dynamic vehicle scheduling for working service network ...a branch and price algorithm for...

Research ArticleDynamic Vehicle Scheduling for Working ServiceNetwork with Dual Demands

Bing Li Wei Xu Hua Xuan and Chunqiu Xu

School of Management Engineering Zhengzhou University Zhengzhou 450001 China

Correspondence should be addressed to Bing Li lbingzzueducn

Received 4 July 2017 Revised 17 August 2017 Accepted 10 September 2017 Published 22 October 2017

Academic Editor Sara Moridpour

Copyright copy 2017 Bing Li et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This study aims to develop some models to aid in making decisions on the combined fleet size and vehicle assignment in workingservice network where the demands include two types (minimum demands and maximum demands) and vehicles themselvescan act like a facility to provide services when they are stationary at one location This type of problem is named as the dynamicworking vehicle scheduling with dual demands (DWVS-DD) and formulated as a mixed integer programming (MIP) Insteadof a large integer program the problem is decomposed into small local problems that are guided by preset control parametersThe approach for preset control parameters is given By introducing them into the MIP formulation the model is reformulatedas a piecewise form Further a piecewise method by updating preset control parameters is proposed for solving the reformulatedmodel Numerical experiments show that the proposed method produces better solution within reasonable computing time

1 Introduction

The vehicle scheduling problems arise when owners andoperators of transportation systems must manage a fleet ofvehicles over space and time to serve current and forecasteddemands

The capacity of a transportation system is directly relatedto the number of available vehicles Determining the optimalnumber of vehicles for a transportation system requires atradeoff among the benefits for meeting demands the own-ership costs of the vehicles and the penalty costs associatedwith not meeting some demands Serving demand resultsin the relocation of vehicles Each vehicle is in a particularlocation and each task demand requires a vehicle in aparticular location The assignment of a vehicle to a taskdemand generates revenueThus we consider the problem ofvehicles assignment strategy

The interaction between fleet sizing decisions and vehicleassignment decisions is the focus of this paper There is asubstantial history of research on vehicle assignment prob-lemswith fixed vehicle fleet But the research described in thispaper attempts to integrate vehicle fleet sizing decisions withvehicle assignment decisions

In this paper we consider the dynamic vehicle schedulingfor working service network with dual demands by applyingan optimization modeling approach in which the servicedemand in each terminal includes two type that is minimumdemands and maximum demands We name this type ofproblem as the dynamic working vehicle scheduling withdual demands (DWVS-DD) The objective is to optimize theperformance of the transportation system over the entireplanning horizon The model of problem starts with theclassical mixed integer programming formulation and is thenreformulated as a piecewise form We develop two types ofreformulated models for the issue and present a piecewisemethod by updating preset control parameters

In addition to the integration of the vehicle fleet sizingand the vehicle assignment problem two other factors suchas the working service network and working vehicle increasesignificantly the complexity of the research in this paper

First we must recognize one crucial characteristic ofworking service network at any location of working servicenetwork in space and time the demands include two typesthat is minimum demands and maximum demands Theminimumdemandsmust bemet butmaximumdemands arenot If insufficient vehicles are available to meet maximum

HindawiJournal of Advanced TransportationVolume 2017 Article ID 7217309 13 pageshttpsdoiorg10115520177217309

2 Journal of Advanced Transportation

demand the penalty cost for unmet demand will generateThis characteristic is the cornerstones of themodel developedin this paper

Second vehicles usually provide pickup or delivery ser-vices between various locations in previous studies Howeverin reality vehicles themselves can sometimes act like a facilityto provide real-time services when they are stationary at onelocation The vehicles cannot provide services when they arein motion and the service begins when a vehicle arrives ata location and ends when it departs For instance medicaltreatment vehicles provide first aid services to areas where theestablished medical facility is temporarily insufficient Alsofood trucks provide fast food services in different regionsin different time periods of the day Note that when thesevehicles are in service they behave like traditional facilitiesThe term working vehicle (WV) will be used in this paperto denote this vehicle Applications of problems arise inmany settings ranging from managing emergency vehiclesmedical testing vehicles traveling salesman and militaryforce deployment

Overall the objectives of this research are twofold Thefirst objective is to develop a novel mathematical modelof DWVS-DD In conventional mathematical models theproblem has been formulated as a mixed integer program-ming model In the proposed model the problem is refor-mulated as a piecewise form to find the local problem atevery time period The second objective is to propose anefficient methodology for solving the model Specifically thecontributions of this paper are as follows

(1) We develop the mixed integer programming model(MIP) The model then is reformulated as two novelformulations that is reconstruction model with sin-gle preset incremental parameters (RM-SPIP) andreconstruction model with double preset incremen-tal-decremental parameters (RM-DPIDP)

(2) We propose the coupled correlation model of totalprofit vehicle supply minimum demand and maxi-mum demand The acquisition approaches of singlepreset control parameters are given Meanwhile wepropose the coupled correlationmodel of the revenuepenalty cost vehicle supply minimum demand andmaximum demand The acquisition approaches ofdouble preset control parameters are given

(3) According to the specific structure of the two recon-struction models the piecewise method by updatingpreset control parameters (PM-PCP) is developed

(4) We tested the performance of PM-PCP approach bysolving many instances We compared the qualityof the solutions provided by PM-PCP solving RM-SPIP and RM-DPIDP versus the results of the CPLEXsolving MIP According to our results in most of theinstances our PM-PCP approach has a better per-formance Indeed PM-PCP provides many optimalsolutions in a very short computation time

The remainder of this paper is organized as followsSection 2 of this paper discusses related earlier researchefforts Section 3 is devoted to the mathematical description

of the DWVS-DD In Section 4 we set up the problem asthe classical mathematical programming formulation andpresent then the reformulated models Section 5 explains theacquisition methodology of preset control parameters basedon coupled correlation function We introduce piecewisemethod by updating preset parameters (PM-PCP) for solvingreconstructionmodel in Section 6The computational exper-iments are described in Section 7 and the effectiveness of theproposed method is shown from the computational resultsThe last section concludes with a summary of current workand extensions

2 Literature Review

In this section we review the relevant literatures about vehiclescheduling problems Literature review indicates that vehiclescheduling problems can be divided into three groups that isvehicle routing fleet sizing and fleet assignment The focusof our literature review will primarily be on the model andexact approach since it is also the model and approach wehave taken in this paper

In recent years many researches on vehicle routingoptimization have been carried out Hou et al [1] focused onvehicle routing problem with soft time window constraintAn exact algorithm based on set partition was proposedto solve the balancing the vehicle number and customersatisfaction by Cao et al [2] Li et al [3] studied the integratedproblem with truck scheduling and storage allocation It wasformulated as an integer programming model to minimizemakespan of the whole discharging course and solved by atwo stages tabu search algorithm Dabia et al [4] presenteda branch and price algorithm for time-dependent vehiclerouting problemwith timewindowsHan et al [5] considereda vehicle routing problem with uncertain travel times inwhich a penalty is incurred for each vehicle that exceeds agiven time limit and given robust scenario approach for thevehicle routing problem Muter et al [6] proposed a columngeneration algorithm for the multidepot vehicle routingproblem with interdepot routes Vidal et al [7] proposeda unified hybrid genetic search metaheuristic algorithm tosolve multiattribute vehicle routing problems Battarra etal [8] presented new exact algorithms for the clusteredvehicle routing problem (CluVRP) and provided two exactalgorithms for the problem that is a branch and cut as well asa branch and cut and price

Fleet sizing is one of the most important decisions as itis a major fixed investment for starting any business Manyscholars have conducted the numerous basic studies on fleetsizing problem (FSP) Zak et al studied a fleet sizing problemin a road freight transportation company with heterogeneousfleet [9] Additionally themathematicalmodel of the decisionproblem was formulated in terms of multiple objectivemathematical programming based on queuing theory Afleet sizing problem arising in anchor handling operationsrelated to movement of offshore mobile units is presentedby Shyshou et al [10] A simulation-based prototype wasproposed and the simulation model was implemented inArena 90 (a simulation software package developed byRockwell Software) Rahimi-Vahed et al [11] addressed the

Journal of Advanced Transportation 3

problemof determining the optimal fleet size for three vehiclerouting problems that is multidepot VRP periodic VRPand multidepot periodic VRP And a new Modular HeuristicAlgorithm (MHA) was proposed Ertogral et al [12] exploreda real strategic fleet sizing problem for a furniture and homeaccessory distributor Then a mixed integer linear programwas proposed to determine the total number and types ofowned and rented vehicles for each region under seasonaldemand The study developed an analytical model for thejoint LGV (Laser Guided Vehicles) fleet sizing problem alsotaking into consideration stochastic phenomena and queuingimplications in Ferrara et al [13] Chang et al [14] studied thevehicle fleet sizing problem in semiconductor manufacturingand proposed a formulation and solution method calledSimulation Sequential Metamodeling (SSM) By using anagent-based model of a flexible carsharing system Barriosand Godier [15] explored the trade-offs between fleet size andhired vehicle redistributors with the objective of maximizingthe demand level that can be satisfactorily served Koc etal [16] introduced the fleet size and mix location-routingproblemwith timewindows and developed a powerful hybridevolutionary search algorithm Park and Kim [17] addressedthe fleet sizing of containers and developed an analyticalmodel for the minimum container fleet size

Moreover some literatures on fleet assignment wereaddressed from the viewpoint of optimization models andsolution methods Xia et al [18] studied a comprehensivemodel that addresses fleet deployment speed optimizationand cargo allocation jointly so as to maximize total profits atthe strategic level Pita et al [19] presented a flight schedulingand fleet assignment optimization model and carried out awelfare analysis of the network And the optimization modeland subsequent welfare analysis were applied to the PSOnetwork of Norway Pilla et al [20] developed a two-stagestochastic programming framework to the fleet assignmentmodel and presented the L-shaped method to solve the two-stage stochastic programming problems Liang and Chao-valitwongse [21] presented a network-based mixed integerlinear programming formulation for the aircraftmaintenancewith the weekly fleet assignment and developed a divingheuristic approach Sherali et al [22] proposed a modelthat integrates certain aspects of the schedule design fleetassignment and aircraft-routing process and designed Ben-dersrsquos decomposition-based method The liner shipping fleetrepositioning problem (LSFRP) was formulated as a novelmathematical model and a simulated annealing algorithm isproposed for the LSFRP by Tierney et al [23] Hashemi andSattarvand [24] studied the different management systems ofthe open pit mining equipment including nondispatchingdispatching and blending solutions for the Sungun coppermine A dispatching simulation model with the objectivefunction of minimizing truck waiting times had been devel-oped A Markov decision model is developed to study thevehicle allocation control problem in the automated materialhandling system (AMHS) in semiconductor manufacturingby Lin et al [25] Simao et al [26] developed a model forlarge-scale fleet management and presented an approximatedynamic programming to solve dynamic programs withextremely high-dimensional state variables Topaloglu and

Powell [27] reported how to coordinate the decisions onpricing and fleet assignment of a freight carrier And atractable method to obtain sample path-based directionalderivatives of the objective function with respect to theprices was presented Aimed at the stochastic dynamic fleetscheduling Li et al [28ndash30] further proposed some heuristicapproaches to deal with these problems A new and improvedLipschitz optimization algorithm to obtain a Ε-optimalsolution for solving the transportation fleet maintenance-scheduling problem is proposed by Yao and Huang [31]In this study a procedure based on slope-checking andstep-size comparison mechanisms was given to improve thecomputation efficiency of the Evtushenko algorithm

The focus of this paper is development of some modelsto aid in making decisions on the combined fleet size andvehicle assignment in working service network where thedemands include two types (minimum demands and maxi-mumdemands) and vehicles themselves can act like a facilityto provide services when they are stationary at one locationTwo types of preset control parameters are applied to themodel of the DWVS-DD so that the problem is decoupledinto some local problems for different time periods Furthera piecewise method by updating preset control parameters isproposed for solving the model

3 Problem Formulation

31 Problem Description Let 119866 represent working servicenetwork 119881 is the set of working service station set in thenetwork 119866 We assume that time is divided into a set ofdiscrete time periods 119879 = 119905 | 119905 = 1 119872 where119872 is the length of the planning horizon We also assumethat there exist demands for vehicle work service at terminal119894 119894 isin 119881 in period 119905 119905 isin 119879 When vehicles servedemands the revenues will generate We assume a unitrevenue per served demand in period 119905 denoted as 119886119905 Servingdemand results in the relocation of vehicles between variouslocations It implies the need for redistribution of vehiclesover the working service network from locations at whichthey have become idle to locations at which they can bereusedTheminimum demands andmaximum demands canbe respectively represented as 119876min

119894119905 and 119876max119894119905 at terminal119894 in period 119905 These demands induce vehicles available to

serve them The minimum demands 119876min119894119905 must be met but

maximum demands 119876max119894119905 are not If insufficient vehicles are

available at location 119894 in period 119905 to meet maximum demandthe penalty cost for unmet demand will generate We denotethe unit penalty cost per period for unmet demand by 119887119905The level of demand in units of vehicle loads is assumedto be specified as data We consider 120585 to be these demandswhich can be serviced by one vehicle Let 119889119895119896 be the distancebetween any pair of terminal 119895 and 119896The demand of terminal119896 can be covered by vehicles located in 119895 if only the distance119889119895119896 le 119863 where 119863 is the maximum coverage distanceConsidering the expense of purchasing or renting vehiclewe assume that the fixed costs of using vehicles are constantand denoted by 119888 for using one vehicle The main purposeof the DWVS-DD is to propose working vehicle assignmentplan for serving as many demands as possible in the given


