[K]

983099983103

983101ak

983091

983107983131

(ij)isinA

cijztij

983059D


983092

983108+ bkβt

983100983104

983102

983092


xtn

983059D

t983060=

t983131

m=1

(qmn

983059D



(qtn yt0 y

tn z


(6)





















Routing decisions



Decisionmaking


Order quantities

Actual demand





qtn

983059D




n




realization



xtn

983059D

t983060=

t983131

m=1

983059qmn

983059D


983060=

t983131

m=1

983059qmn0 + 〈Qm

n D〉minus Dnm

983060= xt

n0 + 〈XtnD〉 (7)

here xtn0 =

t983123m=1

qmn0 and Xtn =

t983123m=1

(Qmn minus ene

primem)


tion cost983123


(6) supPisinP EP

983059u983059983059983123


983060βt






inf983131

tisin[T ]

983131

nisin[N ]

αtn

st supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt



n

983154+ bkα

tn





xtn0 =

t983131

m=1


Xtn =

t983131

m=1

(Qmn minus ene


983043Qt

n



(yt0 y

tn z


(8)


ZR =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

yt0 y

tn z

tij isin 01


(i j)isinA

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

983131

(ij)isinA


ytn le yt


983131

j(nj)isinA


983131

j(jn)isinA


983131

(ij)isinA(S)

ztij le983131

nisinS

ytn minus yt


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102

(9)


n ge 983171












misin[Ms]pm =

1 In this case

P =

983099983103

983101P

983055983055983055983055983055983055P983059D=D(m)


983131

misin[Ms]

pm = 1

983100983104

983102 (10)


supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

=983131

misin[Ms]

pm

983061maxkisin[K]

983153akmax

983153τ tn minusxt

n0 minus〈XtnD

(m)〉 xtn0 + 〈Xt

nD(m)〉minus τ t

n

983154+ bkα

tn

983154983062




983131

misin[Ms]


νmnt ge ak

983059τ tn minusxt

n0 minus〈XtnD

(m)〉983060+ bkα


νmnt ge ak

983059xtn0 + 〈Xt

nD(m)〉minus τ t

n

983060+ bkα


qtn0 + 〈QtnD


qtn0 + 〈QtnD











P =

983099983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059Dle DleD

983060= 1 (a)

EP

983059D983060=Ξ (b)

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ (c)

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094


983100983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983102

(11)















983075983055983055983055983055983055983123

iisin[N ]

DiτminusΞiτΣiτ

983055983055983055983055983055

983076le



983075983055983055983055983055983055983123

τisin[T ]

DiτminusΞiτΣiτ

983055983055983055983055983055







constraint (11d)






nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh le 0

〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime





nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]



T tn ge 0

Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime w







n〉 le 0

Qtn +Lt

n minusH tn = 0

LtnH

tn ge 0




MILP









Q=

983099983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983101

Q

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

Q983059983059

D U V983060isin 983142W

983060= 1

EQ

983059D983060=Ξ

EQ

983059U983060leΣ


983100983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983102

(12)



983142W =

983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101

983059D U V


983055983055983055983055983055983055983055983055983055983055983055983055983055


983055983055983055le U

fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092


983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102






qtn

983059D U V

983060= qtn0 + 〈Qt






0n

983044land

983043Qt

1n


983043Qt

2n


as

supQisinQ

EQ

983061maxkisin[K]

983153akmax



2n V 〉



n

983154+ bkα

tn

983154983060le 0

(13)



















Pdagger983059D

dagger= D

(m)983060=

1

Mr

forallmisin [Mr]



983059983056983056983056Dminus Ddagger983056983056983056983060

st (DDdagger)sim P

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1







983052 (14)





(8a)




rtnθ+983131

misin[Mr ]

stnm le 0


nmk〉+ 〈D(m)

Rtnmk〉 leMrs

tnm minus ak (τ

tn minusxt


〈DVt

nmk〉minus 〈DUt

nmk〉+ 〈D(m)

Rt


tn0 minus τ t


minusU tnmk +V t



minusUt

nmk +Vt

nmk = akXtn minusR

t


983043Rt

nmk


983043Rt

nmk


983059R

t

nmk

983060


983059R

t

nmk

983060


U tnmkV

tnmkU

t

nmkVt


rtn ge 0stn isinRMs









tij le yt

j

and983123

nisin[N ] qtn

983059D

tminus1983060le yt










2015)






efficiently




min gprimeφ



(y0Y Z)isinZR

(15)







(16)



as


iisin[I]


i Z〉)ψi

st983131

iisin[I]

aiψi = g

(17)









min η

st983131

iisin[I]



983131

iisin[I]



(y0Y Z)isinZR

(18)

If G1 = ψ|983123



to (18)




ψ|983123



lowastZlowast)














































as follows

P =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059dle Dnt le d

983060= 1

EP

983059Dnt

983060= micro

EP


EP

983091

983107

983055983055983055983055983055983055

983131

tisin[T ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


EP

983091

983107

983055983055983055983055983055983055

983131

nisin[N ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102


= 20 The




















is to minimize 1NT

983123nisin[N ]





min1

McNT

983131

misin[Mc]

983131

tisin[T ]

983131

nisin[N ]

I(m)nt

st I(m)nt ge x

(m)nt minus τ


I(m)nt ge minusx

(m)nt





983043yt0 y

tn z


983044isinZR




we set Mc = 100












timerdquo






over four periods


30 911 042 185 265 1044 9235 553 027 148 064 849 86

08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62


















(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105

08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155

SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243

35 124 070 068 145 062 112808 27 114 107 099 110 096 752

30 127 092 084 129 078 53735 125 074 071 na 064 395

MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372

08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613


52 Robustness




















(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110

08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141

SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083

35 162 100 086 273 08008 27 137 124 104 125 098

30 161 117 093 168 08535 177 113 090 NA 085

MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140

08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139







6 Conclusions














micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi




micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0



(xi)αi gt


micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0



S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070



micro

983061vτiτi


αλi

983062le micro

983061λvτiτi


αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi


983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376





983059D

t983060







tisin [T ]


















2 Satisficing


























ρτ τ (x) = inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτ iτ i(xi)

αi


983070 (2)
















microu(x) = infη

983069η

983055983055983055983055supPisinP


983070 (4)









ρτ τ (x) = inf

983099983103

983101983131

iisin[I]

αi

983055983055983055983055983055983055supPisinP

EP

983061u

983061vτ iτ i(xi)

αi


983100983104

983102 (5)







ρτ τ (x) = inf

983099983103

983101983131

iisin[I]

αi

983055983055983055983055983055983055Cαi


983100983104

983102

where

Cαi(xi) = αi ln

983061supPisinP

EP

983061exp

983061vτ iτ i(xi)

αi

983062983062983062











be simplified as

ρτ τ (x) = inf

983099983105983105983103

983105983105983101

983131

iisin[I]

αi

983055983055983055983055983055983055983055983055

supPisinP

EP

983061maxkisin[K]


983052983062le 0


983100983105983105983104

983105983105983102



inf983131

tisin[T ]

983131

nisin[N ]

αtn +

983131

tisin[T ]

βt

st supPisinP

EP

983061maxkisin[K]

983153akmax


n

983059D

t983060 xt

n

983059D

t983060minus τ t

n

983154+ bkα

tn


supPisinP

EP

983091

983107maxkisin[K]

983099983103

983101ak

983091

983107983131

(ij)isinA

cijztij

983059D


983092

983108+ bkβt

983100983104

983102

983092


xtn

983059D

t983060=

t983131

m=1

(qmn

983059D



(qtn yt0 y

tn z


(6)





















Routing decisions



Decisionmaking


Order quantities

Actual demand





qtn

983059D




n




realization



xtn

983059D

t983060=

t983131

m=1

983059qmn

983059D


983060=

t983131

m=1

983059qmn0 + 〈Qm

n D〉minus Dnm

983060= xt

n0 + 〈XtnD〉 (7)

here xtn0 =

t983123m=1

qmn0 and Xtn =

t983123m=1

(Qmn minus ene

primem)


tion cost983123


(6) supPisinP EP

983059u983059983059983123


983060βt






inf983131

tisin[T ]

983131

nisin[N ]

αtn

st supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt



n

983154+ bkα

tn





xtn0 =

t983131

m=1


Xtn =

t983131

m=1

(Qmn minus ene


983043Qt

n



(yt0 y

tn z


(8)


ZR =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

yt0 y

tn z

tij isin 01


(i j)isinA

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

983131

(ij)isinA


ytn le yt


983131

j(nj)isinA


983131

j(jn)isinA


983131

(ij)isinA(S)

ztij le983131

nisinS

ytn minus yt


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102

(9)


n ge 983171












misin[Ms]pm =

1 In this case

P =

983099983103

983101P

983055983055983055983055983055983055P983059D=D(m)


983131

misin[Ms]

pm = 1

983100983104

983102 (10)


supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

=983131

misin[Ms]

pm

983061maxkisin[K]

983153akmax

983153τ tn minusxt

n0 minus〈XtnD

(m)〉 xtn0 + 〈Xt

nD(m)〉minus τ t

n

983154+ bkα

tn

983154983062




983131

misin[Ms]


νmnt ge ak

983059τ tn minusxt

n0 minus〈XtnD

(m)〉983060+ bkα


νmnt ge ak

983059xtn0 + 〈Xt

nD(m)〉minus τ t

n

983060+ bkα


qtn0 + 〈QtnD


qtn0 + 〈QtnD











P =

983099983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059Dle DleD

983060= 1 (a)

EP

983059D983060=Ξ (b)

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ (c)

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094


983100983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983102

(11)















983075983055983055983055983055983055983123

iisin[N ]

DiτminusΞiτΣiτ

983055983055983055983055983055

983076le



983075983055983055983055983055983055983123

τisin[T ]

DiτminusΞiτΣiτ

983055983055983055983055983055







constraint (11d)






nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh le 0

〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime





nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]



T tn ge 0

Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime w







n〉 le 0

Qtn +Lt

n minusH tn = 0

LtnH

tn ge 0




MILP









Q=

983099983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983101

Q

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

Q983059983059

D U V983060isin 983142W

983060= 1

EQ

983059D983060=Ξ

EQ

983059U983060leΣ


983100983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983102

(12)



983142W =

983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101

983059D U V


983055983055983055983055983055983055983055983055983055983055983055983055983055


983055983055983055le U

fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092


983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102






qtn

983059D U V

983060= qtn0 + 〈Qt






0n

983044land

983043Qt

1n


983043Qt

2n


as

supQisinQ

EQ

983061maxkisin[K]

983153akmax



2n V 〉



n

983154+ bkα

tn

983154983060le 0

(13)



















Pdagger983059D

dagger= D

(m)983060=

1

Mr

forallmisin [Mr]



983059983056983056983056Dminus Ddagger983056983056983056983060

st (DDdagger)sim P

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1







983052 (14)





(8a)




rtnθ+983131

misin[Mr ]

stnm le 0


nmk〉+ 〈D(m)

Rtnmk〉 leMrs

tnm minus ak (τ

tn minusxt


〈DVt

nmk〉minus 〈DUt

nmk〉+ 〈D(m)

Rt


tn0 minus τ t


minusU tnmk +V t



minusUt

nmk +Vt

nmk = akXtn minusR

t


983043Rt

nmk


983043Rt

nmk


983059R

t

nmk

983060


983059R

t

nmk

983060


U tnmkV

tnmkU

t

nmkVt


rtn ge 0stn isinRMs









tij le yt

j

and983123

nisin[N ] qtn

983059D

tminus1983060le yt










2015)






efficiently




min gprimeφ



(y0Y Z)isinZR

(15)







