interactive fuzzy stochastic two-level linear programming with simple recourse

Information Sciences xxx (2014) xxx–xxx

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Interactive fuzzy stochastic two-level linear programmingwith simple recourse

http://dx.doi.org/10.1016/j.ins.2014.03.0200020-0255/� 2014 Published by Elsevier Inc.

⇑ Corresponding author.E-mail address: [email protected] (M. Sakawa).

Please cite this article in press as: M. Sakawa et al., Interactive fuzzy stochastic two-level linear programming with simple reInform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.03.020

Masatoshi Sakawa ⇑, Hideki Katagiri, Takeshi MatsuiDepartment of System Cybernetics, Faculty of Engineering, Hiroshima University, Higashi-Hiroshima 739-8527, Japan

a r t i c l e i n f o

Article history:Received 28 September 2009Received in revised form 26 April 2010Accepted 8 March 2014Available online xxxx

Keywords:Two-level linear programmingRandom variableSimple recourseInteractive fuzzy programming

a b s t r a c t

In this paper, assuming cooperative behavior of the decision makers, two-level linear pro-gramming problems involving random variables in constraints are considered. Using theconcept of simple recourse, the formulated stochastic two-level simple recourse problemsare transformed into deterministic two-level programming ones. Taking into accountvagueness of judgments of the decision makers, interactive fuzzy programming is pre-sented. In the proposed interactive method, after determining the fuzzy goals of the deci-sion makers at both levels, a satisfactory solution is derived efficiently by updating thesatisfactory degree of the decision maker at the upper level with considerations of overallsatisfactory balance between both levels. An illustrative numerical example is provided todemonstrate the feasibility and efficiency of the proposed method.

� 2014 Published by Elsevier Inc.

1. Introduction

In actual decision making situations, we must often make a decision on the basis of vague information or uncertain data.For such decision making problems involving uncertainty, there exist two typical approaches: probability-theoretic ap-proach and fuzzy-theoretic one. Stochastic programming, as an optimization method based on the probability theory, havebeen developed in various ways [47,48,6,27], including two-stage programming [2,3,9,10,12,20,49,50] and chance con-strained programming [7,8,27,47]. Fuzzy mathematical programming representing the vagueness in decision making situa-tions by fuzzy concepts have been studied by many researchers [22,23]. Fuzzy multiobjective linear programming, firstproposed by Zimmermann [51], have been also developed by numerous researchers, and an increasing number of successfulapplications has been appearing [14,18,23–25,40,45,46,52]. In particular, after reformulating stochastic multiobjective linearprogramming problems using several models for chance constrained programming, Sakawa et al. [26,28,29] presented aninteractive fuzzy satisficing method to derive a satisficing solution for the decision maker (DM) as a generalization of theirprevious results [23,37–40].

However, decision making problems in decentralized organizations are often formulated as two-level programming prob-lems with a DM at the upper level (DM1) and another DM at the lower level (DM2) [32]. Under the assumption that theseDMs do not have motivation to cooperate mutually, the Stackelberg solution [5,19,42,43] is adopted as a reasonable solutionfor the situation. On the other hand, in the case of a project selection problem in the administrative office of a company andits autonomous divisions, the situation that these DMs can cooperate with each other seems to be natural rather than thenoncooperative situation. Assuming that the DMs essentially cooperate with each other, Lai [13] and Shih et al. [41]

course,

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2 M. Sakawa et al. / Information Sciences xxx (2014) xxx–xxx

proposed solution concepts for two-level linear programming problems. In their methods, the DMs identify membershipfunctions of the fuzzy goals for their objective functions, and in particular, the DM at the upper level also specifies thoseof the fuzzy goals for the decision variables. The DM at the lower level solves a fuzzy programming problem with a constraintwith respect to a satisfactory degree of the DM at the upper level. Unfortunately, there is a possibility that their method leadsa final solution to an undesirable one because of inconsistency between the fuzzy goals of the objective function and those ofthe decision variables. In order to overcome the problem in their methods, by eliminating the fuzzy goals for the decisionvariables, Sakawa et al. have proposed interactive fuzzy programming for two-level or multi-level linear programming prob-lems to obtain a satisfactory solution for the DMs [33,34]. The subsequent works on two-level or multi-level programmingunder fuzziness have been developed [1,15,21,30,31,35,36,44].