Time period 1 Time period MTime period t

1 11

QＧＣＨkt Q

ＧＲkt

dik le D

QＧＣＨit Q

ＧＲit

QＧＣＨ

k tQ

ＧＲ

k t

k

QＧＣＨkM Q

ＧＲkM

QＧＣＨi1

QＧＲi1

QＧＣＨ

kMQ

ＧＲ

kM

middot middot middot middot middot middot

middot middot middot

i

N N N

dik le D

k

k

i

dik le D

dik le D

i

k

Figure 1 Dynamic working service network with dual demands

planning horizon at the highest possible profit Owning orleasing a fleet of vehicles is generally quite costly so it isnatural to try to optimize the size of the required fleet Weemphasize the tradeoff among the investment for establishinga suitable fleet (ie the fixed cost) the benefits for meetingdemands (ie the revenue for serving demands) and the lossof benefits for failing to satisfy demands (ie the penalty costfor unmet demands) Dynamic working service networkwithdual demands is shown in Figure 1

32 Problem Definition and Notation We assume that plan-ning horizon is divided into a set of discrete instants 119879 = 119905 |119905 = 1 119872 where119872 is the length of the planning horizonA network is represented by graph 119866 = (119881 119864) where 119881 is theset of terminal 119864 is the set of link in the network We presentthe complete notation for the problem here

Quickly summarizing the notation we have the followingdecision variables

119909119894119895119905 is the number of working vehicles dispatchedfrom terminal 119894 to terminal 119895 in period 119905 119894 119895 isin 119873119905 isin 119879119880 is the vehicle fleet size

The revenues and costs associated with operating thesystem are as follows

119886119905 is the revenue for one unit of met demands inperiod 119905119887119905 is the penalty cost for one unit of unmet demand inperiod 119905119888 is the fixed costs for one vehicle

In addition the demand for vehicles is given by thefollowing

119876min119894119905 is the minimum demand for working service at

location 119894 in period 119905 Theminimum demandmust besatisfied by vehicles dispatched from other terminals119876max119894119905 is the maximum demand for working service at

location 119894 in period 119905Themaximumdemandmay not

be met But the penalty cost for unmet demand willgenerate

Finally the parameters are needed to describe the system

120585 is these demands which can be serviced by onevehicle119889119895119896 is the distance between any pair of terminals 119895 and119896119863 is the maximum coverage distance for any vehicleat any terminal120572119895119896 is 0-1 parameter 120572119895119896 = 1 if the distance 119889119895119896 le 119863or 120572119895119896 = 0 otherwise

We assume that all parameters are deterministic andknown

33 Model Formulation Given the above notations for theparameters and decision variables we present the formula-tion as follows

[MIP] max 119865 (119909)= 120585sum119905isin119879

sum119894isin119866

sum119895isin119866

119886119905119909119894119895119905

minus sum119905isin119879

sum119895isin119866

119887119905(119876max119895119905 minus 120585sum

119894isin119866

119909119894119895119905) minus 119888sdot 119880

(1)

subject to sum119894isin119866

sum119895isin119866

119909119894119895119905 le 119880 forall119905 isin 119879 (2)

120585sum119894isin119866

119909119894119895119905 ge 119876min119895119905+1 forall119895 isin 119866 forall119905 isin 119879 (3)

120585sum119894isin119866

119909119894119895119905 le sum119896isin119866

120572119895119896119876max119896119905+1

forall119895 isin 119866 forall119905 isin 119879(4)


120572119895119896 = 1 119889119895119896 le 119863 forall119895 119896 isin 1198660 119889119895119896 gt 119863 forall119895 119896 isin 119866 (5)

sum119894isin119866

119909119894119895119905 = sum119894isin119866

119909119895119894(119905+1) forall119905 isin 119879 (6)

119880 119909119895119894119905 ge 0 and integer

forall119894 119895 isin 119866 forall119905 isin 119879 (7)

The objective function (1) includes terms for revenuespenalty costs for unmet demand and ownership cost forvehicles It intends to maximize the total revenue of thesystem throughout the planning horizon Constraint (2)restricts that the total number of working vehicle used cannotexceed the fleet size Constraints (3) ensure that theminimumdemand must be met Constraints (4) impose an upper limitfor the service capacity of the working vehicle at each locationin each time period Constraints (5) are coverage restrictionand indicate the coverage relations between the demandnodes 119895 and candidate locations 119896 that is 120572119895119896 = 1 if 119889119895119896 ⩽ 119863or 120572119895119896 = 0 otherwise Constraints (6) are conservation of flowconstraints for vehicles at each location in each time periodConstraints (7) ensure that119880 and 119909119895119894119905 are always nonnegativeand integer The nominal model of DWVS-DD can be solvedas a mixed integer program (MIP) by CPLEX solver

4 Reconstruction of the Model Using PresetControl Parameters

We use 119864[119865119905+1(119909)] to denote an expected value relative to119909119905+1 The expectation functional 119864[119865119905+1(119909)] is called the stage119905 expected recourse function Nowwe introduce the expectedrecourse function into theMIPThe objective function can beexpressed in the recursive form by

max119865119905 (119909) = 120585sum119894isin119866

sum119895isin119866

119886119905119909119894119895119905


119887119905(119876max119895119905 minus 120585sum

119894isin119866

119909119894119895119905) minus 119888 sdot 119880+ 119864 [119865119905+1 (119909)]

(8)

41 Reconstruction of the Model with Single PresetIncremental Parameters

411 Preset Total Incremental Profit Parameters We denote120575119895119905+1 as the total contribution of adding one loaded vehicleat terminal 119895 starting at 119905 + 1 time period through the restplanning horizon Because 120575119895119905+1 depict the marginal profit ofadditional vehicle we call them as preset incremental profitparameter (PIPP)

412 Reconstruction Model with Single Preset IncrementalParameters It is common sense that the total expected profitsin each terminal at each time period depend on the numberof available vehicles there Thus the total expected profits

in each terminal grow linearly with the number of availablevehicles Here we make substitution of expected recoursefunction by a linear function of marginal profit and vehiclenumber

We can now define the state of DWVS-DD at 119905th timeperiod that is 119880119905 | 119880119895119905 119895 isin 119873 Note that the stateof DWVS-DD at 119905th time period is given by the totalvehicle supply in each terminal By replacing the expectationrecourse function 119864[119865119905+1(119909)] with preset total incrementalprofit parameters 120575119895119905+1 the modified stage 119905 + 1 expectedrecourse function becomes

119864 [119865119905+1 (119909)] = sum119895isin119866

120575119895+1119880119895119905+1 (9)

We also note that

119880119895119905+1 = sum119894isin119866

119909119894119895119905 (10)

Substituting them into 119864[119865119905+1(119909)] gives119864 [119865119905+1 (119909)] = sum

119895isin119866

120575119895119905+1sum119894isin119866

119909119894119895119905 (11)

Substituting (11) into formula (8) we arrive at the localproblem which is the problem to be solved at every timeperiod The new formulation of the local problem at singletime period is represented as follows called reconstructionmodel with single preset incremental parameters (RM-SPIP)

[RM-SPIP] max 119865119905 (119909)= 120585sum119894isin119866

sum119895isin119866

119886119905119909119894119895119905


119887119905(119876max119895119905 minus 120585sum

119894isin119866

119909119894119895119905)minus 119888 sdot 119880 + sum

119895isin119866

120575119895119905+1sum119894isin119866

119909119894119895119905= sum119894isin119866

sum119895isin119866

(119886119905120585 + 119887119905120585 + 120575119895119905+1) 119909119894119895119905minus sum119895isin119866

119887119905119876max119895119905 minus 119888 sdot 119880

st Constraints (2)ndash(7)

(12)

42 Reconstruction of the Model with Double PresetIncremental-Decremental Parameters

421 Preset Incremental Revenue Parameters Let 120575+119895119905+1 bethe revenue of adding one vehicle to servicing demand atterminal 119895 starting at 119905 + 1 time period through the restplanning horizon Because 120575+119895119905+1 depict the marginal revenueof additional vehicle we call them as preset incrementalrevenue parameter (PIRP)


422 Preset Decremental Cost Parameters The sameapproach is adopted We denote 120575minus119895119905+1 as the effect on penaltycost of adding one vehicle to servicing demand at terminal119895 starting at 119905 + 1 time period through the rest planninghorizon Because 120575minus119895119905+1 depict the marginal penalty cost ofadditional vehicle we call them as preset decremental costparameter (PDCP)

423 Reconstructing Model with Double Preset Incremental-Decremental Parameters By replacing the expectationrecourse function119864[119865119905+1(119909)]with double preset incremental-decremental parameters 120575+119895119905+1 and 120575minus119895119905+1 the modified stage119905 + 1 expected recourse function becomes

119864 [119865119905+1 (119909)] = sum119895isin119866

120575+119895119905+1119880119895119905+1 + sum119895isin119866

120575minus119895119905+1119880119895119905+1 (13)

And 119880119895119905+1 = sum119894isin119866 119909119894119895119905 119864[119865119905+1(119909)] can be written as

119864 [119865119905+1 (119909)] = sum119895isin119866

120575+119895119905+1sum119894isin119866

119909119894119895119905 + sum119895isin119866

120575minus119895119905+1sum119894isin119866

119909119894119895119905 (14)

Substituting (14) into formula (8) we arrive at the localproblem for each time periodThemodel can be reformulatedin piecewise form as follows We name this new formas reconstruction model with double preset incremental-decremental parameters (RM-DPIDP)

[RM-DPIDP] max 119865119905 (119909)= 120585sum119894isin119866

sum119895isin119866

119886119905119909119894119895119905


119887119905(119876max119895119905 minus 120585sum

119894isin119866

119909119894119895119905)minus 119888 sdot 119880 + sum

119895isin119866

120575+119895119905+1sum119894isin119866

119909119894119895119905+ sum119895isin119866

120575minus119895119905+1sum119894isin119866

119909119894119895119905= sum119894isin119866

sum119895isin119866

(119886119905120585 + 120575+119895119905+1) 119909119894119895119905minus sum119895isin119866

119887119905119876max119895119905

+ sum119894isin119866

sum119895isin119866

(119887119905120585 + 120575minus119895119905+1) 119909119894119895119905minus 119888 sdot 119880


(15)

5 Approach for Preset Control Parameter

The preset control parameter results in decoupling the prob-lems for different time period In section we will develop aninteractive procedure to provide approximations of the presetcontrol parameters

s

Virtual time period1

Ui1

Uk 1

UN1

k

QＧＣＨk1 Q

ＧＲk1

dik le D

dik le D

QＧＲi1Q

ＧＣＨi1

QＧＣＨ

k1Q

ＧＲ

k1

U11

Uk1k

i

N

Figure 2 Dynamic working network for virtual time period

51 Procedure of the Sampling Data

511 Determining Initial Vehicle Distribution

Step 11 Since all minimum demand and maximum demandare available for the first time we have 119876min

1198941 and 119876max1198941

Step 12 Add a virtual source terminal 119878 into working servicenetwork119866(119881 119864) In Figure 2 there are only the outbound arcsfor source terminal 119878

The formulation of local problem for virtual time periodis denoted as LP-VTP This model includes the followingobjective function and constraints

[LP-VTP] max 119865 (119909) = 120585sum119894isin119866

1199091199041198940 minus 119888 sdot 119880subject to sum

119894isin119866

1199091199041198940 = 119880120585 sdot 1199091199041198940 ge 119876min

1198941 forall119894 isin 119866120585 sdot 1199091199041198940 le sum

119896isin119866

120572119894119896 sdot 119876max1198961 forall119894 isin 119866

120572119894119896 = 1 119889119894119896 le 119863 forall119894 119896 isin 1198660 119889119894119896 gt 119863 forall119894 119896 isin 119866

119880 1199091199041198940 ge 0 and integer forall119894 isin 119866

(16)

Because we pose this local problem in the format of aninteger linear program CPLEX solver can be used Optimalsolutions1199091199041198940 can be obtained by usingCPLEX for solving LP-VTP

Step 13 Initial vehicle distribution is obtained according to1198801198941 = 1199091199041198940


512 Sampling Data with Solving Local Problem By solvinglocal problem at each time period the data can be obtainedThe procedure of sampling data procedure is explained asfollows

Step 21 The state vector that is 119880119905 | 119880119894119905 119894 isin 119866 is updatedby equation 119880119894119905 = sum119895isin119866 119909119895119894(119905minus1)Step 22 Sinceminimum demand andmaximum demand aredeterministic and known for the whole planning horizon wehave 119876min

119895119905+1 and 119876max119895119905+1

Step 23 Solve local problem (LP) one for each time periodby CPLEX solver to obtain optimal solution 119909119894119895119905 The formu-lation of the LP is shown as follows

[LP] max 119865119905= 120585sum119894isin119866

119886119905119880119894119905 minus sum119894isin119866

119887119905 (119876max119894119905 minus 120585119880119894119905)

minus 119888sum119894isin119866

119880119894119905subject to sum

119895isin119866

119909119894119895119905 le 119880119894119905 forall119894 isin 119866120585sum119894isin119866

119909119894119895119905 ge 119876min119895119905+1 forall119895 isin 119866

120585sum119894isin119866

119909119894119895119905 le sum119896isin119866

120572119895119896119876max119896119905+1 forall119895 isin 119866


119909119894119895119905 ge 0 and integer forall119894 119895 isin 119866

(17)

Step 24 Taking optimal solution 119909119894119895119905 into the objective func-tion of local problem 119865119894119905 119865+119894119905 119865minus119894119905 can be obtained where 119865119894119905119865+119894119905 119865minus119894119905 denotes respectively total profit revenue andpenalty cost for each time period

Step 25 Record these data that is 119880119894119905 119876min119894119905 119876max

119894119905 119865119894119905 119865+119894119905 119865minus119894119905 52 Modeling for Coupled Correlation

521 Coupled Correlation Function with Single Preset Con-trol Parameters (CCF-SPCP) The total profit function is afunction of vehicle supplyminimumdemand andmaximumdemand We generate a quadratic polynomial function forthe effect of vehicle supply minimum demand and maxi-mum demand on the total profits in each terminal wherethe function is used to approximate the incremental profitparameter of RM-SPIP The quadratic polynomial functionfor approximating the single preset incremental parameter ofRM-SPIP has the following form

