multistage stochastic programming with fuzzy probability distribution

Fuzzy Sets and Systems 160 (2009) 3239–3249www.elsevier.com/locate/fss

Multistage stochastic programming with fuzzy probabilitydistribution

Fouad Ben Abdelaziza,∗, Hatem Masrib

aEngineering System Management Graduate Program, College of Engineering, American University of Sharjah, P.O. Box 26666, Sharjah, UAEbLARODEC Laboratory, Institut Supérieur de Gestion, University of Tunis, 41 rue de la liberté, Le Bardo 2000, Tunisia

Available online 11 November 2008

Abstract

In this paper, we introduce the multistage stochastic programwith fuzzy probability distribution.We focus on the case where fuzzyprobability distribution is defined by (triangular) fuzzy numbers.We extend Ben Abdelaziz andMasri [Stochastic programming withfuzzy linear partial information on probability distribution, European Journal Operational Research 162 (2005) 619–629] solutionstrategy, for the two-stage stochastic program with fuzzy probability distribution, to solve the multistage model. The proposedsolution strategy is based on two transformation steps. In the first step, the fuzzy transformation step, we propose to use the �-cutdefuzzification technique. The level � relates to the DM credibility degree on information sources. This step ends with a certaintyequivalent program. In the second step, the stochastic transformation step, we decompose the certainty equivalent program basedon a minimax approach. The obtained problem is then solved using a modified version of the nested decomposition method. Themodification on the nested decomposition method concerns the way in which we generate optimal constraints. The modified nesteddecomposition algorithm may be used to solve the multistage problem with interval probability distribution.© 2008 Elsevier B.V. All rights reserved.

Keywords: Stochastic programming; Fuzzy number; �-Cut technique; Nested decomposition method

1. Introduction

Let us consider the multistage stochastic program with finite horizon (T stages) with the following decision-observation process [1]:

x1, w2, x2, w3, x3, . . . , wT , xT

where x1, x2, x3, . . . , xT are the decision vectors in the subsequent periods, and w2, w3, . . . , wT are the sequenceof random vectors. We suppose that the information on the possible values of the random vectors w2, w3, . . . , wT isexpressed by a discrete time stochastic process defined on some discrete probability space.In many practical situations, discrete probability distributions of future events may be predicted using existing

statistics and expert judgement. For long run decision making problems, these probabilities are in most cases subjectiveand difficult to estimate.Uncertainty about probability distribution was considered in many operations research problems. Hartfiel and Seneta

[2], for example, addressed the case where transition probabilities for a Markov chain problem belong to intervals.

∗Corresponding author.E-mail addresses: [email protected] (F. Ben Abdelaziz), [email protected] (H. Masri).

0165-0114/$ - see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.fss.2008.10.010

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mailto:[email protected]

3240 F. Ben Abdelaziz, H. Masri / Fuzzy Sets and Systems 160 (2009) 3239–3249

Stage t-1

Stage t k

Dt+1(k)

at-1(k)

Stage t+1

Stage 1

Stage T

Stage T-1

Stage 2 1

1 Kt

K2

Fig. 1. Scenario tree.

In stochastic programming, uncertainty on probability distribution was first modelled via moment constraints [3,4].Moment constraints deal with the case where a range of possible values for these probabilities is given (intervalprobability distribution). Recently, Ben Abdelaziz and Masri [5] studied the two-stage stochastic program with fuzzyprobability distribution. Fuzzy probability distribution relates to the casewhere over the interval of possible probabilitiessome values are more possible than others. In contrast with the two-stage case and as far as we know, this paper is thefirst that addresses the multistage stochastic program with fuzzy probability distribution.In themultistage stochastic programwith fuzzy probability distributionwe have two level of uncertainty: randomness