Realizing the importance of considering not only the fuzziness but also the randomness of coefficients of objective func-tions or constraints in mathematical programming, some researchers developed two-stage or multi-stage fuzzy stochasticprogramming [11,16,17]. However, there is no study which focuses on the simultaneous consideration of two-level decisionmaking situations and fuzzy stochastic programming approaches.

Under these circumstances, we propose a novel fuzzy stochastic two-level programming model which incorporates inter-active two-level fuzzy programming into two-stage stochastic programming. In two-level programming under a cooperativerelationship between the two DMs, the upper-level DM needs to select a solution that takes a balance between his/her ownobjective function value and the lower-level DM’s objective function value. In addition, it is significant to properly representthe imprecision of the satisfaction of DMs with respect to the goals of objective function values. From these viewpoints, inthe proposed interactive method, after determining the fuzzy goals of the DMs at both levels, a satisfactory solution is de-rived efficiently by updating the satisfactory degree of the DM at the upper level with considerations of overall satisfactorybalance between the both level DMs. The proposed method has an advantage that the problem for deriving a satisfactorysolution can be strictly solved by some convex programming techniques like the sequential quadratic programming method.A numerical example of two-level production planning problems is provided to illustrate the feasibility and efficiency of theproposed method.

2. Stochastic two-level linear programming problems

In this paper, we deal with two-level linear programming problems involving random variables in the right-hand side ofconstraints formulated as:

PleaseInform

minimizefor DM1

z1ðx1; x2Þ ¼ c11x1 þ c12x2

minimizefor DM2

z2ðx1; x2Þ ¼ c21x1 þ c22x2

subject to A1x1 þ A2x2 ¼ �bx1 P 0; x2 P 0

9>>>>>=>>>>>;: ð1Þ

In this formulation, x1 is an n1 dimensional decision variable column vector for the decision maker at the upper level(DM1), x2 is an n2 dimensional decision variable column vector for the decision maker at the lower level (DM2), c11 andc21 are n1 dimensional coefficient row vectors, c12 and c22 are n2 dimensional coefficient row vectors, A1 is an m� n1 coef-ficient matrix, A2 is an m� n2 coefficient matrix, z1ðx1; x2Þ is the objective function for DM1, z2ðx1; x2Þ is the objective func-tion for DM2, and �b is an m dimensional random variable column vector whose elements are independent of each other.

It is significant to note here that we are often faced with optimization problems involving randomness like (1). For in-stance, consider an upper level company (DM1) in charge of upper processes of n1 types and a cooperating lower level com-pany (DM2) in charge of lower processes of n2 types in the production of m products. Then, there may exist a two-leveloptimization problem where each of DMs wants to minimize its own objective function under the situation that for eachdecision variable vector xl representing the production level of DMl, the unit production cost coefficient vector for upper pro-cesses of DM1; c11, that for lower processes of DM1; c12, that for upper processes of DM2; c21, that for lower processes ofDM1; c22, the unit production amount coefficient vector for upper processes of the ith product, ai1, and that for lower pro-cesses of the ith product, ai2, are known while each demand for the ith product, bi; i ¼ 1;2; . . . ;m varies randomly.

When chance constrained programming approaches [7,8,27,47] are taken to deal with mathematical programming prob-lems with random variables, it is implicitly assumed that the realized values of the random variable coefficients cannot beobserved at all until the decision is made. However, in real-world decision making problems, we are often faced with situ-ations where the realized values of the random variables are gradually observed; firstly a DM must make a decision beforehe/she knows the realized values of random variables, and secondly, the penalty of violation of constraints is incorporated tocompensate the violation. Such a decision making methodology is called the two-stage program, which was originally intro-duced by Beale [3] as a simple recourse model, and extended as more generalized recourse models [49,50] including thefixed recourse model and the complete recourse model.