[CCF-SPCP] 119865119894 (119880119894 119876min119894 119876max

119894 )= 1205721119894 (119880119894)2 + 1205722119894 (119876min

119894 )2 + 1205723119894 (119876max119894 )2 + 1205724119894 119880119894119876min

119894

+ 1205725119894 119880119894119876max119894 + 1205726119894 119876min

119894 119876max119894 + 1205727119894 119880119894 + 1205728119894 119876min

119894

+ 1205729119894 119876max119894 + 12057210119894 forall119894 isin 119866

(18)

Above coupled correlation function with single presetcontrol parameters is denoted as CCF-SPCP

522 Coupled Correlation Function with Double Preset Con-trol Parameters (CCF-DPCP) The coupled correlation is setup in a control theoretic setting The pair 119865+119894119905 119865minus119894119905 representsthe system outputs The set 119880119894119905 119876min

119894119905 119876max119894119905 represents the

system inputs Multi-input and multi-output control systemsare set up We generate a quadratic polynomial equationfor the effect of vehicle supply minimum demand andmaximum demand on the revenue and penalty cost in eachterminal where the function is used to approximate theincremental revenue parameter and decremental cost param-eters of RM-DPIDP The quadratic polynomial function forapproximating the double preset incremental-decrementalparameters of RM-DPIDP has the form as follows

[CCF-DPCP] 119865+119894 (119880119894 119876min119894 119876max

119894 )= 1205731119894 (119880119894)2 + 1205732119894 (119876min

119894 )2+ 1205733119894 (119876max

119894 )2 + 1205734119894 119880119894119876min119894

+ 1205735119894 119880119894119876max119894 + 1205736119894 119876min

119894 119876max119894

+ 1205737119894 119880119894 + 1205738119894 119876min119894 + 1205739119894 119876max

119894 + 12057310119894 forall119894 isin 119866

119865minus119894 (119880119894 119876min119894 119876max

119894 )= 1205741119894 (119880119894)2 + 1205742119894 (119876min

119894 )2+ 1205743119894 (119876max

119894 )2 + 1205744119894 119880119894119876min119894

+ 1205745119894 119880119894119876max119894 + 1205746119894 119876min

119894 119876max119894 + 1205747119894 119880119894

+ 1205748119894 119876min119894 + 1205749119894 119876max

119894 + 12057410119894 119894 isin 119866

(19)

Above coupled correlation function with double presetcontrol parameters is denoted as CCF-DPCP

53 Fitting Parameters of Coupled Correlation Function

531 Sampling Data Sets for Fitting Parameters By solvinglocal problem at each time period the data sets can beobtained Solution of local problem that is 119909119894119895119905 is obtainedby CPLEX solver 119865119894119905 119865+119894119905 119865minus119894119905 is carried on by taking 119909119894119895119905 intothe objective function of local problem State vector 119880119894119905 isobtained by updated approach of local problem In order tofit parameters of coupled correlation function these data aresplit into three data sets that is 119865119894119905 119880119894119905 119876min

119894119905 119876max119894119905 | 119894 isin119866 119905 isin 119879 119865+119894119905 119880119894119905 119876min

119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879 and119865minus119894119905 119880119894119905 119876min

119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879


532 Fitting the Parameters of Coupled Correlation FunctionWe use the data set 119865119894119905 119880119894119905 119876min

119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879 to

fit the parameter of RM-SPIP with regression method Thusthese parameters 120572119898119894 | 119894 isin 119866 119898 = 1 10 are derivedThe coupled correlation function CCF-SPCP is obtained

Again using the same approach we introduce thedata sets 119865+119894119905 119880119894119905 119876min


119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879 into the RM-DPIDP

for fitting these parameters 120573119898119894 | 119894 isin 119866 119898 = 1 10and 120573119898119894 | 119894 isin 119866 119898 = 1 10 The coupled correlationfunction CCF-DPCP is obtained

54 Computing the Preset Control Parameters of CoupledCorrelation Function

541 Computing Single Preset Control Parameter with CCF-SPCP The coupled correlation function CCF-SPCP impliesthe effect of vehicle supply 119880119895 minimum demand 119876min

119895 andmaximum demand 119876max

119895 on the profits 119865119895 As the derivativeof 119865119895(119880119895 119876min

119895 119876max119895 ) with respect to 119880119895 shows the effect of

adding one vehicle in 119895 terminal at 119905 + 1 time period throughthe rest the planning horizon 120575119895119905+1 is given by

120575119895119905+1 = 120597119865119895 (119880119895 119876min119895 119876max

119895 )12059711988011989510038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

(20)

Accordingly the solution approach of preset incrementalprofit parameter (PIPP) is shown in following formula

120575119895119905+1 = 120597119865119895 (119880119895 119876min119895 119876max

119895 )120597119880119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205721119895119880119895 + 1205724119895119876min119895 + 1205725119895119876max

119895

+ 1205727119895 10038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205721119895119880119895119905+1 + 1205724119895119876min119895119905+1 + 1205725119895119876max

119895119905+1 + 1205727119895

(21)

542 Computing Double Preset Control Parameter with CCF-DPCP We use the same approach to compute the doublepreset control parameter of coupled correlation functionCCF-DPCP The coupled correlation function implies theeffect of vehicle supply 119880119895 minimum demand 119876min


119895 on revenue 119865+119895 and penalty cost 119865minus119895 As the derivative of 119865+119895 (119880119895 119876min

119895 119876max119895 ) with respect to 119880119895

shows the effect of adding one vehicle in 119895 terminal at 119905 + 1time period through the rest the planning horizon 120575+119895119905+1 isgiven by

120575+119895119905+1 = 120597119865+119895 (119880119895 119876min

119895 119876max119895 )120597119880119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

(22)

Accordingly the solution approach of preset incrementalrevenue parameter (PIRP) is shown in the following formula

120575+119895119905+1 = 120597119865+119895 (119880119895 119876min

119895 119876max119895 )

12059711988011989510038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205731119895119880119895 + 1205734119895119876min119895 + 1205735119895119876max

119895

+ 1205737119895 10038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205731119895119880119895119905+1 + 1205734119895119876min119895119905+1 + 1205735119895119876max

119895119905+1 + 1205737119895

(23)

Correspondingly as the derivative of 119865minus119895 (119880119895 119876min119895 119876max

119895 )with respect to 119880119895 nicely depicts the effect of adding onevehicle in 119895 terminal at 119905 + 1 time period through the rest theplanning horizon 120575minus119895119905+1 is written as

120575minus119895119905+1 = 120597119865minus119895 (119880119895 119876min

119895 119876max119895 )120597119880119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

(24)

Accordingly the solution approach of preset decrementalcost parameter (PDCP) is shown in the following formula

120575minus119895119905+1 = 120597119865minus119895 (119880119895 119876min

119895 119876max119895 )120597119880119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205741119895119880119895 + 1205744119895119876min119895 + 1205745119895119876max

119895

+ 1205747119895 10038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205741119895119880119895119905+1 + 1205744119895119876min119895119905+1 + 1205745119895119876max

119895119905+1 + 1205747119895

(25)

6 Piecewise Method by Updating PresetControl Parameters

In this section we develop a solution approach based onupdating preset control parameters An overview of theframework of piecewise method by updating preset controlparameters (PM-PCP) is explained as follows

Stage 1 (sampling data sets)

Step 11 Add a virtual source terminal into working servicenetwork The formulation of local problem for virtual timeperiod is written Optimal solutions 1199091199041198940 can be obtained byusing CPLEX for solving local problem Then initial vehicledistribution 1198801198941 | 119894 isin 119866 is obtainedStep 12 By solving local problem (LP) at each time periodwith CPLEX solver optimal solutions 119909119894119895119905 can be obtainedTaking optimal solution 119909119894119895119905 into the objective function of


local problem 119865119894119905 119865+119894119905 119865minus119894119905 can be obtained State vector 119880119894119905is obtained by updated approach of local problem

Step 13 Record these data sets 119880119894119905 119876min119894119905 119876max

119894119905 119865119894119905 119865+119894119905 119865minus119894119905 ineach terminal at each time period

Step 14 As such repeat Steps 11 to 13 for the whole planninghorizon In order to fit parameters of coupled correlationfunction these data are split into three data sets that is119865119894119905 119880119894119905 119876min

119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879 119865+119894119905 119880119894119905 119876min

119894119905 119876max119894119905 |119894 isin 119866 119905 isin 119879 and 119865minus119894119905 119880119894119905 119876min

119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879

Stage 2 (coupled correlation formulation)

Step 21 Using the data set 119865119894119905 119880119894119905 119876min119894119905 119876max

119894119905 | 119894 isin 119866 119905 isin 119879to fit these parameters of CCF-SPCP by regression methodthe coupled correlation function CCF-SPCP is formed

Step 22 Using the data sets 119865+119894119905 119880119894119905 119876min119894119905 119876max

119894119905 | 119894 isin 119866 119905 isin119879 and 119865+119894119905 119880119894119905 119876min119894119905 119876max

119894119905 | 119894 isin 119866 119905 isin 119879 to fit theseparameters of CCF-DPCP by regressionmethod the coupledcorrelation function CCF-DPCP is formed

Stage 3 (piecewise method guided by preset control parame-ters)

Step 31 Single preset control parameters of RM-SPIP 120575119895119905+1are computed by formula (21)

Step 32 Taking incremental profit parameters 120575119895119905+1 into theobjective function of MIP the piecewise form of the model(RM-SPIP) is given The new solution 119909new119894119895119905 is obtained byresolving RM-SPIP using CPLEX solver for the beginningof the 1st period until the end of an appropriate planninghorizon119872 Further 119880new

119894119905 are obtained

Step 33 Double preset control parameters of RM-DPIDP120575+119895119905+1 and 120575minus119895119905+1 are computed by formulas (23) and (25)

Step 34 Taking preset incremental revenue parameter 120575+119895119905+1and preset decremental cost parameter 120575minus119895119905+1 into the objec-tive function of MIP the piecewise form of the model (RM-DPIDP) is given The new solution 119909new119894119895119905 is obtained byresolving RM-DPIDP using CPLEX solver for the beginningof the 1st period until the end of an appropriate planninghorizon119872 Further 119880new


7 Numerical Study

In this section we try to evaluate the quality of the PM-PCPmethod in terms of traditionalmeasure such as objectivefunction and execution time Section 71 describes the exper-imental design and Sections 72ndash75 report the numericalresults

71 Instances and Test Settings This section describes the dataused in the numerical testing of the models For each vehiclethe region and the time of first availability have to be knownIn this data set theworking service network is composed of 10

terminals and of fixed-length links joining them The lengthof the planning horizon is 50-time periodThe length of eachtime period is constant 60-time unit

All vehicles are assumed to be of the same type and alldemands can be met from that type of vehicleTheminimumdemands at each terminal are assumed to follow Poissondistributions withmean 250Themaximum demands at eachterminal are assumed to follow Poisson distributions withmean 400 The revenue for one unit of met demands is 40dollar The penalty cost for one unit of unmet demand is 18dollar The fixed cost for owning or leasing vehicle is 50000dollar per vehicle The demands which can be serviced byvehicle are 100 units per vehicle The distance between anypair of terminal are assumed to uniform distributions withmean 300Themaximum coverage distance for any vehicle atany terminal is 500 meters

In the following the PM-PCP program for RM-SPIP andRM-DPIDP is coded by using MATLAB 2014 Edition APentium IV 34GHz processor with 2GBmemory is used forthe computation For solving the MIP CPLEX solver is alsoused We compare the three models using the test instancesand evaluate the performance of the MIP model RM-SPIPmodel and RM-DPIDP model

72 Performance Evaluation In this section the major crite-rion in assessing the performance of the models MIP RM-SPIP and RM-DPIDP is the profit generated by revenuesfor assigning vehicles penalty costs for unmet demand andownership costs for owning vehicle in planning horizon ThePM-PCP procedure is coded by usingMATLAB 2014 Editionto solve the RM-SPIP and RM-DPIDP The MIP model issolved by CPLEX

In the experiment we test the performance of the solutionprocedure on working service network At each iteration theobjective function value for each time period is recordedWhen the models MIP RM-SPIP and RM-DPIDP are com-pared the difference in total profit is very clear The RM-SPIP and RM-DPIDP model can generate higher the totalprofit than MIP model Furthermore we observe that thesolution obtained fromRM-DPIDP outperforms the solutionapproaches from RM-SPIP The results obtained by RM-DPDIP RM-SPIP and MIP are displayed in Figure 3

73 Evolution of the Preset Control Parameters The presetcontrol parameters are important for the RM-SPIP modeland RM-DPIDP model In this section we indicate theevolution of the preset control parameters for whole planninghorizon For the RM-SPIP model and RM-DPIDP modelthe following preset control parameters are reported presetincremental profit parameter (PIPP) for RM-SPIP modeland preset incremental revenue parameter (PIRP) and presetdecremental cost parameter (PDCP) for RM-DPIDP modelFigure 4 shows the evolution of three types preset controlparameters through 50-time period

74 Numerical Results on Instances for Different Length ofPlanning Horizon In this section we use two measures ofperformance The first one is the OPT which is the value of


Table 1 Performance for MIP RM-SPIP and RM- DPIDP model applied to different working service station size

Number of service station OPT difference ($) CPU time (s)MIP RM-SPIP RM-DPIDP MIP RM-SPIP RM-DPIDP