(16)



as


iisin[I]


i Z〉)ψi

st983131

iisin[I]

aiψi = g

(17)









min η

st983131

iisin[I]



983131

iisin[I]



(y0Y Z)isinZR

(18)

If G1 = ψ|983123



to (18)




ψ|983123



lowastZlowast)














































as follows

P =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059dle Dnt le d

983060= 1

EP

983059Dnt

983060= micro

EP


EP

983091

983107

983055983055983055983055983055983055

983131

tisin[T ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


EP

983091

983107

983055983055983055983055983055983055

983131

nisin[N ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102


= 20 The




















is to minimize 1NT

983123nisin[N ]





min1

McNT

983131

misin[Mc]

983131

tisin[T ]

983131

nisin[N ]

I(m)nt

st I(m)nt ge x

(m)nt minus τ


I(m)nt ge minusx

(m)nt





983043yt0 y

tn z


983044isinZR




we set Mc = 100












timerdquo






over four periods


30 911 042 185 265 1044 9235 553 027 148 064 849 86

08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62


















(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105

08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155

SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243

35 124 070 068 145 062 112808 27 114 107 099 110 096 752

30 127 092 084 129 078 53735 125 074 071 na 064 395

MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372

08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613


52 Robustness




















(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110

08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141

SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083

35 162 100 086 273 08008 27 137 124 104 125 098

30 161 117 093 168 08535 177 113 090 NA 085

MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140

08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139







6 Conclusions














micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi




micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0



(xi)αi gt


micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0



S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070



micro

983061vτiτi


αλi

983062le micro

983061λvτiτi


αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi


983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376


























ρτ τ (x) = inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτ iτ i(xi)

αi


983070 (2)
















microu(x) = infη

983069η

983055983055983055983055supPisinP


983070 (4)









ρτ τ (x) = inf

983099983103

983101983131

iisin[I]

αi

983055983055983055983055983055983055supPisinP

EP

983061u

983061vτ iτ i(xi)

αi


983100983104

983102 (5)







ρτ τ (x) = inf

983099983103

983101983131

iisin[I]

αi

983055983055983055983055983055983055Cαi


983100983104

983102

where

Cαi(xi) = αi ln

983061supPisinP

EP

983061exp

983061vτ iτ i(xi)

αi

983062983062983062











be simplified as

ρτ τ (x) = inf

983099983105983105983103

983105983105983101

983131

iisin[I]

αi

983055983055983055983055983055983055983055983055

supPisinP

EP

983061maxkisin[K]


983052983062le 0


983100983105983105983104

983105983105983102



inf983131

tisin[T ]

983131

nisin[N ]

αtn +

983131

tisin[T ]

βt

st supPisinP

EP

983061maxkisin[K]

983153akmax


n

983059D

t983060 xt

n

983059D

t983060minus τ t

n

983154+ bkα

tn


supPisinP

EP

983091

983107maxkisin[K]

983099983103

983101ak

983091

983107983131

(ij)isinA

cijztij

983059D


983092

983108+ bkβt

983100983104

983102

983092


xtn

983059D

t983060=

t983131

m=1

(qmn

983059D



(qtn yt0 y

tn z


(6)





















Routing decisions



Decisionmaking


Order quantities

Actual demand





qtn

983059D




n




realization



xtn

983059D

t983060=

t983131

m=1

983059qmn

983059D


983060=

t983131

m=1

983059qmn0 + 〈Qm

n D〉minus Dnm

983060= xt

n0 + 〈XtnD〉 (7)

here xtn0 =

t983123m=1

qmn0 and Xtn =

t983123m=1

(Qmn minus ene

primem)


tion cost983123


(6) supPisinP EP

983059u983059983059983123


983060βt






inf983131

tisin[T ]

983131

nisin[N ]

αtn

st supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt



n

983154+ bkα

tn





xtn0 =

t983131

m=1


Xtn =

t983131

m=1

(Qmn minus ene


983043Qt

n



(yt0 y

tn z


(8)


ZR =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

yt0 y

tn z

tij isin 01


(i j)isinA

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

983131

(ij)isinA


ytn le yt


983131

j(nj)isinA


983131

j(jn)isinA


983131

(ij)isinA(S)

ztij le983131

nisinS

ytn minus yt


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102

(9)


n ge 983171












misin[Ms]pm =

1 In this case

P =

983099983103

983101P

983055983055983055983055983055983055P983059D=D(m)


983131

misin[Ms]

pm = 1

983100983104

983102 (10)


supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

=983131

misin[Ms]

pm

983061maxkisin[K]

983153akmax

983153τ tn minusxt

n0 minus〈XtnD

(m)〉 xtn0 + 〈Xt

nD(m)〉minus τ t

n

983154+ bkα

tn

983154983062




983131

misin[Ms]


νmnt ge ak

983059τ tn minusxt

n0 minus〈XtnD

(m)〉983060+ bkα


νmnt ge ak

983059xtn0 + 〈Xt

nD(m)〉minus τ t

n

983060+ bkα


qtn0 + 〈QtnD


qtn0 + 〈QtnD











P =

983099983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059Dle DleD

983060= 1 (a)

EP

983059D983060=Ξ (b)

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ (c)

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094


983100983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983102

(11)















983075983055983055983055983055983055983123

iisin[N ]

DiτminusΞiτΣiτ

983055983055983055983055983055

983076le



983075983055983055983055983055983055983123

τisin[T ]

DiτminusΞiτΣiτ

983055983055983055983055983055







constraint (11d)






nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh le 0

〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime





nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]



T tn ge 0

Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime w







n〉 le 0

Qtn +Lt

n minusH tn = 0

LtnH

tn ge 0




MILP









Q=

983099983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983101

Q

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

Q983059983059

D U V983060isin 983142W

983060= 1

EQ

983059D983060=Ξ

EQ

983059U983060leΣ


983100983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983102

(12)



983142W =

983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101

983059D U V


983055983055983055983055983055983055983055983055983055983055983055983055983055


983055983055983055le U

fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092


983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102






qtn

983059D U V

983060= qtn0 + 〈Qt






0n

983044land

983043Qt

1n


983043Qt

2n


as

supQisinQ

EQ

983061maxkisin[K]

983153akmax



2n V 〉



n

983154+ bkα

tn

983154983060le 0

(13)



















Pdagger983059D

dagger= D

(m)983060=

1

Mr

forallmisin [Mr]



983059983056983056983056Dminus Ddagger983056983056983056983060

st (DDdagger)sim P

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1







983052 (14)





(8a)




rtnθ+983131

misin[Mr ]

stnm le 0


nmk〉+ 〈D(m)

Rtnmk〉 leMrs

tnm minus ak (τ

tn minusxt


〈DVt

nmk〉minus 〈DUt

nmk〉+ 〈D(m)

Rt


tn0 minus τ t


minusU tnmk +V t



minusUt

nmk +Vt

nmk = akXtn minusR

t


983043Rt

nmk


983043Rt

nmk


983059R

t

nmk

983060


983059R

t

nmk

983060


U tnmkV

tnmkU

t

nmkVt


rtn ge 0stn isinRMs









tij le yt

j

and983123

nisin[N ] qtn

983059D

tminus1983060le yt










2015)






efficiently




min gprimeφ



(y0Y Z)isinZR

(15)







(16)



as


iisin[I]


i Z〉)ψi

st983131

iisin[I]

aiψi = g

(17)









min η

st983131

iisin[I]



983131

iisin[I]



(y0Y Z)isinZR

(18)

If G1 = ψ|983123



to (18)




ψ|983123



lowastZlowast)














































as follows

P =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059dle Dnt le d

983060= 1

EP

983059Dnt

983060= micro

EP


EP

983091

983107

983055983055983055983055983055983055

983131

tisin[T ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


EP

983091

983107

983055983055983055983055983055983055

983131

nisin[N ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102


= 20 The




















is to minimize 1NT

983123nisin[N ]





min1

McNT

983131

misin[Mc]

983131

tisin[T ]

983131

nisin[N ]

I(m)nt

st I(m)nt ge x

(m)nt minus τ


I(m)nt ge minusx

(m)nt





983043yt0 y

tn z


983044isinZR




we set Mc = 100












timerdquo






over four periods


30 911 042 185 265 1044 9235 553 027 148 064 849 86

08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62


















(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105

08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155

SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243

35 124 070 068 145 062 112808 27 114 107 099 110 096 752

30 127 092 084 129 078 53735 125 074 071 na 064 395

MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372

08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613


52 Robustness




















(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110

08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141

SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083

35 162 100 086 273 08008 27 137 124 104 125 098

30 161 117 093 168 08535 177 113 090 NA 085

MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140

08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139







6 Conclusions














micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi




micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0



(xi)αi gt


micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0



S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070



micro

983061vτiτi


αλi

983062le micro

983061λvτiτi


αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi


983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376














microu(x) = infη

983069η

983055983055983055983055supPisinP


983070 (4)









ρτ τ (x) = inf

983099983103

983101983131

iisin[I]

αi

983055983055983055983055983055983055supPisinP

EP

983061u

983061vτ iτ i(xi)

αi


983100983104

983102 (5)







ρτ τ (x) = inf

983099983103

983101983131

iisin[I]

αi

983055983055983055983055983055983055Cαi


983100983104

983102

where

Cαi(xi) = αi ln

983061supPisinP

EP

983061exp

983061vτ iτ i(xi)

αi

983062983062983062











be simplified as

ρτ τ (x) = inf

983099983105983105983103

983105983105983101

983131

iisin[I]

αi

983055983055983055983055983055983055983055983055

supPisinP

EP

983061maxkisin[K]


983052983062le 0


983100983105983105983104

983105983105983102



inf983131

tisin[T ]

983131

nisin[N ]

αtn +

983131

tisin[T ]

βt

st supPisinP

EP

983061maxkisin[K]

983153akmax


n

983059D

t983060 xt

n

983059D

t983060minus τ t

n

983154+ bkα

tn


supPisinP

EP

983091

983107maxkisin[K]

983099983103

983101ak

983091

983107983131

(ij)isinA

cijztij

983059D


983092

983108+ bkβt

983100983104

983102

983092


xtn

983059D

t983060=

t983131

m=1

(qmn

983059D



(qtn yt0 y

tn z


(6)





















Routing decisions



Decisionmaking


Order quantities

Actual demand





qtn

983059D




n




realization



xtn

983059D

t983060=

t983131

m=1

983059qmn

983059D


983060=

t983131

m=1

983059qmn0 + 〈Qm

n D〉minus Dnm

983060= xt

n0 + 〈XtnD〉 (7)

here xtn0 =

t983123m=1

qmn0 and Xtn =

t983123m=1

(Qmn minus ene

primem)


tion cost983123


(6) supPisinP EP

983059u983059983059983123


983060βt






inf983131

tisin[T ]

983131

nisin[N ]

αtn

st supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt



n

983154+ bkα

tn





xtn0 =

t983131

m=1


Xtn =

t983131

m=1

(Qmn minus ene


983043Qt

n



(yt0 y

tn z


(8)


ZR =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

yt0 y

tn z

tij isin 01


(i j)isinA

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

983131

(ij)isinA


ytn le yt


983131

j(nj)isinA


983131

j(jn)isinA


983131

(ij)isinA(S)

ztij le983131

nisinS

ytn minus yt


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102

(9)


n ge 983171












misin[Ms]pm =

1 In this case

P =

983099983103

983101P

983055983055983055983055983055983055P983059D=D(m)