Since the simple recourse model is the most fundamental and practical among recourse models in the sense that theshortage or surplus of products can be directly compensated by the purchase of equivalent alternative products or the dis-posal of products, in this paper, we adopt the simple recourse model together with the consideration of the imprecise natureof DM’s judgment for the goals of objective function values.

cite this article in press as: M. Sakawa et al., Interactive fuzzy stochastic two-level linear programming with simple recourse,. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.03.020


M. Sakawa et al. / Information Sciences xxx (2014) xxx–xxx 3

3. Two-level simple recourse problems

In this section, we consider the situation where the decision makers need to make decisions before they know the realizedvalues of random variables, and then the constraint’s violation as a result of the realization of the random variables is com-pensated. As a two-stage programming problem which is a more general version of the simple recourse model, problem (1)can be reformulated as

PleaseInform

minimizefor DM1

z01ðx1; x2Þ ¼ c11x1 þ c12x2 þ R1ðx1; x2Þ

minimizefor DM2


subject to x1 P 0; x2 P 0

9>>=>>;; ð2Þ

where

Rlðx1; x2Þ ¼ E minyðqlyÞjWy ¼ bðxÞ � A1x1 þ A2x2; y P 0

� �ð3Þ

is called the expectation of recourse for DMl. q is a 2m dimensional constant row vector, and y is a 2m dimensional columnvectors. b is a vector whose elements are the realization biðxÞ of the random variable �bi. W is an m� 2m matrix, called re-course matrix. In particular, W is called complete fixed recourse matrix if all the elements of W are constants and if it holdsthat

frjr ¼Wy; y P 0g– ;; 8r 2 Rm:

This implies that, whatever the first-stage decision x and the realization bðxÞ of �b turn out to be, the second-stage program(3) is always feasible. As a special case, W is called simple recourse matrix if W is expressed as W ¼ ðI;�IÞwhere I is the iden-tity matrix of order m.

In this way, problem (2) can be rewritten as the following two-level simple recourse problem:

minimizefor DM1


minimizefor DM2


subject to x1 P 0; x2 P 0; yþ P 0

9>>=>>;: ð4Þ

In this formulation, Rlðx1; x2Þ defined by (3) is rewritten as

Rlðx1; x2Þ ¼ E minyþ ;y�ðqþl yþ þ q�l y�Þjyþ � y� ¼ bðxÞ � A1x1 � A2x2; y P 0

� �;

where qþl ¼ ðqþ1 ; qþ2 ; . . . ; qþmÞ; q�l ¼ ðq�1 ; q�2 ; . . . ; q�mÞ; yþ ¼ ðyþ1 ; yþ2 ; . . . ; yþmÞT and y� ¼ ðy�1 ; y�2 ; . . . ; y�mÞ

T . Since each element of yþ

means the shortage of each product and each element of y� means the surplus of each product, each element of qþl is re-garded as the unit cost to compensate the shortage of each product and each element of q�l is regarded as the unit costto dispose the surplus of each product. Hence, the assumption qþl þ q�l P 0 seems natural because we could improve theobjective function value infinitely by increasing yþi and y�i infinitely if qþli þ q�li < 0 for some i.

From the assumption, complementary relations

yþi > 0! y�i ¼ 0; y�i > 0! yþi ¼ 0; i ¼ 1;2; . . . ;m

hold for optimal recourse variable vectors yþ and y�. Therefore, the following equations are obtained for i ¼ 1;2; . . . ;m:

yþi ¼ biðxÞ � ai1x1 � ai2x2; y�i ¼ 0 if biðxÞP ai1x1 þ ai2x2

yþi ¼ 0; y�i ¼ ai1x1 þ ai2x2 � biðxÞ if biðxÞ < ai1x1 þ ai2x2

where biðxÞ is the realized value of �bi.If �bi; i ¼ 1;2; . . . ;m are mutually independent, the expectation of the recourse can be calculated as:

