Expert Systems with Applications 40 (2013) 1400–1406

Contents lists available at SciVerse ScienceDirect

Expert Systems with Applications

journal homepage: www.elsevier .com/locate /eswa

Interactive fuzzy random cooperative two-level linear programming throughlevel sets based probability maximization

Masatoshi Sakawa ⇑, Takeshi MatsuiFaculty of Engineering, Hiroshima University, Higashi-Hiroshima 739-8527, Japan

a r t i c l e i n f o a b s t r a c t

Keywords:Two-level linear programming problemsFuzzy random variablesLevel setsProbability maximizationInteractive decision making

0957-4174/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.eswa.2012.08.048

⇑ Corresponding author.E-mail address: sakawa@hiroshima-u.ac.jp (M. Sak

In this paper, assuming cooperative behavior of the decision makers, two-level linear programmingproblems under fuzzy random environments are considered. To deal with the formulated fuzzy randomtwo-level linear programming problems, a-level sets of fuzzy random variables are introduced and ana-stochastic two-level linear programming problem is defined for guaranteeing the degree of realizationof the problem. Taking into account vagueness of judgments of decision makers, fuzzy goals are intro-duced and the a-stochastic two-level linear programming problem is transformed into the problem tomaximize the satisfaction degree for each fuzzy goal. Through probability maximization, the transformedstochastic two-level programming problem can be reduced to a deterministic one. Interactive fuzzy pro-gramming to derive a satisfactory solution for the decision maker at the upper level in consideration ofthe cooperative relation between decision makers is presented. An illustrative numerical example is pro-vided to demonstrate the feasibility and efficiency of the proposed method.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

In the real world, we often encounter situations where there aretwo or more decision makers in an organization with a hierarchicalstructure, and they make decisions in turn or at the same time soas to optimize their objective functions. In particular, consider acase where there are two decision makers; one of the decisionmakers first makes a decision, and then the other who knows thedecision of the opponent makes a decision. Such a situation isformulated as a two-level programming problem (Sakawa &Nishizaki, 2009). In the context of two-level programming, thedecision maker at the upper level first specifies a strategy, and thenthe decision maker at the lower level specifies a strategy so as tooptimize the objective with full knowledge of the action of thedecision maker at the upper level. In conventional multi-levelmathematical programming models employing the solution con-cept of Stackelberg equilibrium, it is assumed that there is no com-munication among decision makers, or they do not make anybinding agreement even if there exists such communication (Bialas& Karwan, 1984; Nishizaki & Sakawa, 2000; Shimizu, Ishizuka, &Bard, 1997; Simaan & Cruz, 1973). Compared with this, for decisionmaking problems in such as decentralized large firms with divi-sional independence, it is quite natural to suppose that there existscommunication and some cooperative relationship among thedecision makers (Sakawa & Nishizaki, 2009).

ll rights reserved.

awa).

Assuming that decisions of decision makers in all levels aresequential and all of the decision makers essentially cooperatewith each other, Lai (1996) and Shih, Lai, and Lee (1996) proposedsolution concepts for two-level linear programming problems. Intheir methods, the decision makers identify membership functionsof the fuzzy goals for their objective functions, and in particular,the decision maker at the upper level also specifies those of thefuzzy goals for the decision variables. The decision maker at thelower level solves a fuzzy programming problem with a constraintwith respect to a satisfactory degree of the decision maker at theupper level. Unfortunately, there is a possibility that their methodleads a final solution to an undesirable one because of inconsis-tency between the fuzzy goals of the objective function and thoseof the decision variables. In order to overcome the problem in theirmethods, by eliminating the fuzzy goals for the decision variables,Sakawa, Nishizaki, and Uemura (1998, 2000) have proposed inter-active fuzzy programming for two-level or multi-level linear pro-gramming problems to obtain a satisfactory solution for decisionmakers. Extensions to two-level linear fractional programmingproblems (Sakawa, Nishizaki, & Uemura, 2001), decentralizedtwo-level linear programming problems (Sakawa & Nishizaki,2002; Sakawa, Nishizaki, & Uemura, 2002), two-level linear frac-tional programming problems with fuzzy parameters (Sakawaet al., 2000), and two-level nonconvex programming problemswith fuzzy parameters (Sakawa & Nishizaki, 2002) were provided.Further extensions to two-level linear programming problemswith random variables, called stochastic two-level linear program-ming problems (Sakawa & Katagiri, 2010) and two-level integer

Fig. 1. An example of the membership function leC ljk

ð�Þ of a fuzzy random variableeC ljk .