on the parameters and fuzziness on the values of the probability distribution. Ben Abdelaziz and Masri [5] proposed atwo steps solution strategy to solve the two-stage version of the multistage model where they deal with the two levelsof uncertainty in a separate manner. In the first step, called fuzzy transformation step, they utilized the �-cut techniqueto defuzzify the fuzzy probability distribution. In the second step, called stochastic transformation step, they solved theobtained certainty equivalent program using a modified version of the L-shaped method.In this paper, we extend the two-stage solution strategy of Ben Abdelaziz and Masri [5] to solve the multistage

stochastic program with fuzzy probability distribution by considering their two transformation steps. The fuzzy trans-formation step is almost the same as in [5], while in the stochastic transformation step, the certainty equivalent programis obtained by considering the risk attitude of the decisionmaker. The obtained program is solved using amodified nesteddecomposition method. This modification concerns the way of generating optimality cuts in the nested decompositionmethod [6].In Section 2, we review the main concepts of the multistage stochastic programming theory. In Section 3, we define

the concept of fuzzy probability distribution. In Section 4, we introduce and solve the multistage stochastic programwith fuzzy probability distribution. In Section 5, we illustrate, by an example, different results of the paper.

2. Multistage stochastic program

The multistage stochastic program with finite horizon (T stages) may be pictured by a scenario tree (see Fig. 1). Let kbe one state of the Kt possible states of wt at the t th stage (i.e. k ∈ {wt

1, . . . , wtKt

}). We denote by at−1(k) the ancestor

state at stage t − 1 of the state k and Dt+1(k) the descendants states of the state k at the period t + 1.

F. Ben Abdelaziz, H. Masri / Fuzzy Sets and Systems 160 (2009) 3239–3249 3241

At each node k of the t th stage a decision xtk has to be taken. Taking a first decision x1 and observing the value ofthe random vector w2, we can make decisions on x2k , k = 1, . . . , K2 so that the constraints on x1 and x2k are fulfilled.In the last stage, we observe a random variable wT and make decisions on xTk , k = 1, . . . , KT . Hence, when decidingon xtk , we have to take into account the past history of the system and the joint conditional probability distribution ofthe future realization of the random variables.The multistage stochastic program with finite horizon can be written as follows [7]:

Min c1x1 +K2∑

k2=1

p2k2

⎡⎣c2k2x2k2 + · · · +

∑kT ∈DT (kT−1)

pTkT [cTkT x

TkT ] . . .

⎤⎦s.t. T t−1

k xt−1at (k) + Wt

k xtk = htk, k = 1, . . . , Kt , t = 1, . . . , T

xtk �0, k = 1, . . . , Kt , t = 1, . . . , T (1)

where ctk , htk , T

tk , W

tk are fixed matrices with appropriate dimensions, for all stages t = 1, . . . , T and all states

k = 1, . . . , Kt . For t = 1, the right member T 0x0 = 0. ptk , k = 1, . . . , Kt ; t = 2, . . . , T , is the conditional probabilitythat the sequence of events leading to the kth state at the t th stage.

The size of the event tree grows exponentially with the number of realizations on each stage and consequently thesize of the multistage program [1]. Two families of algorithms are used to solve multistage stochastic program. Thefirst family is based on stochastic dynamic programming, considering the multistage program as a dynamic program.The second family is based on the nested decomposition method, considering the multistage program as a large scalelinear program with a special structure. The nested decomposition method was first presented for linear programs[8], then extended to stochastic programming by Birge [6] and implemented by Gassmann [9] in the MSLiP softwarepackage.In the multistage stochastic program, the conditional probabilities ptk , k = 1, . . . , Kt ; t = 2, . . . , T , are supposed

to have a predefined exact value. In some situations these probabilities are difficult to evaluate and only an estima-tion of their possible value may be available. Therefore, these probabilities may be considered as fuzzy probabilitydistributions.

3. Fuzzy probability distribution

Let us denote by (�, 2�, P) a discrete probability space with � = {w1, . . . , wK } a finite set of possible states ofnature, 2� the power set of � and P the vector of probabilities pk = P({w = wk}).