E minyþ ;y�ðqþl yþ þ q�l y�Þ

� �¼Xm

i¼1

qþli

Z þ1

aixbi � aixð ÞdFiðbiÞ þ

Xm

i¼1

q�li

Z aix

�1aix� bið ÞdFiðbiÞ

¼Xm

i¼1

qþli E �bi� �

�Xm

i¼1

ðqþli þ q�li ÞZ aix

�1bidFiðbiÞ �

Xm

i¼1

qþli aixþXm

i¼1

ðqþli þ q�li ÞaixFi aixð Þ

where Fið�Þ is the probability distribution function of �bi and aix ¼ ai1x1 þ ai2x2.Consequently, (4) is equivalent to the following problem:

minimizefor DM1

Z1ðx1; x2Þ

minimizefor DM2

Z2ðx1; x2Þ


9>>=>>;; ð5Þ




where

PleaseInform

Zlðx1; x2Þ ¼Xm

i¼1

qþli E �bi� �

þXn1

j¼1

cl1j �Xm

i¼1

ai1jqþli

!x1j þ

Xn2

j¼1

cl2j �Xm

i¼1

ai2jqþli

!x2j

þXm

i¼1

ðqþli þ q�li Þ ðaixÞFiðaixÞ �Z aix

�1bidFiðbiÞ

� �:

It should be noted here that (5) is a two-level convex programming problem since each of Zlðx1; x2Þ; l ¼ 1;2 is convex.

4. Interactive fuzzy programming

In general, it seems natural that the DMs have fuzzy goals for their objective functions when they take fuzziness of humanjudgments into consideration. For each of the objective functions Zlðx1; x2Þ; l ¼ 1;2, assume that the DMs have fuzzy goalssuch as ‘‘Zlðx1; x2Þ should be substantially less than or equal to some specific value.’’ This type of statement can be quantifiedby eliciting a corresponding membership function. Fig. 1 illustrates a possible shape of a monotone decreasing membershipfunction.

Having elicited the membership functions llð�Þ; l ¼ 1;2 which represent the fuzzy goals of the DMs at both levels, prob-lem (5) can be rewritten as:

maximizefor DM1

l1 Z1ðx1; x2Þð Þ

maximizefor DM2

l2 Z2ðx1; x2Þð Þ


9>>=>>;: ð6Þ

To derive a satisfactory solution to the membership function maximization problem (6), we first find the maximizingdecision of the fuzzy decision proposed by Bellman and Zadeh [4]. Namely, the following problem is solved for obtaininga solution which maximizes the smaller degree of satisfaction between those of the two DMs:

maximize minl¼1;2fllðZlðx1; x2ÞÞg


); ð7Þ

or equivalently,

maximize vsubject to v � l1ðZ1ðx1; x2ÞÞ 6 0

v � l2ðZ2ðx1; x2ÞÞ 6 0x1 P 0; x2 P 0

9>>>=>>>;: ð8Þ

Assuming llð�Þ; l ¼ 1;2 are nonincreasing and concave, it can be shown that v � llðZlðx1; x2ÞÞ is a convex function asfollows:

ðkv1 þ ð1� kÞv2Þ � llðZlðkx1 þ ð1� kÞx2ÞÞ 6 ðkv1 þ ð1� kÞv2Þ � llðkZlðx1Þ þ ð1� kÞZlðx2ÞÞ6 ðkv1 þ ð1� kÞv2Þ � ðkllðZlðx1ÞÞ þ ð1� kÞllðZlðx2ÞÞÞ¼ kðv1 � llðZlðx1ÞÞÞ þ ð1� kÞðv2 � llðZlðx2ÞÞÞ;

Fig. 1. An example of a membership function llðZlðx1; x2ÞÞ.




where x1 ¼ x11; x

12

� and x2 ¼ x2

1; x22

� . Hence, (8) is a convex programming problem and it can be solved by some traditional

convex programming technique such as the sequential quadratic programming method.Now, if DM1 is satisfied with the optimal solution ðx�1; x�2Þ to (8), it follows that the optimal solution ðx�1; x�2Þ becomes a

satisfactory solution; however, DM1 is not always satisfied with the solution ðx�1; x�2Þ. It is quite natural to assume thatDM1 specifies the minimal satisfactory level d 2 ð0;1Þ for the membership function l1ðZ1ðx1; x2ÞÞ subjectively.