3 232527 365392 577442 4474 6891 90065 282930 362955 617750 4621 6890 91098 311313 463953 635832 4758 7174 900210 325420 507720 736300 5028 7021 943413 399359 528024 819622 5100 7579 937415 457170 627995 830250 9374 7776 985418 466665 707605 978812 5803 8108 1015220 518180 713780 1049600 6020 8053 1006923 603231 802696 1103402 6205 8759 1072425 668450 915075 1244350 6606 9054 1008028 719057 1013297 1303392 6786 9033 1139330 707980 1031880 1404500 7051 9877 1068733 844143 1189408 1508782 7667 10430 1215735 846770 1264195 1710050 7956 10524 1248938 988489 1301029 1809572 8008 11451 1251640 1034820 1462020 1981000 8721 11893 1338643 1102095 1538160 2105762 9208 12094 1397145 1142130 1675355 2277350 9145 13085 1437948 1294961 1780801 2407352 10067 13360 1472150 1308700 1904200 2599100 10428 14401 15468

Tota

l pro

fit

RM-DPDIP model

MIP model

RM-SPIP model

times104

2

3

4

5

6

7

8

5 4015 20 25 30 35 45 50100Time period

Figure 3 Comparison of models RM-DPDIP RM-SPIP and MIP

the objective function obtained by the MIP and the optimalvalue obtained by RM-SPIP and RM-DPIDP The secondmeasure of performance is the CUP time to run CPLEXsolver forMIPmodel and the PM-PCPprogram for RM-SPIPmodel and RM-DPIDP model

Dynamic working vehicle scheduling with dual demandsservice network (DWVS-DD) for different length of planninghorizon is respectively solved by models MIP RM-SPIPand RM-DPIDP For small time period size (up to 5 timeperiod) the solving RM-SPIP and RM-DPIDP model cangenerally result in slightly higher total profits than that ofMIPmodel Nevertheless for bigger time period size (up to 50 time

period) the solution of RM-SPIP and RM-DPIDPmodel canobviously maintain higher total profits than that of MIP TheOPT performance is shown in Figure 5

Additional measures are the CPU time The requiredCPU time is reported to indicate the usefulness of modelsMIP RM-SPIP and RM-DPIDP These times include theprocessing time needed to solve the RM-SPIP and RM-DPIDP model by PM-PCP program and solve the MIPmodel by CPLEX program The computational results of theperformance of the models are shown in Figure 6

75 Numerical Results on Instances for Working Service Sta-tion Size In this section two measures of performanceare adopted The first one is the OPT difference which isthe difference between the value of the objective functionobtained by MIP model and the optimal value obtainedby RM-SPIP and RM-DPIDP model The second measureof performance is the CUP time difference which is thedifference between the CPU time to find the optimal solutionof MIP model by using CPLEX solver and the CPU timeto run the PM-PCP program for RM-SPIP and RM-DPIDPmodel

When the models MIP RM-SPIP and RM-DPIDP arecompared the difference in total profit is very clear Mean-while the OPT difference will increase with the workingservice station size In other words with increasing workingservice station size the OPT difference will also increaseTheresults for the OPT difference of different working servicestation size are listed in Table 1

Furthermore we have to look at the following affect inCPU time difference Here DWVS-DD size is described by


Pres

et in

crem

enta

l pro

fit p

aram

eter

times103

4

45

5

55

6

65

7

5 10 15 20 25 30 35 40 45 500Time period

(a) Single preset parameters (preset incremental profit parameter)

Pres

et in

crem

enta

l rev

enue

par

amet

er

times103

55

6

65

7

75

8

85

5 10 15 20 25 4035 45 50300Time period

(b) Double preset parameters (preset incremental revenue parameter)

Pres

et d

ecre

men

tal c

ost p

aram

eter

times103

1012141618202224262830

5 10 15 20 25 30 35 40 45 500Time period

(c) Double preset parameters (preset decremental cost parameter)

Figure 4 Dynamic change of preset increment parameters

Cum

ulat

ive t

otal

pro

fit RM-DPIDP

RM-SPIP

MIP

times105

0

1

2

3

4

5

6

7

8

5 10 15 20 25 30 35 40 45 500Time period

Figure 5 The OPT performance of 3 models for different length ofplanning horizon

working service station size The CUP time difference fordifferent DWVS-DD size is described in Table 1

Using polynomial curve fitting to the OPT data canprovide good results The results are shown in Figure 7Referring to the results obtained by using the PM-PCP

Com

pute

r tim

e

RM-DPIDP

RM-SPIP

MIP

0100200300400500600700800900

5 10 15 20 25 30 35 40 45 500Time period

Figure 6 CPU time of 3 models for different length of planninghorizon

program for RM-SPIP and RM-DPIDP model we observethat the quality of the OPT value is improved In comparisonthe performance of PM-PCP program for RM-SPIP and RM-DPIDP is very significant when the scale of the problembecomes relatively large


OPT

diff

eren

ce

Fitting curve for OPT of MIPFitting curve for OPT of RM-SPIPFitting curve for OPT of RM-DPIDP

times106

0

05

1

15

2

25

3

5 10 15 20 25 30 35 40 45 500Working service station size

Figure 7 Fitting curve of OPT for 3 models

Fitting curve for CPU time of MIPFitting curve for CPU time of RM-SPIPFitting curve for CPU time of RM-DPIDP

CPU

tim

e

times102

4

6

8

10

12

14

16

18


Figure 8 Fitting curve of CUP time for 3 models

Furthermore using polynomial curve fitting to the CUPtime data can also provide good results The results aredisplayed in Figure 8 In comparison along with the increasein scale of the problem CPU time of PM-PCP program forRM-SPIP and RM-DPIDP slightly increases

8 Conclusions

In this paper a mixed integer programming model has beendeveloped for DWVS-DD Instead of a large integer programthe problem is decomposed into small local problems thatare guided by preset control parameters The preset controlparameters result in decoupling the local problems for dif-ferent time periods Then we propose two types of presetcontrol parameters namely single preset control parameters(SPCP) and double preset control parameters (DPCP) Byintroducing them into the MIP model the models are then

reformulated as a piecewise form namely RM-SPIP andRM-DPIDP According to the specific structure of the RM-SPIP and RM-DPIDP piecewise method by updating presetcontrol parameters (PM-PCP) is developed

The primary goal of this paper is to set up a newmodel ofthe DWVS-DD and solve it in an effective and efficient wayTests have been conducted to examine the performance of thePM-PCP program for the proposed new model

Future research can focus on multiple vehicle and servicetypes The assumption of multiple vehicle and service typesadds considerable complexity to the problem of DWVS-DDIn spite of this we have shown that the PM-PCP approach canhandle very big problems and provide high-quality integersolutions

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (Grant no U1604150) and Humani-ties amp Social Sciences Research Foundation of Ministry ofEducation of China (Grant no 15YJC630148) The support isgratefully acknowledged

References

[1] Y M Hou Z H Jia X Tian and F F Wei ldquoResearch on vehiclerouting problem with soft time windowsrdquo Journal of SystemsEngineering vol 30 no 2 pp 240ndash250 2015

[2] X X Cao J F Tang and L L Liu ldquoAn accurate algorithmbased on set partitioning for airport shuttle vehicle schedulingproblemrdquo Systems Engineering Theory and Practice vol 33 no7 pp 1682ndash1689 2013

[3] K Li L X Tang and S F Chen ldquoModeling and optimizationof spatial allocation and vehicle scheduling problem in multicontainer yardrdquo System EngineeringTheory and Practice vol 34no 1 pp 115ndash121 2014

[4] S Dabia S Ropke T Van Woensel and T De Kok ldquoBranchand price for the time-dependent vehicle routing problem withtime windowsrdquo Transportation Science vol 47 no 3 pp 380ndash396 2011

[5] J Han C Lee and S Park ldquoA robust scenario approachfor the vehicle routing problem with uncertain travel timesrdquoTransportation Science vol 48 no 3 pp 373ndash390 2014

[6] I Muter J-F Cordeau and G Laporte ldquoA branch-and-pricealgorithm for the multidepot vehicle routing problem withinterdepot routesrdquo Transportation Science vol 48 no 3 pp425ndash441 2014

[7] T Vidal T G Crainic M Gendreau and C Prins ldquoA unifiedsolution framework for multi-attribute vehicle routing prob-lemsrdquo European Journal of Operational Research vol 234 no3 pp 658ndash673 2014

[8] M Battarra G s Erdogan and D Vigo ldquoExact algorithms forthe clustered vehicle routing problemrdquoOperations Research vol62 no 1 pp 58ndash71 2014


[9] J Zak A Redmer and P Sawicki ldquoMultiple objective optimiza-tion of the fleet sizing problem for road freight transportationrdquoJournal of Advanced Transportation vol 45 no 4 pp 321ndash3472011

[10] A Shyshou I Gribkovskaia and J Barcelo ldquoA simulation studyof the fleet sizing problem arising in offshore anchor handlingoperationsrdquo European Journal of Operational Research vol 203no 1 pp 230ndash240 2010

[11] A Rahimi-Vahed T G Crainic M Gendreau and W ReildquoFleet-sizing for multi-depot and periodic vehicle routingproblems using a modular heuristic algorithmrdquo Computers ampOperations Research vol 53 pp 9ndash23 2015

[12] K Ertogral A Akbalik and S Gonzalez ldquoModelling andanalysis of a strategic fleet sizing problem for a furnituredistributorrdquo European Journal of Industrial Engineering vol 11no 1 pp 49ndash77 2017

[13] A Ferrara E Gebennini and A Grassi ldquoFleet sizing of laserguided vehicles and pallet shuttles in automated warehousesrdquoInternational Journal of Production Economics vol 157 no 1 pp7ndash14 2014

[14] K-H Chang Y-H Huang and S-P Yang ldquoVehicle fleetsizing for automated material handling systems to minimizecost subject to time constraintsrdquo IIE Transactions (Institute ofIndustrial Engineers) vol 46 no 3 pp 301ndash312 2014

[15] J A Barrios and J D Godier ldquoFleet sizing for flexible carsharingsystems simulation-based approachrdquo Transportation ResearchRecord vol 2416 pp 1ndash9 2014

[16] C Koc T Bektas O Jabali and G Laporte ldquoThe fleet size andmix location-routing problemwith timewindows formulationsand a heuristic algorithmrdquo European Journal of OperationalResearch vol 248 no 1 pp 33ndash51 2016

[17] S J Park and D S Kim ldquoContainer fleet-sizing for parttransportation and storage in a two-level supply chainrdquo Journalof the Operational Research Society vol 66 no 9 pp 1442ndash14532015

[18] J Xia K X Li H Ma and Z Xu ldquoJoint planning of fleetdeployment speed optimization and cargo allocation for linershippingrdquo Transportation Science vol 49 no 4 pp 922ndash9382015

[19] J P Pita N Adler and A P Antunes ldquoSocially-oriented flightscheduling and fleet assignment model with an application toNorwayrdquo Transportation Research Part B Methodological vol61 pp 17ndash32 2014

[20] V L Pilla J M Rosenberger V Chen N Engsuwan and S Sid-dappa ldquoAmultivariate adaptive regression splines cutting planeapproach for solving a two-stage stochastic programming fleetassignment modelrdquo European Journal of Operational Researchvol 216 no 1 pp 162ndash171 2012

[21] Z Liang and W A Chaovalitwongse ldquoA network-based modelfor the integrated weekly aircraft maintenance routing and fleetassignment problemrdquo Transportation Science vol 47 no 4 pp493ndash507 2012

[22] H D Sherali K-H Bae and M Haouari ldquoAn integratedapproach for airline flight selection and timing fleet assign-ment and aircraft routingrdquo Transportation Science vol 47 no4 pp 455ndash476 2013

[23] K Tierney B Askelsdottir R M Jensen and D PisingerldquoSolving the liner shipping fleet repositioning problem withcargo flowsrdquo Transportation Science vol 49 no 3 pp 652ndash6742015

[24] A S Hashemi and J Sattarvand ldquoSimulation based investi-gation of different fleet management paradigms in open pit

mines-a case study of Sungun copper minerdquo Archives of MiningSciences vol 60 no 1 pp 195ndash208 2015

[25] J T Lin C H Wu and C W Huang ldquoDynamic vehicleallocation control for automated material handling systemin semiconductor manufacturingrdquo Computers amp OperationsResearch vol 40 no 10 pp 2329ndash2339 2013

[26] H P Simao J Day A P George T Gifford J Nienowand W B Powell ldquoAn approximate dynamic programmingalgorithm for large-scale fleet management A case applicationrdquoTransportation Science vol 43 no 2 pp 178ndash197 2009

[27] H Topaloglu and W Powell ldquoIncorporating pricing decisionsinto the stochastic dynamic fleet management problemrdquo Trans-portation Science vol 41 no 3 pp 281ndash301 2007

[28] B Li H Xuan and J Li ldquoAlternating solution strategies of bi-level programming model for stochastic dynamic fleet schedul-ing problem with variable period and storage propertiesrdquoKongzhi yu JueceControl and Decision vol 30 no 5 pp 807ndash814 2015

[29] B Li H Xuan and J Li ldquoSolving strategies for the stochasticdynamic fleet scheduling problem based on leading of parame-tersrdquo Journal of Systems Engineering vol 31 no 4 pp 545ndash5562016

[30] B Li and H Xuan ldquoSolving strategy for stochastic dynamicfleet scheduling with station operation coordinationrdquo Kongzhiyu JueceControl and Decision vol 32 no 1 pp 71ndash78 2017

[31] M-J Yao and J-Y Huang ldquoScheduling of transportation fleetmaintenance service by an improved Lipschitz optimizationalgorithmrdquoOptimization Methods amp Software vol 29 no 3 pp592ndash609 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of


International Journal of

RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



demand the penalty cost for unmet demand will generateThis characteristic is the cornerstones of themodel developedin this paper

Second vehicles usually provide pickup or delivery ser-vices between various locations in previous studies Howeverin reality vehicles themselves can sometimes act like a facilityto provide real-time services when they are stationary at onelocation The vehicles cannot provide services when they arein motion and the service begins when a vehicle arrives ata location and ends when it departs For instance medicaltreatment vehicles provide first aid services to areas where theestablished medical facility is temporarily insufficient Alsofood trucks provide fast food services in different regionsin different time periods of the day Note that when thesevehicles are in service they behave like traditional facilitiesThe term working vehicle (WV) will be used in this paperto denote this vehicle Applications of problems arise inmany settings ranging from managing emergency vehiclesmedical testing vehicles traveling salesman and militaryforce deployment