In practice, the hypothesis that all the pk values must be precise numbers in the unit interval [0, 1] is not realistic.Usually, due to a lack of information, we may not be able to evaluate precisely the probability pk . In many cases,it is much easier to determine the smallest (�

k) and the highest (�k) possible value of it (with 0��

k� pk ��k �1).

Therefore, we can characterize the probability pk by the interval [�k, �k]. In this case, information on possible states

of the nature is given in terms of interval valued probabilities [10].In the interval [�

k, �k], we may expect that the central value is more possible than extreme values. Therefore, the

probability pk can be represented by a fuzzy number.A fuzzy number pk is a fuzzy set defined on� and characterized by its membership function �k(x), �k : � −→ [0, 1],

�k(x) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

0 if x��k

lk(x) if �k

< x�dk1 if dk < x�dkrk(x) if dk < x��k0 if �k < x

where lk(x) and rk(x) are, respectively, nondecreasing and nonincreasing functions that characterize the left and rightside of the fuzzy number pk . The interval [dk, dk] represents the very possible value of pk .

From all possible form of fuzzy numbers, the trapezoidal and triangular fuzzy numbers are very common in theliterature. This is essentially induced by the linear form of the left and right side functions.


Definition 1. A fuzzy number pk is called a trapezoidal fuzzy number if

lk(x) =x − �

k

dk − �k

if �k

< x�dk

rk(x) = �k − x

�k − dkif dk < x��k

Using the notation in [11], we write, pk = (dk, dk, �k, �k).

Definition 2. A fuzzy number pk is called a triangular fuzzy number if it is a trapezoidal fuzzy number and dk = dk =dk . Using the notation in [11], we write, pk = (dk, �k, �k).

A triangular fuzzy number pk = (dk, �k, �k) is composed by a collection of nested intervals constructed by taking�-cuts [12]. An �-cut leads (0��1) to a range of possible values for the probability:

pk[�] := {0� pk �1 : �dk − (1 − �)�k� pk ��dk + (1 − �)�k}

Therefore, � together with the pk values is a discrete fuzzy probability distribution. We write P for fuzzy P and wehave pk = P({w = wk}), k = 1, . . . , K [12].The uncertainty is on some of the pk values, but we know that we have a discrete probability distribution. For that,

we propose the following restriction on the extreme values �kand �k to ensure the existence of a discrete probability

distribution:

Proposition 3. If∑K

k=1 �k�1 and

∑Kk=1 �k �1 then for all � ∈ [0, 1], there exist at least one pk ∈ pk[�], for each

k = 1, . . . , K such that P is a discrete probability distribution [13].

4. Multistage stochastic program with fuzzy probability distribution

We call multistage stochastic program with fuzzy probability distribution the following program:

Min c1x1 +K2∑

k2=1

p2k2

⎡⎣c2k2x2k2 + · · · +

∑kT ∈DT (kT−1)

pTkT [cTkT x

TkT ] . . .

⎤⎦s.t. T t−1

k xt−1a(k) + Wt

k xtk = htk, k = 1, . . . , Kt , t = 1, . . . , T

xtk �0, k = 1, . . . , Kt , t = 1, . . . , T (2)

where the conditional probabilities p tk = (dtk, �

tk, �

tk), k = 1, . . . , Kt ; t = 2, . . . , T , are triangular fuzzy numbers

such that∑

j∈Dt (k) �tj�1 and

∑j∈Dt (k) �

tj �1 for all k = 1, . . . , Kt−1, t = 2, . . . , T , and all the other parameters are

defined as in problem (1).To solve problem (2), we propose to follow the two steps solution strategy proposed by Ben Abdelaziz and Masri

[5] for general stochastic programming with fuzzy linear partial information on probability distribution. The first stepis called the fuzzy transformation step and the second step is called the stochastic transformation step.The purpose of the fuzzy transformation step is to defuzzify the conditional fuzzy probabilities ptk = (dtk, �

tk, �

tk),

k = 1, . . . , Kt ; t = 2, . . . , T . For that a general �-cut on these fuzzy numbers is applied where � is the DM credibilitydegree on information sources. If � is near 1 then the DM is more confident about the expert estimation (ptk = dtk).