983131

misin[Ms]

pm = 1

983100983104

983102 (10)


supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

=983131

misin[Ms]

pm

983061maxkisin[K]

983153akmax

983153τ tn minusxt

n0 minus〈XtnD

(m)〉 xtn0 + 〈Xt

nD(m)〉minus τ t

n

983154+ bkα

tn

983154983062




983131

misin[Ms]


νmnt ge ak

983059τ tn minusxt

n0 minus〈XtnD

(m)〉983060+ bkα


νmnt ge ak

983059xtn0 + 〈Xt

nD(m)〉minus τ t

n

983060+ bkα


qtn0 + 〈QtnD


qtn0 + 〈QtnD











P =

983099983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059Dle DleD

983060= 1 (a)

EP

983059D983060=Ξ (b)

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ (c)

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094


983100983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983102

(11)















983075983055983055983055983055983055983123

iisin[N ]

DiτminusΞiτΣiτ

983055983055983055983055983055

983076le



983075983055983055983055983055983055983123

τisin[T ]

DiτminusΞiτΣiτ

983055983055983055983055983055







constraint (11d)






nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh le 0

〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime





nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]



T tn ge 0

Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime w







n〉 le 0

Qtn +Lt

n minusH tn = 0

LtnH

tn ge 0




MILP









Q=

983099983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983101

Q

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

Q983059983059

D U V983060isin 983142W

983060= 1

EQ

983059D983060=Ξ

EQ

983059U983060leΣ


983100983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983102

(12)



983142W =

983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101

983059D U V


983055983055983055983055983055983055983055983055983055983055983055983055983055


983055983055983055le U

fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092


983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102






qtn

983059D U V

983060= qtn0 + 〈Qt






0n

983044land

983043Qt

1n


983043Qt

2n


as

supQisinQ

EQ

983061maxkisin[K]

983153akmax



2n V 〉



n

983154+ bkα

tn

983154983060le 0

(13)



















Pdagger983059D

dagger= D

(m)983060=

1

Mr

forallmisin [Mr]



983059983056983056983056Dminus Ddagger983056983056983056983060

st (DDdagger)sim P

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1







983052 (14)





(8a)




rtnθ+983131

misin[Mr ]

stnm le 0


nmk〉+ 〈D(m)

Rtnmk〉 leMrs

tnm minus ak (τ

tn minusxt


〈DVt

nmk〉minus 〈DUt

nmk〉+ 〈D(m)

Rt


tn0 minus τ t


minusU tnmk +V t



minusUt

nmk +Vt

nmk = akXtn minusR

t


983043Rt

nmk


983043Rt

nmk


983059R

t

nmk

983060


983059R

t

nmk

983060


U tnmkV

tnmkU

t

nmkVt


rtn ge 0stn isinRMs









tij le yt

j

and983123

nisin[N ] qtn

983059D

tminus1983060le yt










2015)






efficiently




min gprimeφ



(y0Y Z)isinZR

(15)







(16)



as


iisin[I]


i Z〉)ψi

st983131

iisin[I]

aiψi = g

(17)









min η

st983131

iisin[I]



983131

iisin[I]



(y0Y Z)isinZR

(18)

If G1 = ψ|983123



to (18)




ψ|983123



lowastZlowast)














































as follows

P =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059dle Dnt le d

983060= 1

EP

983059Dnt

983060= micro

EP


EP

983091

983107

983055983055983055983055983055983055

983131

tisin[T ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


EP

983091

983107

983055983055983055983055983055983055

983131

nisin[N ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102


= 20 The




















is to minimize 1NT

983123nisin[N ]





min1

McNT

983131

misin[Mc]

983131

tisin[T ]

983131

nisin[N ]

I(m)nt

st I(m)nt ge x

(m)nt minus τ


I(m)nt ge minusx

(m)nt





983043yt0 y

tn z


983044isinZR




we set Mc = 100












timerdquo






over four periods


30 911 042 185 265 1044 9235 553 027 148 064 849 86

08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62


















(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105

08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155

SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243

35 124 070 068 145 062 112808 27 114 107 099 110 096 752

30 127 092 084 129 078 53735 125 074 071 na 064 395

MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372

08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613


52 Robustness




















(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110

08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141

SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083

35 162 100 086 273 08008 27 137 124 104 125 098

30 161 117 093 168 08535 177 113 090 NA 085

MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140

08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139







6 Conclusions














micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi




micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0



(xi)αi gt


micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0



S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070



micro

983061vτiτi


αλi

983062le micro

983061λvτiτi


αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi


983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376







ρτ τ (x) = inf

983099983103

983101983131

iisin[I]

αi

983055983055983055983055983055983055Cαi


983100983104

983102

where

Cαi(xi) = αi ln

983061supPisinP

EP

983061exp

983061vτ iτ i(xi)

αi

983062983062983062











be simplified as

ρτ τ (x) = inf

983099983105983105983103

983105983105983101

983131

iisin[I]

αi

983055983055983055983055983055983055983055983055

supPisinP

EP

983061maxkisin[K]


983052983062le 0


983100983105983105983104

983105983105983102



inf983131

tisin[T ]

983131

nisin[N ]

αtn +

983131

tisin[T ]

βt

st supPisinP

EP

983061maxkisin[K]

983153akmax


n

983059D

t983060 xt

n

983059D

t983060minus τ t

n

983154+ bkα

tn


supPisinP

EP

983091

983107maxkisin[K]

983099983103

983101ak

983091

983107983131

(ij)isinA

cijztij

983059D


983092

983108+ bkβt

983100983104

983102

983092


xtn

983059D

t983060=

t983131

m=1

(qmn

983059D



(qtn yt0 y

tn z


(6)





















Routing decisions



Decisionmaking


Order quantities

Actual demand





qtn

983059D




n




realization



xtn

983059D

t983060=

t983131

m=1

983059qmn

983059D


983060=

t983131

m=1

983059qmn0 + 〈Qm

n D〉minus Dnm

983060= xt

n0 + 〈XtnD〉 (7)

here xtn0 =

t983123m=1

qmn0 and Xtn =

t983123m=1

(Qmn minus ene

primem)


tion cost983123


(6) supPisinP EP

983059u983059983059983123


983060βt






inf983131

tisin[T ]

983131

nisin[N ]

αtn

st supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt



n

983154+ bkα

tn





xtn0 =

t983131

m=1


Xtn =

t983131

m=1

(Qmn minus ene


983043Qt

n



(yt0 y

tn z


(8)


ZR =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

yt0 y

tn z

tij isin 01


(i j)isinA

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

983131

(ij)isinA


ytn le yt


983131

j(nj)isinA


983131

j(jn)isinA


983131

(ij)isinA(S)

ztij le983131

nisinS

ytn minus yt


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102

(9)


n ge 983171












misin[Ms]pm =

1 In this case

P =

983099983103

983101P

983055983055983055983055983055983055P983059D=D(m)


983131

misin[Ms]

pm = 1

983100983104

983102 (10)


supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

=983131

misin[Ms]

pm

983061maxkisin[K]

983153akmax

983153τ tn minusxt

n0 minus〈XtnD

(m)〉 xtn0 + 〈Xt

nD(m)〉minus τ t

n

983154+ bkα

tn

983154983062




983131

misin[Ms]


νmnt ge ak

983059τ tn minusxt

n0 minus〈XtnD

(m)〉983060+ bkα


νmnt ge ak

983059xtn0 + 〈Xt

nD(m)〉minus τ t

n

983060+ bkα


qtn0 + 〈QtnD


qtn0 + 〈QtnD











P =

983099983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059Dle DleD

983060= 1 (a)

EP

983059D983060=Ξ (b)

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ (c)

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094


983100983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983102

(11)















983075983055983055983055983055983055983123

iisin[N ]

DiτminusΞiτΣiτ

983055983055983055983055983055

983076le



983075983055983055983055983055983055983123

τisin[T ]

DiτminusΞiτΣiτ

983055983055983055983055983055







constraint (11d)






nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh le 0

〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime





nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]



T tn ge 0

Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime w







n〉 le 0

Qtn +Lt

n minusH tn = 0

LtnH

tn ge 0




MILP









Q=

983099983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983101

Q

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

Q983059983059

D U V983060isin 983142W

983060= 1

EQ

983059D983060=Ξ

EQ

983059U983060leΣ


983100983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983102

(12)



983142W =

983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101

983059D U V


983055983055983055983055983055983055983055983055983055983055983055983055983055


983055983055983055le U

fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092


983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102






qtn

983059D U V

983060= qtn0 + 〈Qt






0n

983044land

983043Qt

1n


983043Qt

2n


as

supQisinQ

EQ

983061maxkisin[K]

983153akmax



2n V 〉



n

983154+ bkα

tn

983154983060le 0

(13)



















Pdagger983059D

dagger= D

(m)983060=

1

Mr

forallmisin [Mr]



983059983056983056983056Dminus Ddagger983056983056983056983060

st (DDdagger)sim P

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1







983052 (14)





(8a)




rtnθ+983131

misin[Mr ]

stnm le 0


nmk〉+ 〈D(m)

Rtnmk〉 leMrs

tnm minus ak (τ

tn minusxt


〈DVt

nmk〉minus 〈DUt

nmk〉+ 〈D(m)

Rt


tn0 minus τ t


minusU tnmk +V t



minusUt

nmk +Vt

nmk = akXtn minusR

t


983043Rt

nmk


983043Rt

nmk


983059R

t

nmk

983060


983059R

t

nmk

983060


U tnmkV

tnmkU

t

nmkVt


rtn ge 0stn isinRMs









tij le yt

j

and983123

nisin[N ] qtn

983059D

tminus1983060le yt










2015)






efficiently




min gprimeφ



(y0Y Z)isinZR

(15)







(16)



as


iisin[I]


i Z〉)ψi

st983131

iisin[I]

aiψi = g

(17)









min η

st983131

iisin[I]



983131

iisin[I]



(y0Y Z)isinZR

(18)

If G1 = ψ|983123



to (18)




ψ|983123



lowastZlowast)














































as follows

P =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059dle Dnt le d

983060= 1

EP

983059Dnt

983060= micro

EP


EP

983091

983107

983055983055983055983055983055983055

983131

tisin[T ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


EP

983091

983107

983055983055983055983055983055983055

983131

nisin[N ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102


= 20 The




















is to minimize 1NT

983123nisin[N ]





min1

McNT

983131

misin[Mc]

983131

tisin[T ]

983131

nisin[N ]

I(m)nt

st I(m)nt ge x

(m)nt minus τ


I(m)nt ge minusx

(m)nt





983043yt0 y

tn z


983044isinZR




we set Mc = 100












timerdquo






over four periods


30 911 042 185 265 1044 9235 553 027 148 064 849 86

08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62


















(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105

08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155

SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243

35 124 070 068 145 062 112808 27 114 107 099 110 096 752

30 127 092 084 129 078 53735 125 074 071 na 064 395

MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372

08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613


52 Robustness




















(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110

08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141

SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083

35 162 100 086 273 08008 27 137 124 104 125 098

30 161 117 093 168 08535 177 113 090 NA 085

MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140

08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139







6 Conclusions














micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi




micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0



(xi)αi gt


micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0



S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070



micro

983061vτiτi


αλi

983062le micro

983061λvτiτi


αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi


983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376




inf983131

tisin[T ]

983131

nisin[N ]

αtn +

983131

tisin[T ]