Consequently, if DM1 is not satisfied with the solution ðx�1; x�2Þ to problem (8), the following problem is formulated:

PleaseInform

maximize l2ðZ2ðx1; x2ÞÞsubject to l1ðZ1ðx1; x2ÞÞP d

x1 P 0; x2 P 0

9>=>;; ð9Þ

where DM2’s membership function l2ðZ2ðx1; x2ÞÞ is maximized under the condition that DM1’s membership functionl1ðZ1ðx1; x2ÞÞ is larger than or equal to the minimal satisfactory level d specified by DM1.

Assuming that llð�Þ; l ¼ 1;2 are nonincreasing and concave, quite similar to (8), it can be shown that (9) is a convex pro-gramming problem. As a result, it is possible to solve (9) by some convex programming technique like the sequential qua-dratic programming method.

If there exists an optimal solution ðx�1; x�2Þ to problem (9), it follows that DM1 obtains a satisfactory solution having a sat-isfactory degree larger than or equal to the minimal satisfactory level specified by DM1’s self. However, the larger the min-imal satisfactory level d is assessed, the smaller the DM2’s satisfactory degree becomes when the membership functions ofDM1 and DM2 conflict with each other. Consequently, a relative difference between the satisfactory degrees of DM1 andDM2 becomes larger, and it follows that the overall satisfactory balance between both DMs is not appropriate.

In order to take account of the overall satisfactory balance between both DMs, DM1 needs to compromise with DM2 onDM1’s own minimal satisfactory level. To do so, the following ratio of the satisfactory degree of DM2 to that of DM1 ishelpful:

D ¼ l2ðZ2ðx1; x2ÞÞl1ðZ1ðx1; x2ÞÞ

which is originally introduced by Lai [13].DM1 is guaranteed to have a satisfactory degree larger than or equal to the minimal satisfactory level for the fuzzy goal

because the corresponding constraint is involved in problem (9). To take into account the overall satisfactory balance be-tween both DMs, DM1 specifies the lower bound Dmin and the upper bound Dmax of the ratio D, and D is evaluated by verifyingwhether or not it is in the interval ½Dmin;Dmax�. The condition that the overall satisfactory balance is appropriate is repre-sented by D 2 ½Dmin;Dmax�.

At the iteration k, let ðxk1; x

k2Þ; Z

kl ¼ Zlðxk

1; xk2Þ;llðZ

kl Þ and Dk ¼ l2ðZ

k2Þ=l1ðZ

k1Þ denote the current solution, DMl’s objective

function value, DMl’s satisfactory degree and the ratio of satisfactory degrees of the two DMs, respectively. The interactiveprocess terminates if the following two conditions are satisfied and DM1 concludes the solution as a satisfactory solution.

[Termination conditions of the interactive process]

Condition 1 DM1’s satisfactory degree is larger than or equal to the minimal satisfactory level d specified by DM1’s self,i.e., l1ðZ

k1ÞP d.

Condition 2 The ratio Dk of satisfactory degrees lies in the closed interval between the lower and the upper bounds spec-ified by DM1, i.e., Dk 2 ½Dmin;Dmax�.

Condition 1 ensures the minimal satisfaction to DM1 in the sense of the attainment of the fuzzy goal, and condition 2 isprovided in order to keep overall satisfactory balance between both DMs. If these two conditions are not satisfied simulta-neously, DM1 needs to update the minimal satisfactory level d. The updating procedures are summarized as follows:

[Procedure for updating the minimal satisfactory level d]

Case 1 If condition 1 is not satisfied, then DM1 decreases the minimal satisfactory level d.Case 2 If the ratio Dk exceeds its upper bound, then DM1 increases the minimal satisfactory level d. Conversely, if the ratio

Dk is below its lower bound, then DM1 decreases the minimal satisfactory level d.Case 3 Although conditions 1 and 2 are satisfied, if DM1 is not satisfied with the obtained solution and judges that it is

desirable to increase the satisfactory degree of DM1 at the expense of the satisfactory degree of DM2, then DM1increases the minimal satisfactory level d. Conversely, if DM1 judges that it is desirable to increase the satisfactorydegree of DM2 at the expense of the satisfactory degree of DM1, then DM1 decreases the minimal satisfactorylevel d.