Overall the objectives of this research are twofold Thefirst objective is to develop a novel mathematical modelof DWVS-DD In conventional mathematical models theproblem has been formulated as a mixed integer program-ming model In the proposed model the problem is refor-mulated as a piecewise form to find the local problem atevery time period The second objective is to propose anefficient methodology for solving the model Specifically thecontributions of this paper are as follows

(1) We develop the mixed integer programming model(MIP) The model then is reformulated as two novelformulations that is reconstruction model with sin-gle preset incremental parameters (RM-SPIP) andreconstruction model with double preset incremen-tal-decremental parameters (RM-DPIDP)

(2) We propose the coupled correlation model of totalprofit vehicle supply minimum demand and maxi-mum demand The acquisition approaches of singlepreset control parameters are given Meanwhile wepropose the coupled correlationmodel of the revenuepenalty cost vehicle supply minimum demand andmaximum demand The acquisition approaches ofdouble preset control parameters are given

(3) According to the specific structure of the two recon-struction models the piecewise method by updatingpreset control parameters (PM-PCP) is developed

(4) We tested the performance of PM-PCP approach bysolving many instances We compared the qualityof the solutions provided by PM-PCP solving RM-SPIP and RM-DPIDP versus the results of the CPLEXsolving MIP According to our results in most of theinstances our PM-PCP approach has a better per-formance Indeed PM-PCP provides many optimalsolutions in a very short computation time

The remainder of this paper is organized as followsSection 2 of this paper discusses related earlier researchefforts Section 3 is devoted to the mathematical description

of the DWVS-DD In Section 4 we set up the problem asthe classical mathematical programming formulation andpresent then the reformulated models Section 5 explains theacquisition methodology of preset control parameters basedon coupled correlation function We introduce piecewisemethod by updating preset parameters (PM-PCP) for solvingreconstructionmodel in Section 6The computational exper-iments are described in Section 7 and the effectiveness of theproposed method is shown from the computational resultsThe last section concludes with a summary of current workand extensions

2 Literature Review

In this section we review the relevant literatures about vehiclescheduling problems Literature review indicates that vehiclescheduling problems can be divided into three groups that isvehicle routing fleet sizing and fleet assignment The focusof our literature review will primarily be on the model andexact approach since it is also the model and approach wehave taken in this paper

In recent years many researches on vehicle routingoptimization have been carried out Hou et al [1] focused onvehicle routing problem with soft time window constraintAn exact algorithm based on set partition was proposedto solve the balancing the vehicle number and customersatisfaction by Cao et al [2] Li et al [3] studied the integratedproblem with truck scheduling and storage allocation It wasformulated as an integer programming model to minimizemakespan of the whole discharging course and solved by atwo stages tabu search algorithm Dabia et al [4] presenteda branch and price algorithm for time-dependent vehiclerouting problemwith timewindowsHan et al [5] considereda vehicle routing problem with uncertain travel times inwhich a penalty is incurred for each vehicle that exceeds agiven time limit and given robust scenario approach for thevehicle routing problem Muter et al [6] proposed a columngeneration algorithm for the multidepot vehicle routingproblem with interdepot routes Vidal et al [7] proposeda unified hybrid genetic search metaheuristic algorithm tosolve multiattribute vehicle routing problems Battarra etal [8] presented new exact algorithms for the clusteredvehicle routing problem (CluVRP) and provided two exactalgorithms for the problem that is a branch and cut as well asa branch and cut and price

Fleet sizing is one of the most important decisions as itis a major fixed investment for starting any business Manyscholars have conducted the numerous basic studies on fleetsizing problem (FSP) Zak et al studied a fleet sizing problemin a road freight transportation company with heterogeneousfleet [9] Additionally themathematicalmodel of the decisionproblem was formulated in terms of multiple objectivemathematical programming based on queuing theory Afleet sizing problem arising in anchor handling operationsrelated to movement of offshore mobile units is presentedby Shyshou et al [10] A simulation-based prototype wasproposed and the simulation model was implemented inArena 90 (a simulation software package developed byRockwell Software) Rahimi-Vahed et al [11] addressed the














1 11

QＧＣＨkt Q

ＧＲkt

dik le D

QＧＣＨit Q

ＧＲit

QＧＣＨ

k tQ

ＧＲ

k t

k

QＧＣＨkM Q

ＧＲkM

QＧＣＨi1

QＧＲi1

QＧＣＨ

kMQ

ＧＲ

kM



i

N N N

dik le D

k

k

i

dik le D

dik le D

i

k

















[MIP] max 119865 (119909)= 120585sum119905isin119879

sum119894isin119866

sum119895isin119866

119886119905119909119894119895119905


sum119895isin119866

119887119905(119876max119895119905 minus 120585sum

119894isin119866

119909119894119895119905) minus 119888sdot 119880

(1)


sum119895isin119866

119909119894119895119905 le 119880 forall119905 isin 119879 (2)

120585sum119894isin119866


120585sum119894isin119866

119909119894119895119905 le sum119896isin119866

120572119895119896119876max119896119905+1




sum119894isin119866

119909119894119895119905 = sum119894isin119866

119909119895119894(119905+1) forall119905 isin 119879 (6)

119880 119909119895119894119905 ge 0 and integer





max119865119905 (119909) = 120585sum119894isin119866

sum119895isin119866

119886119905119909119894119895119905


119887119905(119876max119895119905 minus 120585sum

119894isin119866

119909119894119895119905) minus 119888 sdot 119880+ 119864 [119865119905+1 (119909)]

(8)






119864 [119865119905+1 (119909)] = sum119895isin119866

120575119895+1119880119895119905+1 (9)

We also note that

119880119895119905+1 = sum119894isin119866

119909119894119895119905 (10)


119895isin119866

120575119895119905+1sum119894isin119866

119909119894119895119905 (11)



sum119895isin119866

119886119905119909119894119895119905


119887119905(119876max119895119905 minus 120585sum

119894isin119866

119909119894119895119905)minus 119888 sdot 119880 + sum

119895isin119866

120575119895119905+1sum119894isin119866

119909119894119895119905= sum119894isin119866

sum119895isin119866

(119886119905120585 + 119887119905120585 + 120575119895119905+1) 119909119894119895119905minus sum119895isin119866

119887119905119876max119895119905 minus 119888 sdot 119880


(12)






119864 [119865119905+1 (119909)] = sum119895isin119866

120575+119895119905+1119880119895119905+1 + sum119895isin119866

120575minus119895119905+1119880119895119905+1 (13)


119864 [119865119905+1 (119909)] = sum119895isin119866

120575+119895119905+1sum119894isin119866

119909119894119895119905 + sum119895isin119866

120575minus119895119905+1sum119894isin119866

119909119894119895119905 (14)



sum119895isin119866

119886119905119909119894119895119905


119887119905(119876max119895119905 minus 120585sum

119894isin119866

119909119894119895119905)minus 119888 sdot 119880 + sum

119895isin119866

120575+119895119905+1sum119894isin119866

119909119894119895119905+ sum119895isin119866

120575minus119895119905+1sum119894isin119866

119909119894119895119905= sum119894isin119866

sum119895isin119866

(119886119905120585 + 120575+119895119905+1) 119909119894119895119905minus sum119895isin119866

119887119905119876max119895119905

+ sum119894isin119866

sum119895isin119866

(119887119905120585 + 120575minus119895119905+1) 119909119894119895119905minus 119888 sdot 119880


(15)



s


Ui1

Uk 1

UN1

k

QＧＣＨk1 Q

ＧＲk1

dik le D

dik le D

QＧＲi1Q

ＧＣＨi1

QＧＣＨ

k1Q

ＧＲ

k1

U11

Uk1k

i

N





1198941 and 119876max1198941





119894isin119866

1199091199041198940 = 119880120585 sdot 1199091199041198940 ge 119876min


119896isin119866




(16)






119895119905+1 and 119876max119895119905+1


[LP] max 119865119905= 120585sum119894isin119866

119886119905119880119894119905 minus sum119894isin119866

119887119905 (119876max119894119905 minus 120585119880119894119905)

minus 119888sum119894isin119866

119880119894119905subject to sum

119895isin119866



120585sum119894isin119866

119909119894119895119905 le sum119896isin119866

120572119895119896119876max119896119905+1 forall119895 isin 119866



(17)





[CCF-SPCP] 119865119894 (119880119894 119876min119894 119876max

119894 )= 1205721119894 (119880119894)2 + 1205722119894 (119876min

119894 )2 + 1205723119894 (119876max119894 )2 + 1205724119894 119880119894119876min

119894

+ 1205725119894 119880119894119876max119894 + 1205726119894 119876min

119894 119876max119894 + 1205727119894 119880119894 + 1205728119894 119876min

119894

+ 1205729119894 119876max119894 + 12057210119894 forall119894 isin 119866

(18)





[CCF-DPCP] 119865+119894 (119880119894 119876min119894 119876max

119894 )= 1205731119894 (119880119894)2 + 1205732119894 (119876min

119894 )2+ 1205733119894 (119876max

119894 )2 + 1205734119894 119880119894119876min119894

+ 1205735119894 119880119894119876max119894 + 1205736119894 119876min

119894 119876max119894

+ 1205737119894 119880119894 + 1205738119894 119876min119894 + 1205739119894 119876max

119894 + 12057310119894 forall119894 isin 119866

119865minus119894 (119880119894 119876min119894 119876max

119894 )= 1205741119894 (119880119894)2 + 1205742119894 (119876min

119894 )2+ 1205743119894 (119876max

119894 )2 + 1205744119894 119880119894119876min119894

+ 1205745119894 119880119894119876max119894 + 1205746119894 119876min

119894 119876max119894 + 1205747119894 119880119894

+ 1205748119894 119876min119894 + 1205749119894 119876max

119894 + 12057410119894 119894 isin 119866

(19)




119894119905 119876max119894119905 | 119894 isin119866 119905 isin 119879 119865+119894119905 119880119894119905 119876min


119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879



119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879 to












120575119895119905+1 = 120597119865119895 (119880119895 119876min119895 119876max

119895 )12059711988011989510038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

(20)


120575119895119905+1 = 120597119865119895 (119880119895 119876min119895 119876max

119895 )120597119880119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205721119895119880119895 + 1205724119895119876min119895 + 1205725119895119876max

119895

+ 1205727119895 10038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205721119895119880119895119905+1 + 1205724119895119876min119895119905+1 + 1205725119895119876max

119895119905+1 + 1205727119895

(21)






120575+119895119905+1 = 120597119865+119895 (119880119895 119876min

119895 119876max119895 )120597119880119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

(22)


120575+119895119905+1 = 120597119865+119895 (119880119895 119876min

119895 119876max119895 )

12059711988011989510038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205731119895119880119895 + 1205734119895119876min119895 + 1205735119895119876max

119895

+ 1205737119895 10038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205731119895119880119895119905+1 + 1205734119895119876min119895119905+1 + 1205735119895119876max

119895119905+1 + 1205737119895

(23)



120575minus119895119905+1 = 120597119865minus119895 (119880119895 119876min

119895 119876max119895 )120597119880119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

(24)


120575minus119895119905+1 = 120597119865minus119895 (119880119895 119876min

119895 119876max119895 )120597119880119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205741119895119880119895 + 1205744119895119876min119895 + 1205745119895119876max

119895

+ 1205747119895 10038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205741119895119880119895119905+1 + 1205744119895119876min119895119905+1 + 1205745119895119876max

119895119905+1 + 1205747119895

(25)










119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879 119865+119894119905 119880119894119905 119876min


119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879














7 Numerical Study













3 232527 365392 577442 4474 6891 90065 282930 362955 617750 4621 6890 91098 311313 463953 635832 4758 7174 900210 325420 507720 736300 5028 7021 943413 399359 528024 819622 5100 7579 937415 457170 627995 830250 9374 7776 985418 466665 707605 978812 5803 8108 1015220 518180 713780 1049600 6020 8053 1006923 603231 802696 1103402 6205 8759 1072425 668450 915075 1244350 6606 9054 1008028 719057 1013297 1303392 6786 9033 1139330 707980 1031880 1404500 7051 9877 1068733 844143 1189408 1508782 7667 10430 1215735 846770 1264195 1710050 7956 10524 1248938 988489 1301029 1809572 8008 11451 1251640 1034820 1462020 1981000 8721 11893 1338643 1102095 1538160 2105762 9208 12094 1397145 1142130 1675355 2277350 9145 13085 1437948 1294961 1780801 2407352 10067 13360 1472150 1308700 1904200 2599100 10428 14401 15468

Tota

l pro

fit

RM-DPDIP model

MIP model

RM-SPIP model

times104

2

3

4

5

6

7

8

5 4015 20 25 30 35 45 50100Time period










Pres

et in

crem

enta

l pro

fit p

aram

eter

times103

4

45

5

55

6

65

7

5 10 15 20 25 30 35 40 45 500Time period


Pres

et in

crem

enta

l rev

enue

par

amet

er

times103

55

6

65

7

75

8

85

5 10 15 20 25 4035 45 50300Time period


Pres

et d

ecre

men

tal c

ost p

aram

eter

times103

1012141618202224262830

5 10 15 20 25 30 35 40 45 500Time period



Cum

ulat

ive t

otal

pro

fit RM-DPIDP

RM-SPIP

MIP

times105

0

1

2

3

4

5

6

7

8

5 10 15 20 25 30 35 40 45 500Time period




Com

pute

r tim

e

RM-DPIDP

RM-SPIP

MIP

0100200300400500600700800900

5 10 15 20 25 30 35 40 45 500Time period




OPT

diff

eren

ce


times106

0

05

1

15

2

25

3




CPU

tim

e

times102

4

6

8

10

12

14

16

18




8 Conclusions







Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






















1 11

QＧＣＨkt Q

ＧＲkt

dik le D

QＧＣＨit Q

ＧＲit

QＧＣＨ

k tQ

ＧＲ

k t

k

QＧＣＨkM Q

ＧＲkM

QＧＣＨi1

QＧＲi1

QＧＣＨ

kMQ

ＧＲ

kM



i

N N N

dik le D

k

k

i

dik le D

dik le D

i

k

















[MIP] max 119865 (119909)= 120585sum119905isin119879

sum119894isin119866

sum119895isin119866

119886119905119909119894119895119905


sum119895isin119866

119887119905(119876max119895119905 minus 120585sum

119894isin119866

119909119894119895119905) minus 119888sdot 119880

(1)


sum119895isin119866

119909119894119895119905 le 119880 forall119905 isin 119879 (2)