βt

st supPisinP

EP

983061maxkisin[K]

983153akmax


n

983059D

t983060 xt

n

983059D

t983060minus τ t

n

983154+ bkα

tn


supPisinP

EP

983091

983107maxkisin[K]

983099983103

983101ak

983091

983107983131

(ij)isinA

cijztij

983059D


983092

983108+ bkβt

983100983104

983102

983092


xtn

983059D

t983060=

t983131

m=1

(qmn

983059D



(qtn yt0 y

tn z


(6)





















Routing decisions



Decisionmaking


Order quantities

Actual demand





qtn

983059D




n




realization



xtn

983059D

t983060=

t983131

m=1

983059qmn

983059D


983060=

t983131

m=1

983059qmn0 + 〈Qm

n D〉minus Dnm

983060= xt

n0 + 〈XtnD〉 (7)

here xtn0 =

t983123m=1

qmn0 and Xtn =

t983123m=1

(Qmn minus ene

primem)


tion cost983123


(6) supPisinP EP

983059u983059983059983123


983060βt






inf983131

tisin[T ]

983131

nisin[N ]

αtn

st supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt



n

983154+ bkα

tn





xtn0 =

t983131

m=1


Xtn =

t983131

m=1

(Qmn minus ene


983043Qt

n



(yt0 y

tn z


(8)


ZR =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

yt0 y

tn z

tij isin 01


(i j)isinA

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

983131

(ij)isinA


ytn le yt


983131

j(nj)isinA


983131

j(jn)isinA


983131

(ij)isinA(S)

ztij le983131

nisinS

ytn minus yt


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102

(9)


n ge 983171












misin[Ms]pm =

1 In this case

P =

983099983103

983101P

983055983055983055983055983055983055P983059D=D(m)


983131

misin[Ms]

pm = 1

983100983104

983102 (10)


supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

=983131

misin[Ms]

pm

983061maxkisin[K]

983153akmax

983153τ tn minusxt

n0 minus〈XtnD

(m)〉 xtn0 + 〈Xt

nD(m)〉minus τ t

n

983154+ bkα

tn

983154983062




983131

misin[Ms]


νmnt ge ak

983059τ tn minusxt

n0 minus〈XtnD

(m)〉983060+ bkα


νmnt ge ak

983059xtn0 + 〈Xt

nD(m)〉minus τ t

n

983060+ bkα


qtn0 + 〈QtnD


qtn0 + 〈QtnD











P =

983099983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059Dle DleD

983060= 1 (a)

EP

983059D983060=Ξ (b)

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ (c)

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094


983100983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983102

(11)















983075983055983055983055983055983055983123

iisin[N ]

DiτminusΞiτΣiτ

983055983055983055983055983055

983076le



983075983055983055983055983055983055983123

τisin[T ]

DiτminusΞiτΣiτ

983055983055983055983055983055







constraint (11d)






nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh le 0

〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime





nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]



T tn ge 0

Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime w







n〉 le 0

Qtn +Lt

n minusH tn = 0

LtnH

tn ge 0




MILP









Q=

983099983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983101

Q

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

Q983059983059

D U V983060isin 983142W

983060= 1

EQ

983059D983060=Ξ

EQ

983059U983060leΣ


983100983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983102

(12)



983142W =

983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101

983059D U V


983055983055983055983055983055983055983055983055983055983055983055983055983055


983055983055983055le U

fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092


983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102






qtn

983059D U V

983060= qtn0 + 〈Qt






0n

983044land

983043Qt

1n


983043Qt

2n


as

supQisinQ

EQ

983061maxkisin[K]

983153akmax



2n V 〉



n

983154+ bkα

tn

983154983060le 0

(13)



















Pdagger983059D

dagger= D

(m)983060=

1

Mr

forallmisin [Mr]



983059983056983056983056Dminus Ddagger983056983056983056983060

st (DDdagger)sim P

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1







983052 (14)





(8a)




rtnθ+983131

misin[Mr ]

stnm le 0


nmk〉+ 〈D(m)

Rtnmk〉 leMrs

tnm minus ak (τ

tn minusxt


〈DVt

nmk〉minus 〈DUt

nmk〉+ 〈D(m)

Rt


tn0 minus τ t


minusU tnmk +V t



minusUt

nmk +Vt

nmk = akXtn minusR

t


983043Rt

nmk


983043Rt

nmk


983059R

t

nmk

983060


983059R

t

nmk

983060


U tnmkV

tnmkU

t

nmkVt


rtn ge 0stn isinRMs









tij le yt

j

and983123

nisin[N ] qtn

983059D

tminus1983060le yt










2015)






efficiently




min gprimeφ



(y0Y Z)isinZR

(15)







(16)



as


iisin[I]


i Z〉)ψi

st983131

iisin[I]

aiψi = g

(17)









min η

st983131

iisin[I]



983131

iisin[I]



(y0Y Z)isinZR

(18)

If G1 = ψ|983123



to (18)




ψ|983123



lowastZlowast)














































as follows

P =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059dle Dnt le d

983060= 1

EP

983059Dnt

983060= micro

EP


EP

983091

983107

983055983055983055983055983055983055

983131

tisin[T ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


EP

983091

983107

983055983055983055983055983055983055

983131

nisin[N ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102


= 20 The




















is to minimize 1NT

983123nisin[N ]





min1

McNT

983131

misin[Mc]

983131

tisin[T ]

983131

nisin[N ]

I(m)nt

st I(m)nt ge x

(m)nt minus τ


I(m)nt ge minusx

(m)nt





983043yt0 y

tn z


983044isinZR




we set Mc = 100












timerdquo






over four periods


30 911 042 185 265 1044 9235 553 027 148 064 849 86

08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62


















(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105

08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155

SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243

35 124 070 068 145 062 112808 27 114 107 099 110 096 752

30 127 092 084 129 078 53735 125 074 071 na 064 395

MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372

08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613


52 Robustness




















(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110

08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141

SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083

35 162 100 086 273 08008 27 137 124 104 125 098

30 161 117 093 168 08535 177 113 090 NA 085

MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140

08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139







6 Conclusions














micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi




micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0



(xi)αi gt


micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0



S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070



micro

983061vτiτi


αλi

983062le micro

983061λvτiτi


αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi


983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376





Routing decisions



Decisionmaking


Order quantities

Actual demand





qtn

983059D




n




realization



xtn

983059D

t983060=

t983131

m=1

983059qmn

983059D


983060=

t983131

m=1

983059qmn0 + 〈Qm

n D〉minus Dnm

983060= xt

n0 + 〈XtnD〉 (7)

here xtn0 =

t983123m=1

qmn0 and Xtn =

t983123m=1

(Qmn minus ene

primem)


tion cost983123


(6) supPisinP EP

983059u983059983059983123


983060βt






inf983131

tisin[T ]

983131

nisin[N ]

αtn

st supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt



n

983154+ bkα

tn





xtn0 =

t983131

m=1


Xtn =

t983131

m=1

(Qmn minus ene


983043Qt

n



(yt0 y

tn z


(8)


ZR =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

yt0 y

tn z

tij isin 01


(i j)isinA

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

983131

(ij)isinA


ytn le yt


983131

j(nj)isinA


983131

j(jn)isinA


983131

(ij)isinA(S)

ztij le983131

nisinS

ytn minus yt


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102

(9)


n ge 983171












misin[Ms]pm =

1 In this case

P =

983099983103

983101P

983055983055983055983055983055983055P983059D=D(m)


983131

misin[Ms]

pm = 1

983100983104

983102 (10)


supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

=983131

misin[Ms]

pm

983061maxkisin[K]

983153akmax

983153τ tn minusxt

n0 minus〈XtnD

(m)〉 xtn0 + 〈Xt

nD(m)〉minus τ t

n

983154+ bkα

tn

983154983062




983131

misin[Ms]


νmnt ge ak

983059τ tn minusxt

n0 minus〈XtnD

(m)〉983060+ bkα


νmnt ge ak

983059xtn0 + 〈Xt

nD(m)〉minus τ t

n

983060+ bkα


qtn0 + 〈QtnD


qtn0 + 〈QtnD











P =

983099983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059Dle DleD

983060= 1 (a)

EP

983059D983060=Ξ (b)

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ (c)

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094


983100983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983102

(11)















983075983055983055983055983055983055983123

iisin[N ]

DiτminusΞiτΣiτ

983055983055983055983055983055

983076le



983075983055983055983055983055983055983123

τisin[T ]

DiτminusΞiτΣiτ

983055983055983055983055983055







constraint (11d)






nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh le 0

〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime





nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]



T tn ge 0

Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime w







n〉 le 0

Qtn +Lt

n minusH tn = 0

LtnH

tn ge 0




MILP









Q=

983099983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983101

Q

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

Q983059983059

D U V983060isin 983142W

983060= 1

EQ

983059D983060=Ξ

EQ

983059U983060leΣ


983100983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983102

(12)



983142W =

983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101

983059D U V


983055983055983055983055983055983055983055983055983055983055983055983055983055


983055983055983055le U

fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092


983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102






qtn

983059D U V

983060= qtn0 + 〈Qt






0n

983044land

983043Qt

1n


983043Qt

2n


as

supQisinQ

EQ

983061maxkisin[K]

983153akmax



2n V 〉



n

983154+ bkα

tn

983154983060le 0

(13)



















Pdagger983059D

dagger= D

(m)983060=

1

Mr

forallmisin [Mr]



983059983056983056983056Dminus Ddagger983056983056983056983060

st (DDdagger)sim P

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1







983052 (14)





(8a)




rtnθ+983131

misin[Mr ]

stnm le 0


nmk〉+ 〈D(m)

Rtnmk〉 leMrs

tnm minus ak (τ

tn minusxt


〈DVt

nmk〉minus 〈DUt

nmk〉+ 〈D(m)

Rt


tn0 minus τ t


minusU tnmk +V t



minusUt

nmk +Vt

nmk = akXtn minusR

t


983043Rt

nmk


983043Rt

nmk


983059R

t

nmk

983060


983059R

t

nmk

983060


U tnmkV

tnmkU

t

nmkVt


rtn ge 0stn isinRMs









tij le yt

j

and983123

nisin[N ] qtn

983059D

tminus1983060le yt










2015)






efficiently




min gprimeφ



(y0Y Z)isinZR

(15)







(16)



as


iisin[I]


i Z〉)ψi

st983131

iisin[I]

aiψi = g

(17)









min η

st983131

iisin[I]



983131

iisin[I]



(y0Y Z)isinZR

(18)

If G1 = ψ|983123



to (18)




ψ|983123



lowastZlowast)














































as follows

P =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059dle Dnt le d

983060= 1

EP

983059Dnt

983060= micro

EP


EP

983091

983107

983055983055983055983055983055983055

983131

tisin[T ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


EP

983091

983107

983055983055983055983055983055983055

983131

nisin[N ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102


= 20 The




















is to minimize 1NT

983123nisin[N ]





min1

McNT

983131

misin[Mc]

983131

tisin[T ]

983131

nisin[N ]

I(m)nt

st I(m)nt ge x

(m)nt minus τ


I(m)nt ge minusx

(m)nt





983043yt0 y

tn z


983044isinZR




we set Mc = 100












timerdquo






over four periods


30 911 042 185 265 1044 9235 553 027 148 064 849 86

08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62


















(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105

08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155

SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243

35 124 070 068 145 062 112808 27 114 107 099 110 096 752

30 127 092 084 129 078 53735 125 074 071 na 064 395

MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372

08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613


52 Robustness




















(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110

08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141

SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083

35 162 100 086 273 08008 27 137 124 104 125 098

30 161 117 093 168 08535 177 113 090 NA 085

MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140

08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139







6 Conclusions














micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi




micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0



(xi)αi gt


micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0



S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070



micro

983061vτiτi


αλi

983062le micro

983061λvτiτi


αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi


983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376




inf983131

tisin[T ]