In particular, if condition 1 is not satisfied, there does not exist any feasible solution for problem (9), and therefore DM1has to moderate the minimal satisfactory level.

Now we are ready to propose interactive fuzzy programming for deriving a satisfactory solution by updating the satisfac-tory degree of the DM at the upper level with considerations of overall satisfactory balance among all the levels.




Computational procedure of interactive fuzzy programming

Step 1: Calculate the individual minimum Zl;min by solving the following problems:

Table 1Coeffici

c11

c21

PleaseInform

minimizex1P0;x2P0Zlðx1; x2Þ; l ¼ 1;2: ð10Þ

Step 2: Ask decision makers to determine the membership functions llðZlðx1; x2ÞÞ; l ¼ 1;2.Step 3: Set k :¼ 1. Solve the maximin problem (8) for obtaining a solution which maximizes the smaller degree of satisfac-

tion between those of the two DMs and calculate Zkl ¼ Zlðxk

1; xk2Þ;llðZ

kl Þ; l ¼ 1;2 and Dk ¼ l2ðZ

k2Þ=l1ðZ

k1Þ via the opti-

mal solution ðxk1; x

k2Þ to (8). If DM1 is satisfied with the optimal solution to (8), the optimal solution becomes a

satisfactory solution and the interactive procedure is terminated. Otherwise, ask DM1 to subjectively set the min-imal satisfactory level d 2 ð0;1Þ for the membership function l1ðZ1ðx1; x2ÞÞ. Furthermore, ask DM1 to set the upperbound Dmax and the lower bound Dmin for D.

Step 4: Set k :¼ kþ 1. Solve problem (9) for finding a solution to maximize DM2’s membership function l2ðZ2ðx1; x2ÞÞ underthe condition that DM1’s membership function l1ðZ1ðx1; x2ÞÞ is larger than or equal to the minimal satisfactory leveld. Calculate Zk

l ¼ Zlðxk1; x

k2Þ;llðZ

kl Þ; l ¼ 1;2 and Dk ¼ l2ðZ

k2Þ=l1ðZ

k1Þ via the optimal solution ðxk

1; xk2Þ to (9).

Step 5: If the current solution ðxk1; x

k2Þ satisfies the termination conditions and DM1 accepts it, then the procedure stops and

the current solution is determined to be a satisfactory solution. Otherwise, ask DM1 to update the minimal satisfac-tory level d, and return to Step 4.

It should be noted here that all of problems (8), (9) and (10) to be solved in the interactive fuzzy programming algorithmare convex programming problems, and optimal solutions can be found by some convex programming technique like thesequential quadratic programming method.

5. Numerical example

To demonstrate the feasibility and efficiency of the proposed method, consider the two-level cost minimization probleminvolving random variables.

Suppose that there are two companies in which the parent company and the subsidiary company operate four kinds ofproducts (washing machine, refrigerator, air conditioner, microwave oven), and there are three production lines at each ofboth companies. The objectives of both companies are to minimize the total production costs under the demand constraints,and the decision variables are the operating times of the production lines. Then, such a decision making situation can be for-mulated as the following two-level programming problem:

minimizefor DM1 z1ðx1; x2Þ ¼ c11x1 þ c12x2

minimizefor DM2 z2ðx1; x2Þ ¼ c21x1 þ c22x2

subject to ai1x1 þ ai2x2 ¼ �bi; i ¼ 1;2;3;4x1 ¼ ðx11; x12; x13ÞT P 0

x2 ¼ ðx21; x22; x23ÞT P 0

9>>>>>>=>>>>>>;; ð11Þ

where decision variables xjk; j ¼ 1;2; k ¼ 1;2;3 are the operating times of the kth production line at company j. The coeffi-cient cijk in the objective functions represents the cost of operating the kth production line at company j per hour for pro-ducing the ith product. The coefficient aijk in the constraints represents the production volume of the ith product perhour when operating the kth production line at company j. Tables 1 and 2 show the values of cijk and aijk, respectively.