120585sum119894isin119866


120585sum119894isin119866

119909119894119895119905 le sum119896isin119866

120572119895119896119876max119896119905+1




sum119894isin119866

119909119894119895119905 = sum119894isin119866

119909119895119894(119905+1) forall119905 isin 119879 (6)

119880 119909119895119894119905 ge 0 and integer





max119865119905 (119909) = 120585sum119894isin119866

sum119895isin119866

119886119905119909119894119895119905


119887119905(119876max119895119905 minus 120585sum

119894isin119866

119909119894119895119905) minus 119888 sdot 119880+ 119864 [119865119905+1 (119909)]

(8)






119864 [119865119905+1 (119909)] = sum119895isin119866

120575119895+1119880119895119905+1 (9)

We also note that

119880119895119905+1 = sum119894isin119866

119909119894119895119905 (10)


119895isin119866

120575119895119905+1sum119894isin119866

119909119894119895119905 (11)



sum119895isin119866

119886119905119909119894119895119905


119887119905(119876max119895119905 minus 120585sum

119894isin119866

119909119894119895119905)minus 119888 sdot 119880 + sum

119895isin119866

120575119895119905+1sum119894isin119866

119909119894119895119905= sum119894isin119866

sum119895isin119866

(119886119905120585 + 119887119905120585 + 120575119895119905+1) 119909119894119895119905minus sum119895isin119866

119887119905119876max119895119905 minus 119888 sdot 119880


(12)






119864 [119865119905+1 (119909)] = sum119895isin119866

120575+119895119905+1119880119895119905+1 + sum119895isin119866

120575minus119895119905+1119880119895119905+1 (13)


119864 [119865119905+1 (119909)] = sum119895isin119866

120575+119895119905+1sum119894isin119866

119909119894119895119905 + sum119895isin119866

120575minus119895119905+1sum119894isin119866

119909119894119895119905 (14)



sum119895isin119866

119886119905119909119894119895119905


119887119905(119876max119895119905 minus 120585sum

119894isin119866

119909119894119895119905)minus 119888 sdot 119880 + sum

119895isin119866

120575+119895119905+1sum119894isin119866

119909119894119895119905+ sum119895isin119866

120575minus119895119905+1sum119894isin119866

119909119894119895119905= sum119894isin119866

sum119895isin119866

(119886119905120585 + 120575+119895119905+1) 119909119894119895119905minus sum119895isin119866

119887119905119876max119895119905

+ sum119894isin119866

sum119895isin119866

(119887119905120585 + 120575minus119895119905+1) 119909119894119895119905minus 119888 sdot 119880


(15)



s


Ui1

Uk 1

UN1

k

QＧＣＨk1 Q

ＧＲk1

dik le D

dik le D

QＧＲi1Q

ＧＣＨi1

QＧＣＨ

k1Q

ＧＲ

k1

U11

Uk1k

i

N





1198941 and 119876max1198941





119894isin119866

1199091199041198940 = 119880120585 sdot 1199091199041198940 ge 119876min


119896isin119866




(16)






119895119905+1 and 119876max119895119905+1


[LP] max 119865119905= 120585sum119894isin119866

119886119905119880119894119905 minus sum119894isin119866

119887119905 (119876max119894119905 minus 120585119880119894119905)

minus 119888sum119894isin119866

119880119894119905subject to sum

119895isin119866



120585sum119894isin119866

119909119894119895119905 le sum119896isin119866

120572119895119896119876max119896119905+1 forall119895 isin 119866



(17)





[CCF-SPCP] 119865119894 (119880119894 119876min119894 119876max

119894 )= 1205721119894 (119880119894)2 + 1205722119894 (119876min

119894 )2 + 1205723119894 (119876max119894 )2 + 1205724119894 119880119894119876min

119894

+ 1205725119894 119880119894119876max119894 + 1205726119894 119876min

119894 119876max119894 + 1205727119894 119880119894 + 1205728119894 119876min

119894

+ 1205729119894 119876max119894 + 12057210119894 forall119894 isin 119866

(18)





[CCF-DPCP] 119865+119894 (119880119894 119876min119894 119876max

119894 )= 1205731119894 (119880119894)2 + 1205732119894 (119876min

119894 )2+ 1205733119894 (119876max

119894 )2 + 1205734119894 119880119894119876min119894

+ 1205735119894 119880119894119876max119894 + 1205736119894 119876min

119894 119876max119894

+ 1205737119894 119880119894 + 1205738119894 119876min119894 + 1205739119894 119876max

119894 + 12057310119894 forall119894 isin 119866

119865minus119894 (119880119894 119876min119894 119876max

119894 )= 1205741119894 (119880119894)2 + 1205742119894 (119876min

119894 )2+ 1205743119894 (119876max

119894 )2 + 1205744119894 119880119894119876min119894

+ 1205745119894 119880119894119876max119894 + 1205746119894 119876min

119894 119876max119894 + 1205747119894 119880119894

+ 1205748119894 119876min119894 + 1205749119894 119876max

119894 + 12057410119894 119894 isin 119866

(19)




119894119905 119876max119894119905 | 119894 isin119866 119905 isin 119879 119865+119894119905 119880119894119905 119876min


119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879



119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879 to












120575119895119905+1 = 120597119865119895 (119880119895 119876min119895 119876max

119895 )12059711988011989510038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

(20)


120575119895119905+1 = 120597119865119895 (119880119895 119876min119895 119876max

119895 )120597119880119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205721119895119880119895 + 1205724119895119876min119895 + 1205725119895119876max

119895

+ 1205727119895 10038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205721119895119880119895119905+1 + 1205724119895119876min119895119905+1 + 1205725119895119876max

119895119905+1 + 1205727119895

(21)






120575+119895119905+1 = 120597119865+119895 (119880119895 119876min

119895 119876max119895 )120597119880119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

(22)


120575+119895119905+1 = 120597119865+119895 (119880119895 119876min

119895 119876max119895 )

12059711988011989510038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205731119895119880119895 + 1205734119895119876min119895 + 1205735119895119876max

119895

+ 1205737119895 10038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205731119895119880119895119905+1 + 1205734119895119876min119895119905+1 + 1205735119895119876max

119895119905+1 + 1205737119895

(23)



120575minus119895119905+1 = 120597119865minus119895 (119880119895 119876min

119895 119876max119895 )120597119880119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

(24)


120575minus119895119905+1 = 120597119865minus119895 (119880119895 119876min

119895 119876max119895 )120597119880119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205741119895119880119895 + 1205744119895119876min119895 + 1205745119895119876max

119895

+ 1205747119895 10038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205741119895119880119895119905+1 + 1205744119895119876min119895119905+1 + 1205745119895119876max

119895119905+1 + 1205747119895

(25)










119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879 119865+119894119905 119880119894119905 119876min


119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879














7 Numerical Study













3 232527 365392 577442 4474 6891 90065 282930 362955 617750 4621 6890 91098 311313 463953 635832 4758 7174 900210 325420 507720 736300 5028 7021 943413 399359 528024 819622 5100 7579 937415 457170 627995 830250 9374 7776 985418 466665 707605 978812 5803 8108 1015220 518180 713780 1049600 6020 8053 1006923 603231 802696 1103402 6205 8759 1072425 668450 915075 1244350 6606 9054 1008028 719057 1013297 1303392 6786 9033 1139330 707980 1031880 1404500 7051 9877 1068733 844143 1189408 1508782 7667 10430 1215735 846770 1264195 1710050 7956 10524 1248938 988489 1301029 1809572 8008 11451 1251640 1034820 1462020 1981000 8721 11893 1338643 1102095 1538160 2105762 9208 12094 1397145 1142130 1675355 2277350 9145 13085 1437948 1294961 1780801 2407352 10067 13360 1472150 1308700 1904200 2599100 10428 14401 15468

Tota

l pro

fit

RM-DPDIP model

MIP model

RM-SPIP model

times104

2

3

4

5

6

7

8

5 4015 20 25 30 35 45 50100Time period










Pres

et in

crem

enta

l pro

fit p

aram

eter

times103

4

45

5

55

6

65

7

5 10 15 20 25 30 35 40 45 500Time period


Pres

et in

crem

enta

l rev

enue

par

amet

er

times103

55

6

65

7

75

8

85

5 10 15 20 25 4035 45 50300Time period


Pres

et d

ecre

men

tal c

ost p

aram

eter

times103

1012141618202224262830

5 10 15 20 25 30 35 40 45 500Time period



Cum

ulat

ive t

otal

pro

fit RM-DPIDP

RM-SPIP

MIP

times105

0

1

2

3

4

5

6

7

8

5 10 15 20 25 30 35 40 45 500Time period




Com

pute

r tim

e

RM-DPIDP

RM-SPIP

MIP

0100200300400500600700800900

5 10 15 20 25 30 35 40 45 500Time period




OPT

diff

eren

ce


times106

0

05

1

15

2

25

3




CPU

tim

e

times102

4

6

8

10

12

14

16

18




8 Conclusions







Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











sum119894isin119866

119909119894119895119905 = sum119894isin119866

119909119895119894(119905+1) forall119905 isin 119879 (6)

119880 119909119895119894119905 ge 0 and integer





max119865119905 (119909) = 120585sum119894isin119866

sum119895isin119866

119886119905119909119894119895119905


119887119905(119876max119895119905 minus 120585sum

119894isin119866

119909119894119895119905) minus 119888 sdot 119880+ 119864 [119865119905+1 (119909)]

(8)






119864 [119865119905+1 (119909)] = sum119895isin119866

120575119895+1119880119895119905+1 (9)

We also note that

119880119895119905+1 = sum119894isin119866

119909119894119895119905 (10)


119895isin119866

120575119895119905+1sum119894isin119866

119909119894119895119905 (11)



sum119895isin119866

119886119905119909119894119895119905


119887119905(119876max119895119905 minus 120585sum

119894isin119866

119909119894119895119905)minus 119888 sdot 119880 + sum

119895isin119866

120575119895119905+1sum119894isin119866

119909119894119895119905= sum119894isin119866

sum119895isin119866

(119886119905120585 + 119887119905120585 + 120575119895119905+1) 119909119894119895119905minus sum119895isin119866

119887119905119876max119895119905 minus 119888 sdot 119880


(12)






119864 [119865119905+1 (119909)] = sum119895isin119866

120575+119895119905+1119880119895119905+1 + sum119895isin119866

120575minus119895119905+1119880119895119905+1 (13)


119864 [119865119905+1 (119909)] = sum119895isin119866

120575+119895119905+1sum119894isin119866

119909119894119895119905 + sum119895isin119866

120575minus119895119905+1sum119894isin119866

119909119894119895119905 (14)



sum119895isin119866

119886119905119909119894119895119905


119887119905(119876max119895119905 minus 120585sum

119894isin119866

119909119894119895119905)minus 119888 sdot 119880 + sum

119895isin119866

120575+119895119905+1sum119894isin119866

119909119894119895119905+ sum119895isin119866

120575minus119895119905+1sum119894isin119866

119909119894119895119905= sum119894isin119866

sum119895isin119866

(119886119905120585 + 120575+119895119905+1) 119909119894119895119905minus sum119895isin119866

119887119905119876max119895119905

+ sum119894isin119866

sum119895isin119866

(119887119905120585 + 120575minus119895119905+1) 119909119894119895119905minus 119888 sdot 119880


(15)



s


Ui1

Uk 1

UN1

k

QＧＣＨk1 Q

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dik le D

dik le D

QＧＲi1Q

ＧＣＨi1

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k1Q

ＧＲ

k1

U11

Uk1k

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N





1198941 and 119876max1198941





119894isin119866

1199091199041198940 = 119880120585 sdot 1199091199041198940 ge 119876min


119896isin119866




(16)






119895119905+1 and 119876max119895119905+1


[LP] max 119865119905= 120585sum119894isin119866

119886119905119880119894119905 minus sum119894isin119866

119887119905 (119876max119894119905 minus 120585119880119894119905)

minus 119888sum119894isin119866

119880119894119905subject to sum

119895isin119866



120585sum119894isin119866

119909119894119895119905 le sum119896isin119866

120572119895119896119876max119896119905+1 forall119895 isin 119866



(17)





[CCF-SPCP] 119865119894 (119880119894 119876min119894 119876max

119894 )= 1205721119894 (119880119894)2 + 1205722119894 (119876min

119894 )2 + 1205723119894 (119876max119894 )2 + 1205724119894 119880119894119876min

119894

+ 1205725119894 119880119894119876max119894 + 1205726119894 119876min

119894 119876max119894 + 1205727119894 119880119894 + 1205728119894 119876min

119894

+ 1205729119894 119876max119894 + 12057210119894 forall119894 isin 119866

(18)





[CCF-DPCP] 119865+119894 (119880119894 119876min119894 119876max

119894 )= 1205731119894 (119880119894)2 + 1205732119894 (119876min

119894 )2+ 1205733119894 (119876max

119894 )2 + 1205734119894 119880119894119876min119894

+ 1205735119894 119880119894119876max119894 + 1205736119894 119876min

119894 119876max119894

+ 1205737119894 119880119894 + 1205738119894 119876min119894 + 1205739119894 119876max