983131

nisin[N ]

αtn

st supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt



n

983154+ bkα

tn





xtn0 =

t983131

m=1


Xtn =

t983131

m=1

(Qmn minus ene


983043Qt

n



(yt0 y

tn z


(8)


ZR =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

yt0 y

tn z

tij isin 01


(i j)isinA

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

983131

(ij)isinA


ytn le yt


983131

j(nj)isinA


983131

j(jn)isinA


983131

(ij)isinA(S)

ztij le983131

nisinS

ytn minus yt


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102

(9)


n ge 983171












misin[Ms]pm =

1 In this case

P =

983099983103

983101P

983055983055983055983055983055983055P983059D=D(m)


983131

misin[Ms]

pm = 1

983100983104

983102 (10)


supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

=983131

misin[Ms]

pm

983061maxkisin[K]

983153akmax

983153τ tn minusxt

n0 minus〈XtnD

(m)〉 xtn0 + 〈Xt

nD(m)〉minus τ t

n

983154+ bkα

tn

983154983062




983131

misin[Ms]


νmnt ge ak

983059τ tn minusxt

n0 minus〈XtnD

(m)〉983060+ bkα


νmnt ge ak

983059xtn0 + 〈Xt

nD(m)〉minus τ t

n

983060+ bkα


qtn0 + 〈QtnD


qtn0 + 〈QtnD











P =

983099983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059Dle DleD

983060= 1 (a)

EP

983059D983060=Ξ (b)

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ (c)

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094


983100983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983102

(11)















983075983055983055983055983055983055983123

iisin[N ]

DiτminusΞiτΣiτ

983055983055983055983055983055

983076le



983075983055983055983055983055983055983123

τisin[T ]

DiτminusΞiτΣiτ

983055983055983055983055983055







constraint (11d)






nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh le 0

〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime





nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]



T tn ge 0

Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime w







n〉 le 0

Qtn +Lt

n minusH tn = 0

LtnH

tn ge 0




MILP









Q=

983099983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983101

Q

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

Q983059983059

D U V983060isin 983142W

983060= 1

EQ

983059D983060=Ξ

EQ

983059U983060leΣ


983100983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983102

(12)



983142W =

983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101

983059D U V


983055983055983055983055983055983055983055983055983055983055983055983055983055


983055983055983055le U

fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092


983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102






qtn

983059D U V

983060= qtn0 + 〈Qt






0n

983044land

983043Qt

1n


983043Qt

2n


as

supQisinQ

EQ

983061maxkisin[K]

983153akmax



2n V 〉



n

983154+ bkα

tn

983154983060le 0

(13)



















Pdagger983059D

dagger= D

(m)983060=

1

Mr

forallmisin [Mr]



983059983056983056983056Dminus Ddagger983056983056983056983060

st (DDdagger)sim P

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1







983052 (14)





(8a)




rtnθ+983131

misin[Mr ]

stnm le 0


nmk〉+ 〈D(m)

Rtnmk〉 leMrs

tnm minus ak (τ

tn minusxt


〈DVt

nmk〉minus 〈DUt

nmk〉+ 〈D(m)

Rt


tn0 minus τ t


minusU tnmk +V t



minusUt

nmk +Vt

nmk = akXtn minusR

t


983043Rt

nmk


983043Rt

nmk


983059R

t

nmk

983060


983059R

t

nmk

983060


U tnmkV

tnmkU

t

nmkVt


rtn ge 0stn isinRMs









tij le yt

j

and983123

nisin[N ] qtn

983059D

tminus1983060le yt










2015)






efficiently




min gprimeφ



(y0Y Z)isinZR

(15)







(16)



as


iisin[I]


i Z〉)ψi

st983131

iisin[I]

aiψi = g

(17)









min η

st983131

iisin[I]



983131

iisin[I]



(y0Y Z)isinZR

(18)

If G1 = ψ|983123



to (18)




ψ|983123



lowastZlowast)














































as follows

P =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059dle Dnt le d

983060= 1

EP

983059Dnt

983060= micro

EP


EP

983091

983107

983055983055983055983055983055983055

983131

tisin[T ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


EP

983091

983107

983055983055983055983055983055983055

983131

nisin[N ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102


= 20 The




















is to minimize 1NT

983123nisin[N ]





min1

McNT

983131

misin[Mc]

983131

tisin[T ]

983131

nisin[N ]

I(m)nt

st I(m)nt ge x

(m)nt minus τ


I(m)nt ge minusx

(m)nt





983043yt0 y

tn z


983044isinZR




we set Mc = 100












timerdquo






over four periods


30 911 042 185 265 1044 9235 553 027 148 064 849 86

08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62


















(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105

08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155

SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243

35 124 070 068 145 062 112808 27 114 107 099 110 096 752

30 127 092 084 129 078 53735 125 074 071 na 064 395

MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372

08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613


52 Robustness




















(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110

08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141

SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083

35 162 100 086 273 08008 27 137 124 104 125 098

30 161 117 093 168 08535 177 113 090 NA 085

MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140

08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139







6 Conclusions














micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi




micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0



(xi)αi gt


micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0



S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070



micro

983061vτiτi


αλi

983062le micro

983061λvτiτi


αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi


983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376









misin[Ms]pm =

1 In this case

P =

983099983103

983101P

983055983055983055983055983055983055P983059D=D(m)


983131

misin[Ms]

pm = 1

983100983104

983102 (10)


supPisinP

EP

983061maxkisin[K]

983153akmax

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

=983131

misin[Ms]

pm

983061maxkisin[K]

983153akmax

983153τ tn minusxt

n0 minus〈XtnD

(m)〉 xtn0 + 〈Xt

nD(m)〉minus τ t

n

983154+ bkα

tn

983154983062




983131

misin[Ms]


νmnt ge ak

983059τ tn minusxt

n0 minus〈XtnD

(m)〉983060+ bkα


νmnt ge ak

983059xtn0 + 〈Xt

nD(m)〉minus τ t

n

983060+ bkα


qtn0 + 〈QtnD


qtn0 + 〈QtnD











P =

983099983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059Dle DleD

983060= 1 (a)

EP

983059D983060=Ξ (b)

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ (c)

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094


983100983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983102

(11)















983075983055983055983055983055983055983123

iisin[N ]

DiτminusΞiτΣiτ

983055983055983055983055983055

983076le



983075983055983055983055983055983055983123

τisin[T ]

DiτminusΞiτΣiτ

983055983055983055983055983055







constraint (11d)






nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh le 0

〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime





nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]



T tn ge 0

Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime w







n〉 le 0

Qtn +Lt

n minusH tn = 0

LtnH

tn ge 0




MILP









Q=

983099983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983101

Q

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

Q983059983059

D U V983060isin 983142W

983060= 1

EQ

983059D983060=Ξ

EQ

983059U983060leΣ


983100983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983102

(12)



983142W =

983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101

983059D U V


983055983055983055983055983055983055983055983055983055983055983055983055983055


983055983055983055le U

fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092


983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102






qtn

983059D U V

983060= qtn0 + 〈Qt






0n

983044land

983043Qt

1n


983043Qt

2n


as

supQisinQ

EQ

983061maxkisin[K]

983153akmax



2n V 〉



n

983154+ bkα

tn

983154983060le 0

(13)



















Pdagger983059D

dagger= D

(m)983060=

1

Mr

forallmisin [Mr]



983059983056983056983056Dminus Ddagger983056983056983056983060

st (DDdagger)sim P

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1







983052 (14)





(8a)




rtnθ+983131

misin[Mr ]

stnm le 0


nmk〉+ 〈D(m)

Rtnmk〉 leMrs

tnm minus ak (τ

tn minusxt


〈DVt

nmk〉minus 〈DUt

nmk〉+ 〈D(m)

Rt


tn0 minus τ t


minusU tnmk +V t



minusUt

nmk +Vt

nmk = akXtn minusR

t


983043Rt

nmk


983043Rt

nmk


983059R

t

nmk

983060


983059R

t

nmk

983060


U tnmkV

tnmkU

t

nmkVt


rtn ge 0stn isinRMs









tij le yt

j

and983123

nisin[N ] qtn

983059D

tminus1983060le yt










2015)






efficiently




min gprimeφ



(y0Y Z)isinZR

(15)







(16)



as


iisin[I]


i Z〉)ψi

st983131

iisin[I]

aiψi = g

(17)









min η

st983131

iisin[I]



983131

iisin[I]



(y0Y Z)isinZR

(18)

If G1 = ψ|983123



to (18)




ψ|983123



lowastZlowast)














































as follows

P =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059dle Dnt le d

983060= 1

EP

983059Dnt

983060= micro

EP


EP

983091

983107

983055983055983055983055983055983055

983131

tisin[T ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


EP

983091

983107

983055983055983055983055983055983055

983131

nisin[N ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102


= 20 The




















is to minimize 1NT

983123nisin[N ]





min1

McNT

983131

misin[Mc]

983131

tisin[T ]

983131

nisin[N ]

I(m)nt

st I(m)nt ge x

(m)nt minus τ


I(m)nt ge minusx

(m)nt





983043yt0 y

tn z


983044isinZR




we set Mc = 100












timerdquo






over four periods


30 911 042 185 265 1044 9235 553 027 148 064 849 86

08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62


















(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105

08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155

SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243

35 124 070 068 145 062 112808 27 114 107 099 110 096 752

30 127 092 084 129 078 53735 125 074 071 na 064 395

MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372

08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613


52 Robustness




















(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110

08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141

SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083

35 162 100 086 273 08008 27 137 124 104 125 098

30 161 117 093 168 08535 177 113 090 NA 085

MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140

08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139







6 Conclusions














micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi




micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0



(xi)αi gt


micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0



S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070



micro

983061vτiτi


αλi

983062le micro

983061λvτiτi


αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi


983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376






P =

983099983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059Dle DleD

983060= 1 (a)

EP

983059D983060=Ξ (b)

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ (c)

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094


983100983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983102

(11)















983075983055983055983055983055983055983123

iisin[N ]

DiτminusΞiτΣiτ

983055983055983055983055983055

983076le



983075983055983055983055983055983055983123

τisin[T ]

DiτminusΞiτΣiτ

983055983055983055983055983055







constraint (11d)






nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh le 0

〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime





nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]



T tn ge 0

Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime w







n〉 le 0

Qtn +Lt

n minusH tn = 0

LtnH

tn ge 0




MILP









Q=

983099983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983101

Q

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

Q983059983059

D U V983060isin 983142W

983060= 1

EQ

983059D983060=Ξ

EQ

983059U983060leΣ


983100983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983102

(12)



983142W =

983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101

983059D U V


983055983055983055983055983055983055983055983055983055983055983055983055983055


983055983055983055le U

fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092


983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102






qtn

983059D U V

983060= qtn0 + 〈Qt






0n

983044land

983043Qt

1n


983043Qt

2n


as

supQisinQ

EQ

983061maxkisin[K]

983153akmax



2n V 〉



n

983154+ bkα

tn

983154983060le 0

(13)



















Pdagger983059D

dagger= D

(m)983060=

1

Mr

forallmisin [Mr]