Each random variable �bi; i ¼ 1;2;3;4 represents the demand of the ith product. In this example, �b1;�b2;

�b3 and �b4 are as-sumed to be Gaussian random variables Nð300;42Þ;Nð400;42Þ;Nð200;22Þ and Nð300;12Þ, respectively, where Nðm; s2Þ standsfor a Gaussian random variable with mean m and variance s2.

Introducing the expectation of simple recourse

Rlðx1; x2Þ ¼ E minyþ ;y�ðqþl yþ þ q�l y�Þjyþ � y� ¼ bðxÞ � A1x1 � A2x2; y P 0

� �;

where qþ ¼ ðyþ1 ; . . . ; yþ4 Þ;q� ¼ ðy�1 ; . . . ; y�4 Þ; yþ ¼ ðyþ1 ; . . . ; yþ4 ÞT; y� ¼ ðy�1 ; . . . ; y�4 Þ

T;bðxÞ ¼ ðb1ðxÞ; . . . ; b4ðxÞÞT ;A1 ¼ ða11; . . . ;

a41ÞT and A2 ¼ ða12; . . . ;a42ÞT , into each objective function of this numerical example, the corresponding two-level simple re-course problem can be formulated as follows:

ent values of objective functions.

x11 x12 x13 x21 x22 x23

8.0 6.3 3.1 c12 2.0 7.8 3.37.6 2.4 6.2 c22 4.0 3.5 5.3



Table 2Coefficient values of constraints.

x11 x12 x13 x21 x22 x23

a11 10 4 3 a12 3 8 2a21 4 11 0 a22 10 5 3a31 2 2 4 a32 5 5 2a41 3 3 9 a42 3 3 7

Table 3Coeffici

qþ1qþ2


PleaseInform

minimizefor DM1 z01ðx1; x2Þ ¼ c11x1 þ c12x2 þ R1ðx1; x2Þminimizefor DM2 z02ðx1; x2Þ ¼ c21x1 þ c22x2 þ R2ðx1; x2Þsubject to x1 P 0; x2 P 0

9>=>;: ð12Þ

In this formulation, it can be interpreted that each element of qþl is regarded as the unit cost to compensate the shortage ofeach product and each element of q�l is regarded as the unit cost to dispose the surplus of each product for the lth product.These values of qþl ;q

�l ; l ¼ 1;2;3;4 are given as shown in Table 3.

Through the use of this numerical example, it is now appropriate to illustrate the proposed interactive fuzzy program-ming. The individual minimal values are calculated as:

Z1;min ¼ Z1ðx11;min; x

12;minÞ ¼ 308:548;

Z2;min ¼ Z2ðx11;min; x

12;minÞ ¼ 269:845:

In this example, as the most simple membership function, we adopt the linear membership function defined as:

llðZlðx1; x2ÞÞ ¼1; Zlðx1; x2Þ < Zl;1

Zlðx1 ;x2Þ�Zl;0Zl;1�Zl;0

; Zl;1 6 Zlðx1; x2Þ

(: ð13Þ

Following the Zimmermann’s method [51], the parameter values Zl;1 and Zl;0; l ¼ 1;2 characterizing the linear membershipfunctions are calculated as:

Z1;1 ¼ Z1ðx11;min; x

12;minÞ ¼ 308:548;


22;minÞ ¼ 389:250;


22;minÞ ¼ 32:539;


12;minÞ ¼ 379:732:

Solving the maximin problem yields l1ðZ1ðx11; x

12ÞÞ ¼ l2ðZ2ðx1

1; x12ÞÞ ¼ 0:548, and the ratio of satisfactory degrees D1 is equal

to 1.000 as shown at the column labeled ‘‘1st’’ in Table 4.Since DM1 is not satisfied with this solution, DM1 sets the minimal satisfactory level d 2 ð0;1Þ for l1ðZ1ðx1; x2ÞÞ to 0:700

so that l1ðZ1ðx1; x2ÞÞ will be improved from its current value 0:548. Furthermore, the upper bound and the lower bound ofthe ratio of satisfactory degrees D are set as Dmax ¼ 0:800 and Dmin ¼ 0:700.