M. Sakawa, T. Matsui / Expert Systems with Applications 40 (2013) 1400–1406 1401

programming problems (Sakawa, Katagiri, & Matsui, 2012) havealso been considered. A recent survey paper of Sakawa andNishizaki (2012) is devoted to reviewing and classifying thenumerous major papers in the area of so-called cooperative mul-ti-level programming.

However, to utilize two-level programming for resolution ofconflict in decision making problems in real-world decentralizedorganizations, it is important to realize that simultaneous consid-erations of both fuzziness (Sakawa, 1993, 2000, 2001) andrandomness (Birge & Louveaux, 1997; Sakawa & Kato, 2008;Stancu-Minasian, 1984) would be required. Fuzzy random vari-ables, first introduced by Kwakernaak (1978), have been develop-ing (Kruse & Meyer, 1987; Liu & Liu, 2003; Puri & Ralescu, 1986),and an overview of the developments of fuzzy random variableswas found in Gil, Lopez-Diaz, and Ralescu (2006). Studies on linearprogramming problems with fuzzy random variable coefficients,called fuzzy random linear programming problems, were initiatedby Wang and Qiao (1993) and Qiao, Zhang, and Wang (1994) asseeking the probability distribution of the optimal solution andoptimal value. Optimization models for fuzzy random linear pro-gramming problems were first developed by Luhandjula (1996)and Luhandjula and Gupta (1996) and further developed by Liu(2001a, 2001b) and Rommelfanger (2007). A brief survey of majorfuzzy stochastic programming models including fuzzy randomprogramming was found in the paper by Luhandjula (2006).

Under these circumstances, in this paper, assuming coopera-tive behavior of the decision makers, we consider solution meth-ods for decision making problems in hierarchical organizationsunder fuzzy random environments. To deal with the formulatedtwo-level linear programming problems involving fuzzy randomvariables, a-level sets of fuzzy random variables are introducedand an a-stochastic two-level linear programming problem is de-fined for guaranteeing the degree of realization of the problem.Taking into account vagueness of judgments of decision makers,fuzzy goals are introduced and the a-stochastic two-level linearprogramming problem is transformed into the problem to max-imize the satisfaction degree for each fuzzy goal. Following prob-ability maximization, the transformed stochastic two-levelprogramming problem can be reduced to a deterministic one.Interactive fuzzy programming to obtain a satisfactory solutionfor the decision maker at the upper level in consideration ofthe cooperative relation between decision makers is presented.It is shown that all of the problems to be solved in the proposedinteractive fuzzy programming can be easily solved by the sim-plex method or the combined use of the bisection method andthe simplex method.

2. Fuzzy random two-level linear programming problem

Fuzzy random variables, first introduced by Kwakernaak (1978),have been defined in various ways (Kwakernaak, 1978; Kruse &Meyer, 1987; Liu & Liu, 2003; Puri & Ralescu, 1986). For example,as a special case of fuzzy random variables given by Kruse andMeyer (1987) defined a fuzzy random variable as follows.

Definition 1. Fuzzy random variableLet (X,B,P) be a probabilityspace, FðRÞ the set of fuzzy numbers with compact supports and Xa measurable mapping X! FðRÞ. Then X is a fuzzy randomvariable if and only if given x 2 X; XaðxÞ is a random interval forany a 2 ð0;1�, where XaðxÞ is an a-level set of the fuzzy set X(x).

Although there exist some minor differences in several defini-tions of fuzzy random variables, fuzzy random variables are con-sidered to be random variables whose observed values are fuzzysets.

In this paper, we deal with two-level linear programming prob-lems involving fuzzy random variable coefficients in objectivefunctions formulated as:

minimizefor DM1

z1ðx1; x2Þ ¼ eC 11x1 þ eC 12x2

minimizefor DM2

z2ðx1; x2Þ ¼ eC 21x1 þ eC 22x2

subject to A1x1 þ A2x2 6 bx1 P 0; x2 P 0

9>>>>>>=>>>>>>;

ð1Þ

where x1 is an n1 dimensional decision variable column vector forthe DM at the upper level (DM1), x2 is an n2 dimensional decisionvariable column vector for the DM at the lower level (DM2),z1(x1,x2) is the objective function for DM1 and z2(x1,x2) is the objec-tive function for DM2.