Otherwise, the DMwill give � a value near 0 (i.e. �tk� ptk ��

tk). We may specify for each conditional fuzzy probability

ptk a different credibility degree �tk if the information is coming from different sources with different credibility degreesfor the DM. We note that if we increase the value of �tk from 0 to 1, the interval of possible value of the probabilitydistribution gets smaller. As illustrated in the example at the end of the paper, the value of the optimal solution of theproblem is expected to be smaller when alpha increases.


The defuzzification leads to a range of possible values for the probabilities ptk ,

�tkdtk − (1 − �tk)�

tk� ptk ��tkd

tk + (1 − �tk)�

tk

At each node k of the t − 1th stage, t = 2, . . . , T , let us denote by Ptk the vector of conditional probabilities ptj ,

j ∈ Dt (k). When all probabilities have not fixed value, we still have that the sum of all the individual probabilities isone. Therefore, the set of possible probability distributions �tk obtained by the fuzzy transformation step may be writtenas follows:

�tk =

⎧⎪⎨⎪⎩Ptk := (ptj , j ∈ Dt (k)) : �tj d

tj − (1 − �tj )�

tj� ptj ��tj d

tj + (1 − �tj )�

tj ,∑

j∈Dt (k)ptj = 1, ptj �0, j ∈ Dt (k)

⎫⎪⎬⎪⎭The set �tk is a closed and bounded subset of �N . As for all k = 1, . . . , Kt−1, t = 2, . . . , T ,

∑j∈Dt (k) �

tj�1 and∑

j∈Dt (k) �tj �1, and based on Proposition 3, �tk is a nonempty set.

The fuzzy transformation step ends with the following multistage stochastic program with interval probabilitydistributions:

Min c1x1 +K2∑

k2=1

p2k2

⎡⎣c2k2x2k2 + · · · +

∑kH∈DH (kH−1)

pHkH [cHkH x

HkH ] . . .

⎤⎦s.t. Wt

k xtk = htk − T t−1

k xt−1a(k), k = 1, . . . , Kt , t = 1, . . . , T

Ptk ∈ �tk, k = 1, . . . , Kt−1, t = 2, . . . , T

xtk �0, k = 1, . . . , Kt , t = 2, . . . , T (3)

The purpose of the stochastic transformation step is to decompose and solve problem (3).From the scenario tree form of problem (3), the DM, in a first stage, looks for the optimal decision x1 that gives the

minimum value for the first stage cost c1x1 plus the future expected cost that we denote by R2(x1):

Min c1x1 + R2(x1)

s.t. W 1x1 = h1

x1�0 (4)

To compute the future expected cost R2(x1), we have to specify a probability distribution P2 ∈ �2 such that R2(x1) =∑K2j=1 p

2j R

2j (x

1, w2j ), where R2

j (x1, w2

j ) is the future cost if the first stage decision x1 was chosen and the scenario w2j

occurs:

R2j (x

1, w2j ) = Min c2j x

2j + R3(x2j )

s.t. T 1x1 + W 2j x

2j = h2j

x2j �0

We suppose that the DM has no preference structure over the set of possible probability distributions �2 and that he isaware from the worst value of the objective function. Therefore, R2(x1) is equal to the maximum of the long-run costsubject to the set of possible probability distributions �2:

R2(x1) = MaxP2∈�2

K2∑j=1

p2j R2j (x

1, w2j )

The use of other decision approaches may be possible and depends on the DM preference structure over the sets ofpossible probability distributions.


By inductive reasoning, we can compute future decisions and costs as follows:

Rtk(x

t−1a(k), w

tk) = Min ctk x

tk + Rt+1(xtk)

s.t. T t−1a(k) x

t−1a(k) + Wt

k xtk = htk for k = 1, . . . , Kt , t = 2, . . . , T

xtk �0 (5)

where

Rt+1(xtk) = MaxPt+1k ∈�t+1

k

∑j∈Dt (k)

pt+1j Rt+1

j (xtk, wt+1j )

and RT+1(.) = 0.The stochastic transformation step ends with the deterministic program (4) to the multistage stochastic program

with fuzzy probability distribution (2). To solve problem (4), we need to explicit the exact form of the expected costR2(.), which is difficult to achieve as we cannot usually specify the form of all future costs Rt (.), t = 2, . . . , T . In thefollowing, we will try to describe a mathematical property of the Rt (.), t = 2, . . . , T , functions.