119894 + 12057310119894 forall119894 isin 119866

119865minus119894 (119880119894 119876min119894 119876max

119894 )= 1205741119894 (119880119894)2 + 1205742119894 (119876min

119894 )2+ 1205743119894 (119876max

119894 )2 + 1205744119894 119880119894119876min119894

+ 1205745119894 119880119894119876max119894 + 1205746119894 119876min

119894 119876max119894 + 1205747119894 119880119894

+ 1205748119894 119876min119894 + 1205749119894 119876max

119894 + 12057410119894 119894 isin 119866

(19)




119894119905 119876max119894119905 | 119894 isin119866 119905 isin 119879 119865+119894119905 119880119894119905 119876min


119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879



119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879 to












120575119895119905+1 = 120597119865119895 (119880119895 119876min119895 119876max

119895 )12059711988011989510038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

(20)


120575119895119905+1 = 120597119865119895 (119880119895 119876min119895 119876max

119895 )120597119880119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205721119895119880119895 + 1205724119895119876min119895 + 1205725119895119876max

119895

+ 1205727119895 10038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205721119895119880119895119905+1 + 1205724119895119876min119895119905+1 + 1205725119895119876max

119895119905+1 + 1205727119895

(21)






120575+119895119905+1 = 120597119865+119895 (119880119895 119876min

119895 119876max119895 )120597119880119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

(22)


120575+119895119905+1 = 120597119865+119895 (119880119895 119876min

119895 119876max119895 )

12059711988011989510038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205731119895119880119895 + 1205734119895119876min119895 + 1205735119895119876max

119895

+ 1205737119895 10038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205731119895119880119895119905+1 + 1205734119895119876min119895119905+1 + 1205735119895119876max

119895119905+1 + 1205737119895

(23)



120575minus119895119905+1 = 120597119865minus119895 (119880119895 119876min

119895 119876max119895 )120597119880119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

(24)


120575minus119895119905+1 = 120597119865minus119895 (119880119895 119876min

119895 119876max119895 )120597119880119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205741119895119880119895 + 1205744119895119876min119895 + 1205745119895119876max

119895

+ 1205747119895 10038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205741119895119880119895119905+1 + 1205744119895119876min119895119905+1 + 1205745119895119876max

119895119905+1 + 1205747119895

(25)










119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879 119865+119894119905 119880119894119905 119876min


119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879














7 Numerical Study













3 232527 365392 577442 4474 6891 90065 282930 362955 617750 4621 6890 91098 311313 463953 635832 4758 7174 900210 325420 507720 736300 5028 7021 943413 399359 528024 819622 5100 7579 937415 457170 627995 830250 9374 7776 985418 466665 707605 978812 5803 8108 1015220 518180 713780 1049600 6020 8053 1006923 603231 802696 1103402 6205 8759 1072425 668450 915075 1244350 6606 9054 1008028 719057 1013297 1303392 6786 9033 1139330 707980 1031880 1404500 7051 9877 1068733 844143 1189408 1508782 7667 10430 1215735 846770 1264195 1710050 7956 10524 1248938 988489 1301029 1809572 8008 11451 1251640 1034820 1462020 1981000 8721 11893 1338643 1102095 1538160 2105762 9208 12094 1397145 1142130 1675355 2277350 9145 13085 1437948 1294961 1780801 2407352 10067 13360 1472150 1308700 1904200 2599100 10428 14401 15468

Tota

l pro

fit

RM-DPDIP model

MIP model

RM-SPIP model

times104

2

3

4

5

6

7

8

5 4015 20 25 30 35 45 50100Time period










Pres

et in

crem

enta

l pro

fit p

aram

eter

times103

4

45

5

55

6

65

7

5 10 15 20 25 30 35 40 45 500Time period


Pres

et in

crem

enta

l rev

enue

par

amet

er

times103

55

6

65

7

75

8

85

5 10 15 20 25 4035 45 50300Time period


Pres

et d

ecre

men

tal c

ost p

aram

eter

times103

1012141618202224262830

5 10 15 20 25 30 35 40 45 500Time period



Cum

ulat

ive t

otal

pro

fit RM-DPIDP

RM-SPIP

MIP

times105

0

1

2

3

4

5

6

7

8

5 10 15 20 25 30 35 40 45 500Time period




Com

pute

r tim

e

RM-DPIDP

RM-SPIP

MIP

0100200300400500600700800900

5 10 15 20 25 30 35 40 45 500Time period




OPT

diff

eren

ce


times106

0

05

1

15

2

25

3




CPU

tim

e

times102

4

6

8

10

12

14

16

18




8 Conclusions







Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119864 [119865119905+1 (119909)] = sum119895isin119866

120575+119895119905+1119880119895119905+1 + sum119895isin119866

120575minus119895119905+1119880119895119905+1 (13)


119864 [119865119905+1 (119909)] = sum119895isin119866

120575+119895119905+1sum119894isin119866

119909119894119895119905 + sum119895isin119866

120575minus119895119905+1sum119894isin119866

119909119894119895119905 (14)



sum119895isin119866

119886119905119909119894119895119905


119887119905(119876max119895119905 minus 120585sum

119894isin119866

119909119894119895119905)minus 119888 sdot 119880 + sum

119895isin119866

120575+119895119905+1sum119894isin119866

119909119894119895119905+ sum119895isin119866

120575minus119895119905+1sum119894isin119866

119909119894119895119905= sum119894isin119866

sum119895isin119866

(119886119905120585 + 120575+119895119905+1) 119909119894119895119905minus sum119895isin119866

119887119905119876max119895119905

+ sum119894isin119866

sum119895isin119866

(119887119905120585 + 120575minus119895119905+1) 119909119894119895119905minus 119888 sdot 119880


(15)



s


Ui1

Uk 1

UN1

k

QＧＣＨk1 Q

ＧＲk1

dik le D

dik le D

QＧＲi1Q

ＧＣＨi1

QＧＣＨ

k1Q

ＧＲ

k1

U11

Uk1k

i

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1198941 and 119876max1198941





119894isin119866

1199091199041198940 = 119880120585 sdot 1199091199041198940 ge 119876min


119896isin119866




(16)






119895119905+1 and 119876max119895119905+1


[LP] max 119865119905= 120585sum119894isin119866

119886119905119880119894119905 minus sum119894isin119866

119887119905 (119876max119894119905 minus 120585119880119894119905)

minus 119888sum119894isin119866

119880119894119905subject to sum

119895isin119866



120585sum119894isin119866

119909119894119895119905 le sum119896isin119866

120572119895119896119876max119896119905+1 forall119895 isin 119866



(17)





[CCF-SPCP] 119865119894 (119880119894 119876min119894 119876max

119894 )= 1205721119894 (119880119894)2 + 1205722119894 (119876min

119894 )2 + 1205723119894 (119876max119894 )2 + 1205724119894 119880119894119876min

119894

+ 1205725119894 119880119894119876max119894 + 1205726119894 119876min

119894 119876max119894 + 1205727119894 119880119894 + 1205728119894 119876min

119894

+ 1205729119894 119876max119894 + 12057210119894 forall119894 isin 119866

(18)





[CCF-DPCP] 119865+119894 (119880119894 119876min119894 119876max

119894 )= 1205731119894 (119880119894)2 + 1205732119894 (119876min

119894 )2+ 1205733119894 (119876max

119894 )2 + 1205734119894 119880119894119876min119894

+ 1205735119894 119880119894119876max119894 + 1205736119894 119876min

119894 119876max119894

+ 1205737119894 119880119894 + 1205738119894 119876min119894 + 1205739119894 119876max

119894 + 12057310119894 forall119894 isin 119866

119865minus119894 (119880119894 119876min119894 119876max

119894 )= 1205741119894 (119880119894)2 + 1205742119894 (119876min

119894 )2+ 1205743119894 (119876max

119894 )2 + 1205744119894 119880119894119876min119894

+ 1205745119894 119880119894119876max119894 + 1205746119894 119876min

119894 119876max119894 + 1205747119894 119880119894

+ 1205748119894 119876min119894 + 1205749119894 119876max

119894 + 12057410119894 119894 isin 119866

(19)




119894119905 119876max119894119905 | 119894 isin119866 119905 isin 119879 119865+119894119905 119880119894119905 119876min


119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879



119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879 to












120575119895119905+1 = 120597119865119895 (119880119895 119876min119895 119876max

119895 )12059711988011989510038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

(20)


120575119895119905+1 = 120597119865119895 (119880119895 119876min119895 119876max

119895 )120597119880119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205721119895119880119895 + 1205724119895119876min119895 + 1205725119895119876max

119895

+ 1205727119895 10038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205721119895119880119895119905+1 + 1205724119895119876min119895119905+1 + 1205725119895119876max

119895119905+1 + 1205727119895

(21)






120575+119895119905+1 = 120597119865+119895 (119880119895 119876min

119895 119876max119895 )120597119880119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

(22)


120575+119895119905+1 = 120597119865+119895 (119880119895 119876min

119895 119876max119895 )

12059711988011989510038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205731119895119880119895 + 1205734119895119876min119895 + 1205735119895119876max

119895

+ 1205737119895 10038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205731119895119880119895119905+1 + 1205734119895119876min119895119905+1 + 1205735119895119876max

119895119905+1 + 1205737119895

(23)



120575minus119895119905+1 = 120597119865minus119895 (119880119895 119876min

119895 119876max119895 )120597119880119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

(24)


120575minus119895119905+1 = 120597119865minus119895 (119880119895 119876min

119895 119876max119895 )120597119880119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205741119895119880119895 + 1205744119895119876min119895 + 1205745119895119876max

119895

+ 1205747119895 10038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205741119895119880119895119905+1 + 1205744119895119876min119895119905+1 + 1205745119895119876max

119895119905+1 + 1205747119895

(25)










119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879 119865+119894119905 119880119894119905 119876min


119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879














7 Numerical Study













3 232527 365392 577442 4474 6891 90065 282930 362955 617750 4621 6890 91098 311313 463953 635832 4758 7174 900210 325420 507720 736300 5028 7021 943413 399359 528024 819622 5100 7579 937415 457170 627995 830250 9374 7776 985418 466665 707605 978812 5803 8108 1015220 518180 713780 1049600 6020 8053 1006923 603231 802696 1103402 6205 8759 1072425 668450 915075 1244350 6606 9054 1008028 719057 1013297 1303392 6786 9033 1139330 707980 1031880 1404500 7051 9877 1068733 844143 1189408 1508782 7667 10430 1215735 846770 1264195 1710050 7956 10524 1248938 988489 1301029 1809572 8008 11451 1251640 1034820 1462020 1981000 8721 11893 1338643 1102095 1538160 2105762 9208 12094 1397145 1142130 1675355 2277350 9145 13085 1437948 1294961 1780801 2407352 10067 13360 1472150 1308700 1904200 2599100 10428 14401 15468

Tota

l pro

fit

RM-DPDIP model

MIP model

RM-SPIP model

times104

2

3

4

5

6

7

8

5 4015 20 25 30 35 45 50100Time period










Pres

et in

crem

enta

l pro

fit p

aram

eter

times103

4

45

5

55

6

65

7

5 10 15 20 25 30 35 40 45 500Time period


Pres

et in

crem

enta

l rev

enue

par

amet

er

times103

55

6

65

7

75

8

85

5 10 15 20 25 4035 45 50300Time period


Pres

et d

ecre

men

tal c

ost p

aram

eter

times103

1012141618202224262830

5 10 15 20 25 30 35 40 45 500Time period



Cum

ulat

ive t

otal

pro

fit RM-DPIDP

RM-SPIP

MIP

times105

0

1

2

3

4

5

6

7

8

5 10 15 20 25 30 35 40 45 500Time period




Com

pute

r tim

e

RM-DPIDP

RM-SPIP

MIP

0100200300400500600700800900

5 10 15 20 25 30 35 40 45 500Time period




OPT

diff

eren

ce


times106

0

05

1

15

2

25

3




CPU

tim

e

times102

4

6

8

10

12

14

16

18




8 Conclusions







Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119895119905+1 and 119876max119895119905+1


[LP] max 119865119905= 120585sum119894isin119866

119886119905119880119894119905 minus sum119894isin119866

119887119905 (119876max119894119905 minus 120585119880119894119905)

minus 119888sum119894isin119866

119880119894119905subject to sum

119895isin119866



120585sum119894isin119866

119909119894119895119905 le sum119896isin119866

120572119895119896119876max119896119905+1 forall119895 isin 119866



(17)





[CCF-SPCP] 119865119894 (119880119894 119876min119894 119876max

119894 )= 1205721119894 (119880119894)2 + 1205722119894 (119876min

119894 )2 + 1205723119894 (119876max119894 )2 + 1205724119894 119880119894119876min

119894

+ 1205725119894 119880119894119876max119894 + 1205726119894 119876min

119894 119876max119894 + 1205727119894 119880119894 + 1205728119894 119876min

119894

+ 1205729119894 119876max119894 + 12057210119894 forall119894 isin 119866

(18)





[CCF-DPCP] 119865+119894 (119880119894 119876min119894 119876max

119894 )= 1205731119894 (119880119894)2 + 1205732119894 (119876min

119894 )2+ 1205733119894 (119876max

119894 )2 + 1205734119894 119880119894119876min119894

+ 1205735119894 119880119894119876max119894 + 1205736119894 119876min

119894 119876max119894

+ 1205737119894 119880119894 + 1205738119894 119876min119894 + 1205739119894 119876max

119894 + 12057310119894 forall119894 isin 119866

119865minus119894 (119880119894 119876min119894 119876max

119894 )= 1205741119894 (119880119894)2 + 1205742119894 (119876min

119894 )2+ 1205743119894 (119876max

119894 )2 + 1205744119894 119880119894119876min119894

+ 1205745119894 119880119894119876max119894 + 1205746119894 119876min

119894 119876max119894 + 1205747119894 119880119894

+ 1205748119894 119876min119894 + 1205749119894 119876max

119894 + 12057410119894 119894 isin 119866

(19)