Solving (9) for the specified value of d ¼ 0:700 yields l1ðZ1ðx21; x

22ÞÞ ¼ 0:700;l2ðZ2ðx2

1; x22ÞÞ ¼ 0:396 and D2 ¼ 0:566 as

shown at the column labeled ‘‘2nd’’ in Table 4. Since the ratio of satisfactory degrees D2 does not exceed Dmin ¼ 0:700,the second condition of termination of the interactive process is not fulfilled. Suppose that DM1 considers thatl2ðZ2ðx1; x2ÞÞ should be improved even if l1ðZ1ðx1; x2ÞÞ becomes worse, and DM1 updates the minimal satisfactory level dfrom 0:700 to 0:600 in order to improve l2ðZ2ðx1; x2ÞÞ. Solving (9) for the updated value of d ¼ 0:600 yieldsl1ðZ1ðx3

1; x32ÞÞ ¼ 0:600;l2ðZ2ðx3

1; x32ÞÞ ¼ 0:504 and D3 ¼ 0:841 as shown at the column labeled ‘‘3rd’’ in Table 4.

Since the ratio of satisfactory degrees D3 exceeds Dmax ¼ 0:800, the third condition of termination of the interactive pro-cess is not fulfilled. Suppose that DM1 considers that l1ðZ1ðx1; x2ÞÞ should be improved even if l2ðZ2ðx1; x2ÞÞ becomes worse,and DM1 updates the minimal satisfactory level d from 0:600 to 0:650 in order to improve l1ðZ1ðx1; x2ÞÞ. Solving (9) for theupdated value of d ¼ 0:650 yields l1ðZ1ðx3

1; x32ÞÞ ¼ 0:650;l2ðZ2ðx3

1; x32ÞÞ ¼ 0:461 and D4 ¼ 0:710 as shown at the column la-

beled ‘‘4th’’ in Table 4. Since the current solution satisfies all of the termination conditions of the interactive process andDM1 is satisfied with the current solution, the satisfactory solution is obtained and the interactive algorithm is terminated.

ent values of recourses.

�b1�b2

�b3�b4

�b1�b2

�b3�b4

3.0 3.0 1.5 1.0 q�1 5.0 4.0 2.0 1.52.5 2.0 1.5 1.0 q�2 4.0 4.0 2.0 1.5



Table 4Interaction process.

Interaction 1st 2nd 3rd 4th

d – 0.70 0.600 0.650

l1ðZ1ðxk1; x

k2ÞÞ 0.548 0.700 0.600 0.650

l2ðZ2ðxk1; x

k2ÞÞ 0.548 0.396 0.504 0.461

Dk 1.000 0.566 0.841 0.710


In the proposed interactive fuzzy programming, through a series of update procedures of the minimal satisfactory level d,it is possible to obtain a satisfactory solution where the satisfactory degree of DM1 is guaranteed to be greater than or equalto the minimal satisfactory level d and is well balanced with that of DM2.

6. Conclusion

In this paper, we focused on two-level linear programming problems with random variables in constraints. Through theuse of the simple recourse model, the formulated two-level simple recourse problems are transformed into deterministictwo-level programming ones. Taking into account vagueness of judgments of the DMs, interactive fuzzy programminghas been proposed. In the proposed interactive method, after determining the fuzzy goals of the DMs at both levels, a sat-isfactory solution is derived efficiently by updating the satisfactory degree of the DM at the upper level with considerationsof overall satisfactory balance between both levels. It is significant to note here that the transformed deterministic problemsto derive a satisfactory solution can be easily solved through some convex programming technique like the sequential qua-dratic programming method. An illustrative numerical example was provided to demonstrate the feasibility and efficiency ofthe proposed method.

As a future work, the proposed method may be extended by incorporating the sensitivity analysis. Extensions to otherstochastic programming models will be considered elsewhere. Also extensions to two-level integer programming problemsor multi-level linear programming problems involving random variables will be required in the near future.

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