First, the function RTk (., w

Tk ), as it has the same form as the recourse function in the two-stage stochastic program,

then, it is a piecewise linear convex [7]. Second, all sets �Tk have a polyhedral shape. Let us denote by �T

k the set ofprobability distributions which are represented by the extreme points of the polyhedral set �T

k . The set �Tk is a closed

and a bounded subset of�N then �Tk is finite. RT (.), the maximum of the mathematical expectation over �T

k , is attainedfor some P ∈ �T

k

RT (.) = MaxPTk ∈�T

k

∑j∈DT (.)

pTj RTj (., w

Tj )

�Tk is finite then RT (.) as the maximum of finite number of piecewise linear convex functions is a piecewise linear

convex. By recursive reasoning, we conclude that the function Rt (.) is piecewise linear convex, for t = 2, . . . , T .Based on the piecewise linear convexity of Rt (.), t = 2, . . . , T , we extend the nested decomposition method to solve

problem (4). Let us consider for each stage t = 1, . . . , T and for each scenario k = 1, . . . , Kt , the following masterproblem P(t, k):

Min ctk xtk + �tk

s.t. T t−1a(k) x

t−1a(k) + Wt

k xtk = htk (6-1)

Dtk, j x

tk �dtk, j , j = 1, . . . , r tk (6-2)

Etk, j x

tk + �tk �etk, j , j = 1, . . . , stk (6-3)

xtk �0

where Dtk, j , E

tk, j , d

tk, j and etk, j are vectors with appropriate dimensions related to constraints (6-2) and (6-3) and

are generated by the nested decomposition method to approximate the piecewise linear convex function Rt (.). Thefirst series of constraints (6-2), called feasibility constraints, insures that the chosen solution is feasible for all fu-ture scenarios. The second series of constraints (6-3), called optimality constraints, are an outer linearization of thefunction Rt (.). In the nested decomposition these constraints are built according to the probability distribution of thescenario.Our proposed modification to the nested decomposition method concerns the way that it generates the optimality

constraints. These constraints should be formulated according to the probability distribution that gives the maximumof R(xk), where xk is the solution at the kth iteration. Therefore, in the modified nested decomposition algorithm andbefore we formulate the optimality cut, we solve a linear program to determine the probability distribution that givesthe maximum value of R(xk).

The modified nested decomposition algorithm still based on the fast forward and fast backward procedure [6]. Itmeans that the algorithm proceeds in forward as far as possible until an infeasible master problem is found or when itreaches final stage.


In the following, we present the modified nested decomposition method using the same notations as in [14]:

Step 0: Set t = 1, k = 1, r tk = stk = 0, add the constraint �tk = 0 to P(t, k) for all t and k, and let DIR = FORE. Goto step 1.

Step 1: Solve the current problem, P(t, k). If infeasible and t = 1, then stop, problem (4) is infeasible. If infeasibleand t > 1, then let r t−1

a(k) = r t−1a(k) + 1, let DIR = BACK. Let the infeasibility condition be obtained by a dual

basic solution, ztk, �tk �0, such that

(ztk)T W t + (�tk)

T Dtk �0

but

(ztk)T (htk − T t−1

a(k) xt−1a(k)) + (�tk)

T dtk > 0

Let

Dt−1a(k),r t−1

a(k)= (ztk)

T T t−1a(k)

dt−1a(k),r t−1

a(k)= (ztk)