983059983056983056983056Dminus Ddagger983056983056983056983060

st (DDdagger)sim P

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1







983052 (14)





(8a)




rtnθ+983131

misin[Mr ]

stnm le 0


nmk〉+ 〈D(m)

Rtnmk〉 leMrs

tnm minus ak (τ

tn minusxt


〈DVt

nmk〉minus 〈DUt

nmk〉+ 〈D(m)

Rt


tn0 minus τ t


minusU tnmk +V t



minusUt

nmk +Vt

nmk = akXtn minusR

t


983043Rt

nmk


983043Rt

nmk


983059R

t

nmk

983060


983059R

t

nmk

983060


U tnmkV

tnmkU

t

nmkVt


rtn ge 0stn isinRMs









tij le yt

j

and983123

nisin[N ] qtn

983059D

tminus1983060le yt










2015)






efficiently




min gprimeφ



(y0Y Z)isinZR

(15)







(16)



as


iisin[I]


i Z〉)ψi

st983131

iisin[I]

aiψi = g

(17)









min η

st983131

iisin[I]



983131

iisin[I]



(y0Y Z)isinZR

(18)

If G1 = ψ|983123



to (18)




ψ|983123



lowastZlowast)














































as follows

P =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059dle Dnt le d

983060= 1

EP

983059Dnt

983060= micro

EP


EP

983091

983107

983055983055983055983055983055983055

983131

tisin[T ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


EP

983091

983107

983055983055983055983055983055983055

983131

nisin[N ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102


= 20 The




















is to minimize 1NT

983123nisin[N ]





min1

McNT

983131

misin[Mc]

983131

tisin[T ]

983131

nisin[N ]

I(m)nt

st I(m)nt ge x

(m)nt minus τ


I(m)nt ge minusx

(m)nt





983043yt0 y

tn z


983044isinZR




we set Mc = 100












timerdquo






over four periods


30 911 042 185 265 1044 9235 553 027 148 064 849 86

08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62


















(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105

08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155

SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243

35 124 070 068 145 062 112808 27 114 107 099 110 096 752

30 127 092 084 129 078 53735 125 074 071 na 064 395

MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372

08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613


52 Robustness




















(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110

08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141

SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083

35 162 100 086 273 08008 27 137 124 104 125 098

30 161 117 093 168 08535 177 113 090 NA 085

MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140

08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139







6 Conclusions














micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi




micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0



(xi)αi gt


micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0



S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070



micro

983061vτiτi


αλi

983062le micro

983061λvτiτi


αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi


983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376





constraint (11d)






nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh le 0

〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime





nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]



T tn ge 0

Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime w







n〉 le 0

Qtn +Lt

n minusH tn = 0

LtnH

tn ge 0




MILP









Q=

983099983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983101

Q

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

Q983059983059

D U V983060isin 983142W

983060= 1

EQ

983059D983060=Ξ

EQ

983059U983060leΣ


983100983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983102

(12)



983142W =

983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101

983059D U V


983055983055983055983055983055983055983055983055983055983055983055983055983055


983055983055983055le U

fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092


983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102






qtn

983059D U V

983060= qtn0 + 〈Qt






0n

983044land

983043Qt

1n


983043Qt

2n


as

supQisinQ

EQ

983061maxkisin[K]

983153akmax



2n V 〉



n

983154+ bkα

tn

983154983060le 0

(13)



















Pdagger983059D

dagger= D

(m)983060=

1

Mr

forallmisin [Mr]



983059983056983056983056Dminus Ddagger983056983056983056983060

st (DDdagger)sim P

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1







983052 (14)





(8a)




rtnθ+983131

misin[Mr ]

stnm le 0


nmk〉+ 〈D(m)

Rtnmk〉 leMrs

tnm minus ak (τ

tn minusxt


〈DVt

nmk〉minus 〈DUt

nmk〉+ 〈D(m)

Rt


tn0 minus τ t


minusU tnmk +V t



minusUt

nmk +Vt

nmk = akXtn minusR

t


983043Rt

nmk


983043Rt

nmk


983059R

t

nmk

983060


983059R

t

nmk

983060


U tnmkV

tnmkU

t

nmkVt


rtn ge 0stn isinRMs









tij le yt

j

and983123

nisin[N ] qtn

983059D

tminus1983060le yt










2015)






efficiently




min gprimeφ



(y0Y Z)isinZR

(15)







(16)



as


iisin[I]


i Z〉)ψi

st983131

iisin[I]

aiψi = g

(17)









min η

st983131

iisin[I]



983131

iisin[I]



(y0Y Z)isinZR

(18)

If G1 = ψ|983123



to (18)




ψ|983123



lowastZlowast)














































as follows

P =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059dle Dnt le d

983060= 1

EP

983059Dnt

983060= micro

EP


EP

983091

983107

983055983055983055983055983055983055

983131

tisin[T ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


EP

983091

983107

983055983055983055983055983055983055

983131

nisin[N ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102


= 20 The




















is to minimize 1NT

983123nisin[N ]





min1

McNT

983131

misin[Mc]

983131

tisin[T ]

983131

nisin[N ]

I(m)nt

st I(m)nt ge x

(m)nt minus τ


I(m)nt ge minusx

(m)nt





983043yt0 y

tn z


983044isinZR




we set Mc = 100












timerdquo






over four periods


30 911 042 185 265 1044 9235 553 027 148 064 849 86

08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62


















(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105

08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155

SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243

35 124 070 068 145 062 112808 27 114 107 099 110 096 752

30 127 092 084 129 078 53735 125 074 071 na 064 395

MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372

08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613


52 Robustness




















(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110

08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141

SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083

35 162 100 086 273 08008 27 137 124 104 125 098

30 161 117 093 168 08535 177 113 090 NA 085

MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140

08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139







6 Conclusions














micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi




micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0



(xi)αi gt


micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0



S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070



micro

983061vτiτi


αλi

983062le micro

983061λvτiτi


αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi


983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376








n〉 le 0

Qtn +Lt

n minusH tn = 0

LtnH

tn ge 0




MILP









Q=

983099983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983101

Q

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

Q983059983059

D U V983060isin 983142W

983060= 1

EQ

983059D983060=Ξ

EQ

983059U983060leΣ


983100983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983102

(12)



983142W =

983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101

983059D U V


983055983055983055983055983055983055983055983055983055983055983055983055983055


983055983055983055le U

fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092


983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102






qtn

983059D U V

983060= qtn0 + 〈Qt






0n

983044land

983043Qt

1n


983043Qt

2n


as

supQisinQ

EQ

983061maxkisin[K]

983153akmax



2n V 〉



n

983154+ bkα

tn

983154983060le 0

(13)



















Pdagger983059D

dagger= D

(m)983060=

1

Mr

forallmisin [Mr]



983059983056983056983056Dminus Ddagger983056983056983056983060

st (DDdagger)sim P

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1







983052 (14)





(8a)




rtnθ+983131

misin[Mr ]

stnm le 0


nmk〉+ 〈D(m)

Rtnmk〉 leMrs

tnm minus ak (τ

tn minusxt


〈DVt

nmk〉minus 〈DUt

nmk〉+ 〈D(m)

Rt


tn0 minus τ t


minusU tnmk +V t



minusUt

nmk +Vt

nmk = akXtn minusR

t


983043Rt

nmk


983043Rt

nmk


983059R

t

nmk

983060


983059R

t

nmk

983060


U tnmkV

tnmkU

t

nmkVt


rtn ge 0stn isinRMs









tij le yt

j

and983123

nisin[N ] qtn

983059D

tminus1983060le yt










2015)






efficiently




min gprimeφ



(y0Y Z)isinZR

(15)







(16)



as


iisin[I]


i Z〉)ψi

st983131

iisin[I]

aiψi = g

(17)









min η

st983131

iisin[I]



983131

iisin[I]



(y0Y Z)isinZR

(18)

If G1 = ψ|983123



to (18)




ψ|983123



lowastZlowast)














































as follows

P =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059dle Dnt le d

983060= 1

EP

983059Dnt

983060= micro

EP


EP

983091

983107

983055983055983055983055983055983055

983131

tisin[T ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


EP

983091

983107

983055983055983055983055983055983055

983131

nisin[N ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102


= 20 The




















is to minimize 1NT

983123nisin[N ]





min1

McNT

983131

misin[Mc]

983131

tisin[T ]

983131

nisin[N ]

I(m)nt

st I(m)nt ge x

(m)nt minus τ


I(m)nt ge minusx

(m)nt





983043yt0 y

tn z


983044isinZR




we set Mc = 100












timerdquo






over four periods


30 911 042 185 265 1044 9235 553 027 148 064 849 86

08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62


















(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105

08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155

SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243

35 124 070 068 145 062 112808 27 114 107 099 110 096 752

30 127 092 084 129 078 53735 125 074 071 na 064 395

MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372

08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613


52 Robustness




















(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110

08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141

SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083

35 162 100 086 273 08008 27 137 124 104 125 098

30 161 117 093 168 08535 177 113 090 NA 085

MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140

08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139







6 Conclusions














micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi




micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0



(xi)αi gt


micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0



S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070



micro

983061vτiτi


αλi

983062le micro

983061λvτiτi


αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi


983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376





983142W =

983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101

983059D U V


983055983055983055983055983055983055983055983055983055983055983055983055983055


983055983055983055le U

fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092


983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102






qtn

983059D U V

983060= qtn0 + 〈Qt






0n

983044land

983043Qt

1n


983043Qt

2n


as

supQisinQ

EQ

983061maxkisin[K]

983153akmax



2n V 〉



n

983154+ bkα

tn

983154983060le 0

(13)



















Pdagger983059D

dagger= D

(m)983060=

1

Mr

forallmisin [Mr]



983059983056983056983056Dminus Ddagger983056983056983056983060

st (DDdagger)sim P

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1







983052 (14)





(8a)




rtnθ+983131

misin[Mr ]

stnm le 0


nmk〉+ 〈D(m)

Rtnmk〉 leMrs

tnm minus ak (τ

tn minusxt


〈DVt

nmk〉minus 〈DUt

nmk〉+ 〈D(m)

Rt


tn0 minus τ t


minusU tnmk +V t



minusUt

nmk +Vt

nmk = akXtn minusR

t


983043Rt

nmk


983043Rt

nmk


983059R

t

nmk

983060


983059R

t

nmk

983060


U tnmkV

tnmkU

t

nmkVt


rtn ge 0stn isinRMs









tij le yt

j

and983123

nisin[N ] qtn

983059D

tminus1983060le yt










2015)






efficiently




min gprimeφ



(y0Y Z)isinZR

(15)







(16)



as


iisin[I]


i Z〉)ψi

st983131

iisin[I]

aiψi = g

(17)









min η

st983131

iisin[I]



983131

iisin[I]



(y0Y Z)isinZR

(18)

If G1 = ψ|983123



to (18)




ψ|983123



lowastZlowast)














































as follows

P =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059dle Dnt le d

983060= 1

EP

983059Dnt

983060= micro

EP


EP

983091

983107

983055983055983055983055983055983055

983131

tisin[T ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


EP

983091

983107

983055983055983055983055983055983055

983131

nisin[N ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102


= 20 The




















is to minimize 1NT

983123nisin[N ]





min1

McNT

983131

misin[Mc]

983131

tisin[T ]

983131

nisin[N ]