It should be emphasized here that randomness and fuzziness ofthe coefficients are denoted by the ‘‘dash above’’ and ‘‘wave above’’i.e., ‘‘ � ’’ and ‘‘ �’’, respectively. In this formulation, x1 is an n1

dimensional decision variable column vector for the DM at theupper level (DM1), x2 is an n2 dimensional decision variable col-umn vector for the DM at the lower level (DM2), z1(x1,x2) is theobjective function for DM1 andz2(x1,x2) is the objective functionfor DM2. In (1), eC lj; l ¼ 1;2; j ¼ 1;2 are vectors whose elementseCljk; k ¼ 1;2; . . . ;nj are fuzzy random variables characterized bythe following membership function:

leC ljk

ðsÞ ¼L

�dljk�sbljk

� �; if s 6 �dljk

R s��dljk

cljk

� �; otherwise

8><>:

where L: [0,1) ? [0,1] is a monotone decreasing function definedas L (t) = max{0,k(t)} and R: [0,1)? [0,1] is also a monotonedecreasing function defined as R(t) = max{0,q(t)}. Here, k(t) andq(t) are monotone decreasing functions which satisfy k(0) = 1 andq(0) = 1, respectively. Furthermore, parameters �dljk; bljk and cljk rep-resent a mean value, the left spread and the right spread of eCljk,respectively. In this paper, parameters �dljk are random variables de-fined as �dljk ¼ d1

ljk þ �tld2ljk, using random variables �tl with mean Ml,

l = 1,2. This definition of random variables is one of the simplestrandomization modeling of coefficients using dilation and transla-tion of random variables, as discussed by Stancu-Minasian (1990).

Fig. 1 illustrates an example of the membership function of afuzzy random variable eCljk.

Fuzzy random two-level linear programming problems formu-lated as (1) are often seen in actual decision making situations.For example, consider a supply chain planning (Roghanian, Sadjadi,& Aryanezhad, 2007) where the distribution center (DM1) and theproduction part (DM2) hope to minimize the distribution cost and

1402 M. Sakawa, T. Matsui / Expert Systems with Applications 40 (2013) 1400–1406

the production cost respectively. Since coefficients of these objec-tive functions are often affected by the economic conditions vary-ing at random, they can be regarded as random variables. Inaddition, since observed values of them are often ambiguous andestimated by fuzzy numbers, they are expressed by fuzzy randomvariables. Then, the supply chain planning problem can be formu-lated as a two-level linear programming problem involving fuzzyrandom variable coefficients like (1).

Since each coefficient eCljk is a fuzzy random variable defined asa random variable whose observed values are L-R fuzzy numbers,each objective function eC lx ¼ eC l1x1 þ eC l2x2 is also a fuzzy randomvariable whose observed values are fuzzy numbers characterizedby the following membership function.

leC lxðtÞ ¼

L�dlx�tblx

� �; if t 6 �dlx

R t��dlxclx

� �; otherwise

8><>:

An example of the membership function of the objective func-tion of DMl is shown in Fig. 2.

3. Level sets and fuzzy goals

Observing that (1) involves fuzzy random variables in the objec-tive functions, we first introduce the a-level set of the fuzzy ran-dom variables. The a-level set of the fuzzy random variables eCljk

is defined as a random interval for which the degree of their mem-bership functions exceeds the level a:

eC ljka ¼ sjleC ljk

ðsÞP a; s 2 R� �

; j ¼ 1;2; k ¼ 1;2; . . . ; nj:

For notational convenience, in the following, let eC la ¼ ðeC l1a;eC l2aÞ,

l = 1,2 be an a-level set defined as the Cartesian product of a-level

sets eCljka of fuzzy random variables eCljk; j ¼ 1;2; k ¼ 1;2; . . . ;nj.Now suppose that DM1 decides that the degree of all of the

membership functions of the fuzzy random variables involved in(1) should be greater than or equal to some value a. Then for sucha degree a, (1) can be interpreted as the following stochastic two-level linear programming problem which depends on the coeffi-

cient vectors ðC11;C12Þ 2 ðeC 11a;eC 12aÞ and ðC21;C22Þ 2 ðeC 21a;

eC 22aÞ:

minimizefor DM1

z1ðx1; x2Þ ¼ C11x1 þ C12x2

minimizefor DM2

z2ðx1; x2Þ ¼ C21x1 þ C22x2

subject to A1x1 þ A2x2 6 bx1 P 0; x2 P 0

9>>>>>=>>>>>;: ð2Þ

Fig. 2. An example of the membership function leC l xð�Þ of the objective function of

DMl.