T htk + (�tk)T dtk

Let t = t − 1, k = a(k) and return to step 1.If feasible, update the values of xtk , �tk , and store the value of the complementary basic dual multipliers onconstraints (6-1)–(6-3) as (ztk, �

tk,

tk), respectively. Otherwise, (k = Kt ), if DIR = FORE and t < T , let

t = t + 1 and return to step 1. If t = T , let DIR = BACK. Go to step 2.Step 2: For all scenarios k = 1, . . . , Kt−1 at t − 1, solve the following linear program:

et−1k = Max

Ptk∈�t

k

∑j∈Dt (k)

[(ztj )

T htj +rk∑i=1

(�tj i )T dtji +

sk∑i=1

(tj i )T etji

]ptj

Let Ptk = (ptj , j ∈ Dt (k)) the vector of probability mass that gives the maximum value of the above linear

program. Then, compute Et−1k = ∑

j∈Dt (k) ptj (z

tj )T

t−1k .

If the constraint �t−1k = 0 appears in P(t − 1, k), then remove it, let st−1

k = 1 and add the constraint (6-3) withEt−1k and et−1

k to P(t − 1, k).If et−1

k − Et−1k xt−1

k > �t−1k , then let st−1

k = st−1k + 1 and add a constraint (6-3) to P(t − 1, k).

If t = 2 and no constraints are added to P(1), then stop with x1 optimal.Otherwise, let t = t − 1, k = 1.If t = 1, let DIR = FORE.Go to step 1.

Because of the finite number of extreme probability distributions in the sets �tk , the number of optimality constraints(6-3) that will be added to all scenarios k = 1, . . . , Kt−1 at t − 1 is finite. As the nested decomposition method givesin a finite many steps the optimal solution if it exists [9] then we conclude that also the modified nested decompositionalgorithm finitely converges to an optimal solution, if it exists.In the next section, we illustrate by an example our solution strategy for the multistage stochastic programwith fuzzy

probability distribution.

5. An illustrative example

Let us consider (see Fig. 2) a scenario tree of a multistage stochastic program with fuzzy probability distribution.To get information about the possible occurrence of different scenarios, the DM consults an expert. The expert

estimates that all scenarios may have the same chance of occurrence and then ptk are “around” 0.5. The word aroundis a fuzzy word. The expert affirms that the imprecision in the given estimation of probabilities is about 0.05 for p2k ,k = 1, 2 and about 0.1 for p3k , k = 1, 2, 3, 4. Therefore, we propose to model the probabilities of different scenarios


t=1

(x11,x1

2)(1, 0.2)

x11 - x

12=10

t=2k=1

(x21,1 ,x2

1,2)

(1.5, 0.2)

x21,1 + x1

2 - x21,2=15

t=2k=2

(x22,1 ,x2

2,2)

(1.4, 0.2)x2

2,1 + x12 - x2

2,2=8

t=3k=1

(x31,1 ,x3

1,2)(1.8, 0.2)

x31,1 + x2

1,2 - x31,2=17

t=3k=2

(x32,1 ,x3

2,2)

(1.4, 0.2)x3

2,1 + x21,2 - x3

2,2=10

t=3k=3

(x33,1 ,x3

3,2)(1.4, 0.2)

x33,1 + x2

2,2 - x33,2=10

t=3k=4

(x34,1,x3

4,2)(0.9, 0.2)

x34,1 + x2

2,2 - x34,2=5

~p21

~p22

~p31

~p32

~p33

~p34

Fig. 2. For each node k at stage t, we give the decision variables xtk := (xtk,1, xtk,2) and the corresponding objective function vector c

tk and constraint

T t−1k xt−1

a(k) + Wtk x

tk = htk,1.

using fuzzy triangular numbers as follows p2k = (0.5, 0.45, 0.55), k = 1, 2 and p3k = (0.5, 0.4, 0.6), k = 1, 2, 3, 4.These fuzzy numbers verifies the requirement of Proposition 3.The multistage stochastic program with fuzzy probability distribution under study is as follows:

Min c1x1 +2∑

k2=1

p2k2

⎛⎝c2k2x2k2 +

∑k3∈DT (k2)

p3k3 [c3k3x

3k3 ]

⎞⎠s.t. Wx1 = h1

xt−1a(k),2 + xtk,1 − xtk,2 = htk, k = 1, . . . , K t , t = 2, 3

xtk,2�3, k = 1, . . . , K t , t = 1, 2, 3

xtk �0, k = 1, . . . , K t , t = 1, 2, 3 (7)

where ctk , htk,1 and W are fixed matrices with appropriate dimensions given by Fig. 2.