I(m)nt

st I(m)nt ge x

(m)nt minus τ


I(m)nt ge minusx

(m)nt





983043yt0 y

tn z


983044isinZR




we set Mc = 100












timerdquo






over four periods


30 911 042 185 265 1044 9235 553 027 148 064 849 86

08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62


















(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105

08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155

SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243

35 124 070 068 145 062 112808 27 114 107 099 110 096 752

30 127 092 084 129 078 53735 125 074 071 na 064 395

MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372

08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613


52 Robustness




















(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110

08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141

SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083

35 162 100 086 273 08008 27 137 124 104 125 098

30 161 117 093 168 08535 177 113 090 NA 085

MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140

08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139







6 Conclusions














micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi




micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0



(xi)αi gt


micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0



S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070



micro

983061vτiτi


αλi

983062le micro

983061λvτiτi


αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi


983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376












Pdagger983059D

dagger= D

(m)983060=

1

Mr

forallmisin [Mr]



983059983056983056983056Dminus Ddagger983056983056983056983060

st (DDdagger)sim P

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1







983052 (14)





(8a)




rtnθ+983131

misin[Mr ]

stnm le 0


nmk〉+ 〈D(m)

Rtnmk〉 leMrs

tnm minus ak (τ

tn minusxt


〈DVt

nmk〉minus 〈DUt

nmk〉+ 〈D(m)

Rt


tn0 minus τ t


minusU tnmk +V t



minusUt

nmk +Vt

nmk = akXtn minusR

t


983043Rt

nmk


983043Rt

nmk


983059R

t

nmk

983060


983059R

t

nmk

983060


U tnmkV

tnmkU

t

nmkVt


rtn ge 0stn isinRMs









tij le yt

j

and983123

nisin[N ] qtn

983059D

tminus1983060le yt










2015)






efficiently




min gprimeφ



(y0Y Z)isinZR

(15)







(16)



as


iisin[I]


i Z〉)ψi

st983131

iisin[I]

aiψi = g

(17)









min η

st983131

iisin[I]



983131

iisin[I]



(y0Y Z)isinZR

(18)

If G1 = ψ|983123



to (18)




ψ|983123



lowastZlowast)














































as follows

P =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059dle Dnt le d

983060= 1

EP

983059Dnt

983060= micro

EP


EP

983091

983107

983055983055983055983055983055983055

983131

tisin[T ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


EP

983091

983107

983055983055983055983055983055983055

983131

nisin[N ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102


= 20 The




















is to minimize 1NT

983123nisin[N ]





min1

McNT

983131

misin[Mc]

983131

tisin[T ]

983131

nisin[N ]

I(m)nt

st I(m)nt ge x

(m)nt minus τ


I(m)nt ge minusx

(m)nt





983043yt0 y

tn z


983044isinZR




we set Mc = 100












timerdquo






over four periods


30 911 042 185 265 1044 9235 553 027 148 064 849 86

08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62


















(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105

08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155

SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243

35 124 070 068 145 062 112808 27 114 107 099 110 096 752

30 127 092 084 129 078 53735 125 074 071 na 064 395

MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372

08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613


52 Robustness




















(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110

08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141

SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083

35 162 100 086 273 08008 27 137 124 104 125 098

30 161 117 093 168 08535 177 113 090 NA 085

MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140

08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139







6 Conclusions














micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi




micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0



(xi)αi gt


micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0



S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070



micro

983061vτiτi


αλi

983062le micro

983061λvτiτi


αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi


983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376






rtnθ+983131

misin[Mr ]

stnm le 0


nmk〉+ 〈D(m)

Rtnmk〉 leMrs

tnm minus ak (τ

tn minusxt


〈DVt

nmk〉minus 〈DUt

nmk〉+ 〈D(m)

Rt


tn0 minus τ t


minusU tnmk +V t



minusUt

nmk +Vt

nmk = akXtn minusR

t


983043Rt

nmk


983043Rt

nmk


983059R

t

nmk

983060


983059R

t

nmk

983060


U tnmkV

tnmkU

t

nmkVt


rtn ge 0stn isinRMs









tij le yt

j

and983123

nisin[N ] qtn

983059D

tminus1983060le yt










2015)






efficiently




min gprimeφ



(y0Y Z)isinZR

(15)







(16)



as


iisin[I]


i Z〉)ψi

st983131

iisin[I]

aiψi = g

(17)









min η

st983131

iisin[I]



983131

iisin[I]



(y0Y Z)isinZR

(18)

If G1 = ψ|983123



to (18)




ψ|983123



lowastZlowast)














































as follows

P =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059dle Dnt le d

983060= 1

EP

983059Dnt

983060= micro

EP


EP

983091

983107

983055983055983055983055983055983055

983131

tisin[T ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


EP

983091

983107

983055983055983055983055983055983055

983131

nisin[N ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102


= 20 The




















is to minimize 1NT

983123nisin[N ]





min1

McNT

983131

misin[Mc]

983131

tisin[T ]

983131

nisin[N ]

I(m)nt

st I(m)nt ge x

(m)nt minus τ


I(m)nt ge minusx

(m)nt





983043yt0 y

tn z


983044isinZR




we set Mc = 100












timerdquo






over four periods


30 911 042 185 265 1044 9235 553 027 148 064 849 86

08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62


















(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105

08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155

SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243

35 124 070 068 145 062 112808 27 114 107 099 110 096 752

30 127 092 084 129 078 53735 125 074 071 na 064 395

MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372

08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613


52 Robustness




















(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110

08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141

SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083

35 162 100 086 273 08008 27 137 124 104 125 098

30 161 117 093 168 08535 177 113 090 NA 085

MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140

08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139







6 Conclusions














micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi




micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0



(xi)αi gt


micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0



S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070



micro

983061vτiτi


αλi

983062le micro

983061λvτiτi


αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi


983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376







2015)






efficiently




min gprimeφ



(y0Y Z)isinZR

(15)







(16)



as


iisin[I]


i Z〉)ψi

st983131

iisin[I]

aiψi = g

(17)









min η

st983131

iisin[I]



983131

iisin[I]



(y0Y Z)isinZR

(18)

If G1 = ψ|983123



to (18)




ψ|983123



lowastZlowast)














































as follows

P =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059dle Dnt le d

983060= 1

EP

983059Dnt

983060= micro

EP


EP

983091

983107

983055983055983055983055983055983055

983131

tisin[T ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


EP

983091

983107

983055983055983055983055983055983055

983131

nisin[N ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102


= 20 The




















is to minimize 1NT

983123nisin[N ]





min1

McNT

983131

misin[Mc]

983131

tisin[T ]

983131

nisin[N ]

I(m)nt

st I(m)nt ge x

(m)nt minus τ


I(m)nt ge minusx

(m)nt





983043yt0 y

tn z


983044isinZR




we set Mc = 100












timerdquo






over four periods


30 911 042 185 265 1044 9235 553 027 148 064 849 86

08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62


















(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105

08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155

SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243

35 124 070 068 145 062 112808 27 114 107 099 110 096 752

30 127 092 084 129 078 53735 125 074 071 na 064 395

MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372

08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613


52 Robustness




















(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110

08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141

SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083

35 162 100 086 273 08008 27 137 124 104 125 098

30 161 117 093 168 08535 177 113 090 NA 085

MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140

08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139







6 Conclusions














micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi




micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0



(xi)αi gt


micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0



S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070



micro

983061vτiτi


αλi

983062le micro

983061λvτiτi


αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi


983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376











min η

st983131

iisin[I]



983131

iisin[I]



(y0Y Z)isinZR

(18)

If G1 = ψ|983123



to (18)




ψ|983123



lowastZlowast)














































as follows

P =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059dle Dnt le d

983060= 1

EP

983059Dnt

983060= micro

EP


EP

983091

983107

983055983055983055983055983055983055

983131

tisin[T ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


EP

983091

983107

983055983055983055983055983055983055

983131

nisin[N ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102


= 20 The




















is to minimize 1NT

983123nisin[N ]





min1

McNT

983131

misin[Mc]

983131

tisin[T ]

983131

nisin[N ]

I(m)nt

st I(m)nt ge x

(m)nt minus τ


I(m)nt ge minusx

(m)nt





983043yt0 y

tn z


983044isinZR




we set Mc = 100












timerdquo






over four periods


30 911 042 185 265 1044 9235 553 027 148 064 849 86

08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62


















(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105

08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155

SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243

35 124 070 068 145 062 112808 27 114 107 099 110 096 752

30 127 092 084 129 078 53735 125 074 071 na 064 395

MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372

08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613


52 Robustness




















(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110

08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141

SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083

35 162 100 086 273 08008 27 137 124 104 125 098

30 161 117 093 168 08535 177 113 090 NA 085

MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140

08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139







6 Conclusions














micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi




micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0



(xi)αi gt


micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0



S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070



micro

983061vτiτi


αλi

983062le micro

983061λvτiτi


αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi


983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376































as follows

P =

983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101

P

983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055

P983059dle Dnt le d

983060= 1

EP

983059Dnt

983060= micro

EP


EP

983091

983107

983055983055983055983055983055983055

983131

tisin[T ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


EP

983091

983107

983055983055983055983055983055983055

983131

nisin[N ]

Dnt minusmicro

σ

983055983055983055983055983055983055

983092


983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102


= 20 The




















is to minimize 1NT

983123nisin[N ]





min1

McNT

983131

misin[Mc]

983131

tisin[T ]

983131

nisin[N ]

I(m)nt

st I(m)nt ge x

(m)nt minus τ


I(m)nt ge minusx

(m)nt





983043yt0 y

tn z


983044isinZR




we set Mc = 100












timerdquo






over four periods


30 911 042 185 265 1044 9235 553 027 148 064 849 86

08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62


















(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105

08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155

SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243

35 124 070 068 145 062 112808 27 114 107 099 110 096 752

30 127 092 084 129 078 53735 125 074 071 na 064 395

MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372

08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613


52 Robustness




















(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110

08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141

SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083

35 162 100 086 273 08008 27 137 124 104 125 098

30 161 117 093 168 08535 177 113 090 NA 085

MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140

08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139







6 Conclusions














micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi




micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0



(xi)αi gt


micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0



S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070



micro

983061vτiτi


αλi

983062le micro

983061λvτiτi


αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi


983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376






is to minimize 1NT

983123nisin[N ]





min1

McNT

983131

misin[Mc]

983131

tisin[T ]

983131

nisin[N ]