119894119905 119876max119894119905 | 119894 isin119866 119905 isin 119879 119865+119894119905 119880119894119905 119876min


119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879



119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879 to












120575119895119905+1 = 120597119865119895 (119880119895 119876min119895 119876max

119895 )12059711988011989510038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

(20)


120575119895119905+1 = 120597119865119895 (119880119895 119876min119895 119876max

119895 )120597119880119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205721119895119880119895 + 1205724119895119876min119895 + 1205725119895119876max

119895

+ 1205727119895 10038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205721119895119880119895119905+1 + 1205724119895119876min119895119905+1 + 1205725119895119876max

119895119905+1 + 1205727119895

(21)






120575+119895119905+1 = 120597119865+119895 (119880119895 119876min

119895 119876max119895 )120597119880119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

(22)


120575+119895119905+1 = 120597119865+119895 (119880119895 119876min

119895 119876max119895 )

12059711988011989510038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205731119895119880119895 + 1205734119895119876min119895 + 1205735119895119876max

119895

+ 1205737119895 10038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205731119895119880119895119905+1 + 1205734119895119876min119895119905+1 + 1205735119895119876max

119895119905+1 + 1205737119895

(23)



120575minus119895119905+1 = 120597119865minus119895 (119880119895 119876min

119895 119876max119895 )120597119880119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

(24)


120575minus119895119905+1 = 120597119865minus119895 (119880119895 119876min

119895 119876max119895 )120597119880119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205741119895119880119895 + 1205744119895119876min119895 + 1205745119895119876max

119895

+ 1205747119895 10038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205741119895119880119895119905+1 + 1205744119895119876min119895119905+1 + 1205745119895119876max

119895119905+1 + 1205747119895

(25)










119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879 119865+119894119905 119880119894119905 119876min


119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879














7 Numerical Study













3 232527 365392 577442 4474 6891 90065 282930 362955 617750 4621 6890 91098 311313 463953 635832 4758 7174 900210 325420 507720 736300 5028 7021 943413 399359 528024 819622 5100 7579 937415 457170 627995 830250 9374 7776 985418 466665 707605 978812 5803 8108 1015220 518180 713780 1049600 6020 8053 1006923 603231 802696 1103402 6205 8759 1072425 668450 915075 1244350 6606 9054 1008028 719057 1013297 1303392 6786 9033 1139330 707980 1031880 1404500 7051 9877 1068733 844143 1189408 1508782 7667 10430 1215735 846770 1264195 1710050 7956 10524 1248938 988489 1301029 1809572 8008 11451 1251640 1034820 1462020 1981000 8721 11893 1338643 1102095 1538160 2105762 9208 12094 1397145 1142130 1675355 2277350 9145 13085 1437948 1294961 1780801 2407352 10067 13360 1472150 1308700 1904200 2599100 10428 14401 15468

Tota

l pro

fit

RM-DPDIP model

MIP model

RM-SPIP model

times104

2

3

4

5

6

7

8

5 4015 20 25 30 35 45 50100Time period










Pres

et in

crem

enta

l pro

fit p

aram

eter

times103

4

45

5

55

6

65

7

5 10 15 20 25 30 35 40 45 500Time period


Pres

et in

crem

enta

l rev

enue

par

amet

er

times103

55

6

65

7

75

8

85

5 10 15 20 25 4035 45 50300Time period


Pres

et d

ecre

men

tal c

ost p

aram

eter

times103

1012141618202224262830

5 10 15 20 25 30 35 40 45 500Time period



Cum

ulat

ive t

otal

pro

fit RM-DPIDP

RM-SPIP

MIP

times105

0

1

2

3

4

5

6

7

8

5 10 15 20 25 30 35 40 45 500Time period




Com

pute

r tim

e

RM-DPIDP

RM-SPIP

MIP

0100200300400500600700800900

5 10 15 20 25 30 35 40 45 500Time period




OPT

diff

eren

ce


times106

0

05

1

15

2

25

3




CPU

tim

e

times102

4

6

8

10

12

14

16

18




8 Conclusions







Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879 to












120575119895119905+1 = 120597119865119895 (119880119895 119876min119895 119876max

119895 )12059711988011989510038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

(20)


120575119895119905+1 = 120597119865119895 (119880119895 119876min119895 119876max

119895 )120597119880119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205721119895119880119895 + 1205724119895119876min119895 + 1205725119895119876max

119895

+ 1205727119895 10038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205721119895119880119895119905+1 + 1205724119895119876min119895119905+1 + 1205725119895119876max

119895119905+1 + 1205727119895

(21)






120575+119895119905+1 = 120597119865+119895 (119880119895 119876min

119895 119876max119895 )120597119880119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

(22)


120575+119895119905+1 = 120597119865+119895 (119880119895 119876min

119895 119876max119895 )

12059711988011989510038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205731119895119880119895 + 1205734119895119876min119895 + 1205735119895119876max

119895

+ 1205737119895 10038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205731119895119880119895119905+1 + 1205734119895119876min119895119905+1 + 1205735119895119876max

119895119905+1 + 1205737119895

(23)



120575minus119895119905+1 = 120597119865minus119895 (119880119895 119876min

119895 119876max119895 )120597119880119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

(24)


120575minus119895119905+1 = 120597119865minus119895 (119880119895 119876min

119895 119876max119895 )120597119880119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205741119895119880119895 + 1205744119895119876min119895 + 1205745119895119876max

119895

+ 1205747119895 10038161003816100381610038161003816119876min119895 =119876

min119895119905+1 119876

max119895 =119876

max119895119905+1

119880119895=119880119895119905+1

= 21205741119895119880119895119905+1 + 1205744119895119876min119895119905+1 + 1205745119895119876max

119895119905+1 + 1205747119895

(25)










119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879 119865+119894119905 119880119894119905 119876min


119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879














7 Numerical Study













3 232527 365392 577442 4474 6891 90065 282930 362955 617750 4621 6890 91098 311313 463953 635832 4758 7174 900210 325420 507720 736300 5028 7021 943413 399359 528024 819622 5100 7579 937415 457170 627995 830250 9374 7776 985418 466665 707605 978812 5803 8108 1015220 518180 713780 1049600 6020 8053 1006923 603231 802696 1103402 6205 8759 1072425 668450 915075 1244350 6606 9054 1008028 719057 1013297 1303392 6786 9033 1139330 707980 1031880 1404500 7051 9877 1068733 844143 1189408 1508782 7667 10430 1215735 846770 1264195 1710050 7956 10524 1248938 988489 1301029 1809572 8008 11451 1251640 1034820 1462020 1981000 8721 11893 1338643 1102095 1538160 2105762 9208 12094 1397145 1142130 1675355 2277350 9145 13085 1437948 1294961 1780801 2407352 10067 13360 1472150 1308700 1904200 2599100 10428 14401 15468

Tota

l pro

fit

RM-DPDIP model

MIP model

RM-SPIP model

times104

2

3

4

5

6

7

8

5 4015 20 25 30 35 45 50100Time period










Pres

et in

crem

enta

l pro

fit p

aram

eter

times103

4

45

5

55

6

65

7

5 10 15 20 25 30 35 40 45 500Time period


Pres

et in

crem

enta

l rev

enue

par

amet

er

times103

55

6

65

7

75

8

85

5 10 15 20 25 4035 45 50300Time period


Pres

et d

ecre

men

tal c

ost p

aram

eter

times103

1012141618202224262830

5 10 15 20 25 30 35 40 45 500Time period



Cum

ulat

ive t

otal

pro

fit RM-DPIDP

RM-SPIP

MIP

times105

0

1

2

3

4

5

6

7

8

5 10 15 20 25 30 35 40 45 500Time period




Com

pute

r tim

e

RM-DPIDP

RM-SPIP

MIP

0100200300400500600700800900

5 10 15 20 25 30 35 40 45 500Time period




OPT

diff

eren

ce


times106

0

05

1

15

2

25

3




CPU

tim

e

times102

4

6

8

10

12

14

16

18




8 Conclusions







Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879 119865+119894119905 119880119894119905 119876min


119894119905 119876max119894119905 | 119894 isin 119866 119905 isin 119879














7 Numerical Study













3 232527 365392 577442 4474 6891 90065 282930 362955 617750 4621 6890 91098 311313 463953 635832 4758 7174 900210 325420 507720 736300 5028 7021 943413 399359 528024 819622 5100 7579 937415 457170 627995 830250 9374 7776 985418 466665 707605 978812 5803 8108 1015220 518180 713780 1049600 6020 8053 1006923 603231 802696 1103402 6205 8759 1072425 668450 915075 1244350 6606 9054 1008028 719057 1013297 1303392 6786 9033 1139330 707980 1031880 1404500 7051 9877 1068733 844143 1189408 1508782 7667 10430 1215735 846770 1264195 1710050 7956 10524 1248938 988489 1301029 1809572 8008 11451 1251640 1034820 1462020 1981000 8721 11893 1338643 1102095 1538160 2105762 9208 12094 1397145 1142130 1675355 2277350 9145 13085 1437948 1294961 1780801 2407352 10067 13360 1472150 1308700 1904200 2599100 10428 14401 15468

Tota

l pro

fit

RM-DPDIP model

MIP model

RM-SPIP model

times104

2

3

4

5

6

7

8

5 4015 20 25 30 35 45 50100Time period










Pres

et in

crem

enta

l pro

fit p

aram

eter

times103

4

45

5

55

6

65

7

5 10 15 20 25 30 35 40 45 500Time period


Pres

et in

crem

enta

l rev

enue

par

amet

er

times103

55

6

65

7

75

8

85

5 10 15 20 25 4035 45 50300Time period


Pres

et d

ecre

men

tal c

ost p

aram

eter

times103

1012141618202224262830

5 10 15 20 25 30 35 40 45 500Time period



Cum

ulat

ive t

otal

pro

fit RM-DPIDP

RM-SPIP

MIP

times105

0

1

2

3

4

5

6

7

8

5 10 15 20 25 30 35 40 45 500Time period




Com

pute

r tim

e

RM-DPIDP

RM-SPIP

MIP

0100200300400500600700800900

5 10 15 20 25 30 35 40 45 500Time period




OPT

diff

eren

ce


times106

0

05

1

15

2

25

3




CPU

tim

e

times102

4

6

8

10

12

14

16

18




8 Conclusions







Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












3 232527 365392 577442 4474 6891 90065 282930 362955 617750 4621 6890 91098 311313 463953 635832 4758 7174 900210 325420 507720 736300 5028 7021 943413 399359 528024 819622 5100 7579 937415 457170 627995 830250 9374 7776 985418 466665 707605 978812 5803 8108 1015220 518180 713780 1049600 6020 8053 1006923 603231 802696 1103402 6205 8759 1072425 668450 915075 1244350 6606 9054 1008028 719057 1013297 1303392 6786 9033 1139330 707980 1031880 1404500 7051 9877 1068733 844143 1189408 1508782 7667 10430 1215735 846770 1264195 1710050 7956 10524 1248938 988489 1301029 1809572 8008 11451 1251640 1034820 1462020 1981000 8721 11893 1338643 1102095 1538160 2105762 9208 12094 1397145 1142130 1675355 2277350 9145 13085 1437948 1294961 1780801 2407352 10067 13360 1472150 1308700 1904200 2599100 10428 14401 15468

Tota

l pro

fit

RM-DPDIP model

MIP model

RM-SPIP model

times104

2

3

4

5

6

7

8

5 4015 20 25 30 35 45 50100Time period










Pres

et in

crem

enta

l pro

fit p

aram

eter

times103

4

45

5

55

6

65

7

5 10 15 20 25 30 35 40 45 500Time period


Pres

et in

crem

enta

l rev

enue

par

amet

er

times103

55

6

65

7

75

8

85

5 10 15 20 25 4035 45 50300Time period


Pres

et d

ecre

men

tal c

ost p

aram

eter

times103

1012141618202224262830

5 10 15 20 25 30 35 40 45 500Time period



Cum

ulat

ive t

otal

pro

fit RM-DPIDP

RM-SPIP

MIP

times105

0

1

2

3

4

5

6

7

8

5 10 15 20 25 30 35 40 45 500Time period




Com

pute

r tim

e

RM-DPIDP

RM-SPIP

MIP

0100200300400500600700800900

5 10 15 20 25 30 35 40 45 500Time period




OPT

diff

eren

ce


times106

0

05

1

15

2

25

3




CPU

tim

e

times102

4

6

8

10

12

14

16

18




8 Conclusions







Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Pres

et in

crem

enta

l pro

fit p

aram

eter

times103

4

45

5

55

6

65

7

5 10 15 20 25 30 35 40 45 500Time period


Pres

et in

crem

enta

l rev

enue

par

amet

er

times103

55

6

65

7

75

8

85

5 10 15 20 25 4035 45 50300Time period


Pres

et d

ecre

men

tal c

ost p

aram

eter

times103

1012141618202224262830

5 10 15 20 25 30 35 40 45 500Time period



Cum

ulat

ive t

otal

pro

fit RM-DPIDP

RM-SPIP

MIP

times105

0

1

2

3

4

5

6

7

8

5 10 15 20 25 30 35 40 45 500Time period




Com

pute

r tim

e

RM-DPIDP

RM-SPIP

MIP

0100200300400500600700800900

5 10 15 20 25 30 35 40 45 500Time period




OPT

diff

eren

ce


times106

0

05

1

15

2

25

3




CPU

tim

e

times102

4

6

8

10

12

14

16

18




8 Conclusions







Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










OPT

diff

eren

ce


times106

0

05

1

15

2

25

3




CPU

tim

e

times102

4

6

8

10

12

14

16

18




8 Conclusions







Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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