In the following, we solve problem (7) using the two steps solution method presented in the previous section.The first step, the fuzzy transformation step, concerns the defuzzification of the conditional fuzzy probabilities of

different scenarios in problem (7). The DM is not confident on expert estimation then he decides to choose � = 0. Thesets of possible probability distributions �tk are as follows:

�2 = {P2 := (p21, p21) : p

21 + p22 = 1, 0.45� p2k �0.55 and ptk �0 for k = 1, 2}

�31 = {P31 := (p31, p

32) : p

31 + p32 = 1, 0.4� p3k �0.6 and ptk �0 for k = 1, 2}

�32 = {P32 := (p33, p

34) : p

23 + p24 = 1, 0.4� p2k �0.6 and ptk �0 for k = 3, 4}

The resulting multistage stochastic program with interval probability distribution is

Min c1x1 +2∑

k2=1

p2k2

⎛⎝c2k2x2k2 +

∑k3∈DT (k2)

p3k3 [c3k3x

3k3 ]

⎞⎠s.t. Wx1 = h1

xt−1a(k),2 + xtk,1 − xtk,2 = htk, k = 1, . . . , K t , t = 2, 3


xtk,2�3, k = 1, . . . , K t , t = 1, 2, 3

Ptk ∈ �tk, k = 1, . . . , Kt−1, t = 2, 3

xtk �0, k = 1, . . . , K t , t = 1, 2, 3 (8)

The second step, the stochastic transformation step, targets to decompose and solve problem (8).Using the minimax approach, the decomposition yields to the following equivalent program:

Min x11 + 0.2 x12 + R2(x11 , x12 )

s.t. x11 − x12 = 10

x12 �3

x11 �0, x12 �0 (9)

where

Rt+1(xtk) = MaxPt+1k ∈�t+1

k

∑j∈Dt (k)

pt+1j Rt+1

j (xtk, wt+1j )

and

Rtk(x

t−1a(k), w

tk) = Min ctk x

tk + Rt+1(xtk)

s.t. xt−1a(k),2 + xtk,1 − xtk,2 = htk for k = 1, . . . , Kt , t = 2, 3

xtk,2�3

xtk �0

To solve problem (9), we define, for each stage t = 1, . . . , T and for each scenario k = 1, . . . , Kt , the master problemP(t, k):

Min ctk xtk + �tk

s.t. xt−1a(k),2 + xtk,1 − xtk,2 = htkDtk, j x

tk �dtk, j , j = 1, . . . , r tk, j

Etk, j x

tk + �tk �etk, j , j = 1, . . . , stk, j

x tk �0

In the following, we illustrate the two backward passes and two forward passes of the modified nested decompositionalgorithm to solve problem (9):

Step 0: Add for all P(k, t) problems the constraint �tk = 0. Let DIR = FORE, go to step 1.Step 1: All the P(t, k) problems are feasible with the following primal–dual optimal solutions:

Problem xtk ztkP(1) (10, 0) (1, 0)

P(2, 1) (15, 0) (1.5, 0)P(2, 2) (8, 0) (1.4, 0)P(3, 1) (17, 0) (1.8, 0)P(3, 2) (10, 0) (1.4, 0)P(3, 3) (10, 0) (1.4, 0)P(3, 4) (5, 0) (0.9, 0)

Let DIR = BACK and go to step 2.