I(m)nt

st I(m)nt ge x

(m)nt minus τ


I(m)nt ge minusx

(m)nt





983043yt0 y

tn z


983044isinZR




we set Mc = 100












timerdquo






over four periods


30 911 042 185 265 1044 9235 553 027 148 064 849 86

08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62


















(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105

08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155

SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243

35 124 070 068 145 062 112808 27 114 107 099 110 096 752

30 127 092 084 129 078 53735 125 074 071 na 064 395

MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372

08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613


52 Robustness




















(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110

08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141

SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083

35 162 100 086 273 08008 27 137 124 104 125 098

30 161 117 093 168 08535 177 113 090 NA 085

MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140

08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139







6 Conclusions














micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi




micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0



(xi)αi gt


micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0



S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070



micro

983061vτiτi


αλi

983062le micro

983061λvτiτi


αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi


983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376






timerdquo






over four periods


30 911 042 185 265 1044 9235 553 027 148 064 849 86

08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62


















(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105

08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155

SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243

35 124 070 068 145 062 112808 27 114 107 099 110 096 752

30 127 092 084 129 078 53735 125 074 071 na 064 395

MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372

08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613


52 Robustness




















(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110

08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141

SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083

35 162 100 086 273 08008 27 137 124 104 125 098

30 161 117 093 168 08535 177 113 090 NA 085

MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140

08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139







6 Conclusions














micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi




micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0



(xi)αi gt


micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0



S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070



micro

983061vτiτi


αλi

983062le micro

983061λvτiτi


αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi


983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376







(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105

08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155

SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243

35 124 070 068 145 062 112808 27 114 107 099 110 096 752

30 127 092 084 129 078 53735 125 074 071 na 064 395

MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372

08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613


52 Robustness




















(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110

08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141

SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083

35 162 100 086 273 08008 27 137 124 104 125 098

30 161 117 093 168 08535 177 113 090 NA 085

MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140

08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139







6 Conclusions














micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi




micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0



(xi)αi gt


micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0



S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070



micro

983061vτiτi


αλi

983062le micro

983061λvτiτi


αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi


983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376









(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110

08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141

SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083

35 162 100 086 273 08008 27 137 124 104 125 098

30 161 117 093 168 08535 177 113 090 NA 085

MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140

08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139







6 Conclusions














micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi




micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0



(xi)αi gt


micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0



S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070



micro

983061vτiτi


αλi

983062le micro

983061λvτiτi


αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi


983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376












micro

983061vτiτi

(xi)

αi

983062ge micro

983061vτiτi

(yi)

αi




micro

983061vτiτi

(xi)

αi

983062le micro(0) = 0



(xi)αi gt


micro

983061vτiτi

(xi)

αi

983062ge micro(983171) = micro(0)+ 983171= 983171gt 0



S(x) =

983069α

983055983055983055983055micro983061vτiτi

(xi)

αi


983070



micro

983061vτiτi


αλi

983062le micro

983061λvτiτi


αλi

983062

= micro

983061λαx

i

αλi

vτiτi(xi)

αxi

+(1minusλ)αy

i

αλi

vτiτi(yi)

αyi

983062

le omicro

983061vτiτi

(xi)

αxi


983061vτiτi

(yi)

αyi

983062

le 0


where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376




where o= λαxi α






983069I983123

i=1

αxi α

x isin S(x)


983069I983123

i=1

αyi α

y isin S(y)

983070

= inf

983069I983123

i=1



983070

ge inf

983069I983123

i=1


983070




983083I983131

i=1

αi

983055983055983055983055micro983061vλτiλτi

(λxi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061λvτiτi

(xi)

αi


983084

= inf

983083I983131

i=1

αi

983055983055983055983055micro983061vτiτi

(xi)

αiλ


983084

= inf

983083I983131

i=1

λβi

983055983055983055983055micro983061vτiτi

(xi)

βi


983084

= λρτ τ (x)







9830430 δ




i = 0 Then




i ) foralliisin I


vminusinfin0(xi)




micro



983062= micro



983062= micro



983062minus δ


gtmicro



983062minus δ


ge 0







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376







iisinI


iisinI








vminusinfin0(xi)


δ=min

983069δ

2 α

983061micro


α

983062minus η

983062983070isin (0 δ)





micro


α

983062= micro


α

983062

= micro


α

983062minus δ

αge micro


α



α

983062minus η

983062= ηgt 0











= micro(x)+w

3 Convexity



le λ+(1minusλ)

= 1








983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376










983052= 0




983052

= min

983099983103

983101a

983055983055983055983055983055983055

983131

iisin[I]

αi le 1 micro


α1

983062le 0 micro


αj

983062le 0 j = 2 I αi gt 0 iisin [I]

983100983104

983102

= min

983069a

983055983055983055983055αle 1 micro


α

983062le 0 αgt 0

983070




= micro(x)






micro(w) = micro

983061αw

α+(1minusα)0

983062le αmicro

983061w

α


983061w

α

983062le 0









ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376





ρτ τ (x) = inf

983069I983123

i=1

αi

983055983055983055microu

983059vτiτi

(xi)

αi


983154

= inf

983069I983123

i=1

αi

983055983055983055983055infηisinR

983069η

983055983055983055983055supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0


983070

= inf

983069I983123

i=1

αi


EP

983147u983059

vτiτi(xi)

αiminus ηi


983070

= inf

983069I983123

i=1

αi

983055983055983055983055 supPisinP

EP

983147u983059

vτiτi(xi)

αi


983070


supPisinP

EP

983147u983059

vτiτi(xi)

αiminus η


PisinPEP

983147u983059

vτiτi(xi)

αi



vτiτi(xi)

αi

983060le u

983059vτiτi

(xi)

αiminus η



EP

983147u983059

vτiτi(xi)

αi

983060983148le sup

PisinPEP

983147u983059

vτiτi(xi)

αiminus η

983060983148le 0 QED





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376





supPisinP

EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

= sup EP

983061maxkisin[K]

983153ak max

983153τ tn minusxt



n

983154+ bkα

tn

983154983062

st P983059Dle DleD

983060= 1

EP

983059D983060=Ξ

EP

983059983055983055983055DminusΞ983055983055983055983060leΣ

EP

983093

983095fl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983094



nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge maxkisin[K]

983051ak max

983051τ tn minusxt



n

983052+ bkα

tn

983052forallDleDleD


T tn ge 0


nΣ〉+983131

lisin[L]

983131

hisin[H]

983171lhrtnlh



lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043xtn0 + 〈Xt

nD〉minus τ tn

983044+ bkα




lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

ge ak

983043τ tn minusxt

n0 minus〈XtnD〉

983044+ bkα



T tn ge 0



minDleDleD

983099983103


tnD〉+ 〈T t


lisin[L]

983131

hisin[H]

rtnlhfl

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108

983100983104


n)

+bkαtn minus stn0





tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376







tnD〉+ 〈T t


lisin[L]

983131

hisin[H]


983091

983107olkprime

983091

983107983131

(iτ)isinSh

Diτ minusΞiτ

Σiτ

983092

983108+ plkprime

983092

983108

st DgeD

DleD


max 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

st983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0



〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime



983059U

t

nk minusVt

nk minusFt

nk +Gt

nk

983060

iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n minus akXtn

983044iτ


Ft

nk +Gt

nk = T tn

983131

kprimeisin[Kl]


Ut

nkVt

nkFt

nkGt

nk ge 0






nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376







nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

ge ak (τtn minusxt


983043U t


nk +Gtnk

983044iτminus

983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime =

983043St

n + akXtn

983044iτ


F tnk +Gt

nk = T tn

983131

kprimeisin[Kl]


U tnkV

tnkF

tnkG

tnk ge 0



QED



983052 for any




〈DQtn〉 ge 0


〈DQtn〉 ge 0

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101


n〉 le 0

Qtn +Lt

n minusHtn = 0

LtnH

tn ge 0


be formulated as



n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376





n forallD isinW


〈DQtn〉 leMyt

n


〈DQtn〉 leMyt

n

hArr

983099983105983105983105983105983105983103

983105983105983105983105983105983101

qtn0 + 〈DLt

n〉minus 〈DHt

n〉 leMytn

Qtn minusL

t

n +Ht

n = 0

Lt

nHt

n ge 0

QED



inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376





inf983131

tisin[T ]

983131

nisin[N]

αtn


nΣ〉+983123

lisin[L]

983123hisin[H]


stn0 + 〈DUt

nk〉minus 〈DVt

nk〉+ 〈ΞGt

nk minusFt

nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

minusak

983075t983131

m=1

qmn0

983076minus bkα

tn geminusakτ




nk〉+983131

lisin[L]

983131

hisin[H]

983131

kprimeisin[Kl]

983091


983131

(iτ)isinSh

Ξiτ

Σiτ

983092

983108wtnklhkprime

+ak

983075t983131

m=1

qmn0

983076minus bkα

tn ge akτ


983075U

t

nk minusVt

nk minusFt

nk +Gt

nk minusStn + ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=

983075ak

t983131

m=1

(eneprimem)

983076

iτ


983075U t


nk +Gtnk minusSt

n minus ak

983075t983131

m=1

Qmn

983076983076

iτ

minus983131


983131

lisin[L]

983131

kprimeisin[Kl]

olkprime

Σiτ

wtnklhkprime

=minus983075ak

t983131

m=1

(eneprimem)

983076

iτ


Ft

nk +Gt


F tnk +Gt


983131

kprimeisin[Kl]


983131

kprimeisin[Kl]




qtn0 + 〈DLt

n〉minus 〈DHt


Qtn +Lt


Qtn minusL

t

n +Ht


983043Qt

n




T tnL

tnH

tnL

t

nHt


Ut

nkVt

nkFt

nkGt

nkUtnkV

tnkF

tnkG


wtnklhkprime wt


(yt0 y

tn z



QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376




QED









sup EP

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st EP

983059983056983056983056Dminus Ddagger9830569830569830561

983060le θ

ΠDP= P

ΠD

daggerP= Pdagger

P983059(DD

dagger)isinW timesW

983060= 1

(19)


(m) m isin


983059D=D

983060= P

983059D=D|Ddagger

= D(m)


P983059D=D

983060=983123

misin[Mr]Pm

983059D=D

983060Pdagger

983059D

dagger= D

(m)983060= 1

Mr

983123misin[Mr]

Pm

983059D=D



sup1

Mr

983131

misin[Mr]

EPm

983061maxkisin[K]

983153ak max


983154+ bkα

983154983062

st1

Mr

983131

misin[Mr]

EPm

983059983056983056983056Dminus D(m)

9830569830569830561

983060le θ

Pm

983059D isinW

983060= 1 misin [Mr]



dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376





dual problem is

inf rθ+983131

misin[Mr]

sm

str

Mr

983056983056983056Dminus D(m)

9830569830569830561+ sm ge 1

Mr

983061maxkisin[K]



rge 0sisinRMr

= inf rθ+1primes

st r983056983056983056Dminus D



rge 0sisinRMr


two inequalities

r983056983056983056Dminus D


r983056983056983056Dminus D



supDisinW


983056983056983056Dminus D(m)

9830569830569830561


supDisinW


983056983056983056Dminus D(m)

9830569830569830561


(20)


r983056983056983056Dminus D

(m)9830569830569830561= max983042Rmk983042infin

ler

983111983059Dminus D

(m)983060Rmk

983112




supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376






supDisinW



983111983059Dminus D

(m)983060Rmk


lArrrArr supDisinW

983083min



983111983059Dminus D

(m)983060Rmk



ler

983069supDisinW


983111983059Dminus D

(m)983060Rmk


lArrrArr

983099983105983105983103

983105983105983101

supDisinW


983111983059Dminus D

(m)983060Rmk



lArrrArr

983099983105983105983103

983105983105983101

supDisinW


(m)Rmk





supDisinW



st DgeD

DleD



UmkV mk ge 0




983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983105983105983105983105983105983105983101






UmkV mk ge 0



983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376





983105983105983105983105983105983105983105983105983105983105983105983105983105983101




983043Rmk


983043Eτ

i Rmk


UmkV mk ge 0



References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376




References


52(4) 499-514



103-120





59(8) 1055-1065



1515-1536






9(3) 203-228












248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376







248-271



























55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376







55(1) 71-84










Research 55(5) 828-842









Science 48(1) 1-19








Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376





Research 20-46














tics 6(4) 429-447

























186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376

















186-200


gramming 158(1-2) 291-327








349-360







870-885






69(1) 99-118








Research 62(6) 1358-1376






870-885






69(1) 99-118








Research 62(6) 1358-1376



inventory routing problem under uncertainty · inventory routing problem under uncertainty zheng...

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