Step 2: Solve the following linear programs:

e21,1 = Max (1.8, 0)

(173

)p31 + (1.4, 0)

(103

)p32

s.t. 0.4� p3k �0.6, k = 1, 2

p31 + p32 = 1

p3k �0, k = 1, 2

The vector of optimal solutions P21 = (0.6, 0.4) with e21 = 23.96. Determine the values of E2

1,1 as follows:

E21,1 = 0.6(1.8, 0)

(0 10 0

)+ 0.4(1.4, 0)

(0 10 0

)= (0, 1.64)

Remove the constraints �21 = 0 from the problem P(2, 1) and add the following optimality constraint:

1.64x21,2 + �21�23.96

Solve the following linear programs:

e22,1 = Max (1.4, 0)

(103

)p33 + (0.9, 0)

(53

)p34

s.t. 0.4� p3k �0.6, k = 3, 4

p33 + p34 = 1

p3k �0, k = 3, 4

The vector of optimal solutions P22 = (0.6, 0.4) with e22 = 10.2. Determine the values of E2

2,1 as follows:

E22,1 = 0.6(1.4, 0)

(0 10 0

)+ 0.4(0.9, 0)

(0 10 0

)= (0, 1.2)

Remove the constraints �2k = 0 from the problem P(2, 2) and add the following optimality constraint:

1.2x22,2 + �22�10.2

Go to step 1.Step 1: The solution of the problems P(2, 1) and P(2, 2) with the new added optimal constraints:

Problem x2k �2k z2k 2kP(2, 1) (15, 0) 23.96 (1.5, 0) 1P(2, 2) (8, 0) 10.2 (1.4, 0) 1

Go to step 2.Step 2: Solve the following linear program:

e11 = Max

((1.5, 0)

(153

)+ 23.96

)p21 +

((1.4, 0)

(83

)+ 10.2

)p22

s.t. 0.45� p2k �0.55, k = 1, 2

p21 + p22 = 1

p2k �0, k = 1, 2

The vector of optimal solution P1 = (0.55, 0.45) where e11 = 35.183. Determine the value of E1

1 :

E11 = 0.55(1.5, 0)

(0 10 0

)+ 0.45(1.4, 0)

(0 10 0

)= (0, 1.455)


Remove the constraint �1 = 0 and add the following optimal constraint to P(1):

1.455x12 + �1�35.183

Let DIR = FORE, go to step 1.Step 1: All the P(t, k) problems are feasible with the following primal–dual optimal solutions:

Problem xtk �tk ztk tkP(1) (13, 3) 30.818 (1, 0.255) 1

P(2, 1) (12, 0) 23.96 (1.5, 0) 1P(2, 2) (5, 0) 10.2 (1.4, 0) 1P(3, 1) (17, 0) 0 (1.8, 0) 0P(3, 2) (10, 0) 0 (1.4, 0) 0P(3, 3) (10, 0) 0 (1.4, 0) 0P(3, 4) (5, 0) 0 (0.9, 0) 0

Let DIR = BACK and go to step 2.Step 2: No optimality constraints will be added to P(2, 1), P(2, 2) and also P(1) then stop. x1 = (13, 3) is optimal

with an objective function value equal to 44.418.If the DM credibility degree � on expert information is equal to 1 then problem (7) becomes a multistage stochastic

program and the value of the optimal solution is 41.875. The difference between the two values (44.418 − 41.875) =2.543 may be viewed as the amount to pay for a credible expert.

6. Conclusion

The proposed multistage stochastic programwith fuzzy probability distribution model provides a general frameworkfor integrating uncertainty and fuzziness on probability distribution for multistage stochastic models. The solutionstrategy to the multistage stochastic program with fuzzy probability distribution was generated under two essentialassumptions considering the high level of uncertainty and ambiguity on the studied problem. First, we suppose thatthe DM is able to estimate his credibility degree � on the information sources. Second, we suppose that the DM ispessimistic or cautious toward future events and then he is looking for a robust solution.Moreover, we proposed in this work a new algorithm for multistage stochastic programs with interval probability

distribution based on the nested decomposition method. Our algorithm generates the optimal solution in finite manysteps.

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multistage stochastic programming with fuzzy probability distribution

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