Observe that there exists an infinite number of such problems

depending on the coefficient vector ðC11;C12Þ 2 ðeC 11a;eC 12aÞ and

ðC21;C22Þ 2 ðeC 21a;eC 22aÞ, and the values of ðC11;C12Þ and ðC21;C22Þ

are arbitrary for any ðC11;C12Þ 2 ðeC 11a;eC 12aÞ and ðC21;C22Þ 2

ðeC 21a;eC 22aÞ in the sense that the degree of all of the membership

functions for the fuzzy random variables in (2) exceeds the levela. However, if possible, it would be desirable for DM1 to choose

ðC11;C12Þ 2 ðeC 11a;eC 12aÞ and ðC21;C22Þ 2 ðeC 21a;

eC 22aÞ in (2) to mini-mize the objective functions under the constraints. From such apoint of view, for a certain degree a, it seems to be quite naturalto have (2) reformulated as the following a-stochastic two-levellinear programming problem:

minimizefor DM1

z1ðx1; x2Þ ¼ C11x1 þ C12x2

minimizefor DM2

z2ðx1; x2Þ ¼ C21x1 þ C22x2

subject to A1x1 þ A2x2 6 bx1 P 0; x2 P 0

C1 ¼ ðC11;C12Þ 2 eC 1a; C2 ¼ ðC21; C22Þ 2 eC 2a

9>>>>>>>>=>>>>>>>>;: ð3Þ

Considering vague natures of the decision makers’ judgment, itis natural to assume that decision makers may have vague or fuzzygoals for each of the objective functions in the a-stochastic two-le-vel linear programming problem (3). In a minimization problem, agoal stated by decision makers may be to achieve ‘‘substantiallyless than or equal to some value.’’ This type of statement can bequantified by eliciting a corresponding membership function.Fig. 3 illustrates a possible shape of a monotone decreasing mem-bership function.

Having elicited the membership functions ll(�), l = 1,2 whichwell represent the fuzzy goals of the decision makers at both levels,problem (3) can be transformed as:

maximizefor DM1

l1ðC1xÞ

maximizefor DM2

l2ðC2xÞ

subject to A1x1 þ A2x2 6 bx1 P 0; x2 P 0

C1 2 eC 1a; C2 2 eC 2a

9>>>>>>>>=>>>>>>>>;: ð4Þ

Observing C lx and llðC lxÞ involve random variables, it is significantto note here (4) is a stochastic programming problem.

4. Probability maximization

Observing that (4) contains random variable coefficients, solu-tion methods for ordinary deterministic two-level linear program-ming problems cannot be directly applied. In stochasticprogramming, expectation optimization, variance minimization,probability maximization and fractile criterion optimization (Birge& Louveaux, 1997; Charnes & Cooper, 1959, 1963; Kataoka, 1963;

Fig. 3. An example of a membership function ll(�) of a fuzzy goal.

M. Sakawa, T. Matsui / Expert Systems with Applications 40 (2013) 1400–1406 1403

Sakawa & Kato, 2008; Stancu-Minasian, 1984) are typical optimi-zation models for objective functions involving random variables.For instance, let the objective function represent a profit. If thedecision maker wishes to simply maximize the expected profitwithout caring about the fluctuation of the profit, the expectationoptimization model (Charnes & Cooper, 1963) to optimize theexpectation of the objective function is appropriate. On the otherhand, if the decision maker hopes to decrease the fluctuation ofthe profit as little as possible from the viewpoint of the stabilityof the profit, the variance minimization model (Charnes & Cooper,1963) to minimize the variance of the objective function is useful.In contrast to these two types of optimizing approaches, as satisfic-ing approaches, the probability maximization model (Charnes &Cooper, 1963) and the fractile criterion optimization model or Kat-aoka’s model (Kataoka, 1963) have been proposed. When the deci-sion maker wants to maximize the probability that the profit isgreater than or equal to a certain permissible level, probabilitymaximization model (Charnes & Cooper, 1963) is recommended.In contrast, when the decision maker wishes to optimize such apermissible level as the probability that the profit is greater thanor equal to the permissible level is greater than or equal to a cer-tain threshold, the fractile criterion optimization model will beappropriate.

In this paper, assuming that the decision makers are interestedin the probability that each objective function attains a goal valuerather than the expectation or variance of each membership func-tion, we adopt the probability maximization model (Charnes &Cooper, 1963) as a decision making model. Through probabilitymaximization, problem (4) can be rewritten as:

maximizefor DM1

Prfl1ðC1xÞP h1g

maximizefor DM2

Prfl2ðC2xÞP h2g

subject to A1x1 þ A2x2 6 bx1 P 0; x2 P 0

C1 2 eC 1a; C2 2 eC 2a

9>>>>>>>>=>>>>>>>>;; ð5Þ

where hl is a goal value, called a permissible level, for the member-ship function ll(�).

Now, let CLljka and CR

ljka be s satisfying Lðð�dljk � sÞ=bljkÞ ¼ a and s0

satisfying Rððs0 � �dljkÞ=cljkÞ ¼ a, respectively. Then, the a-level set ofeCljk becomes a closed interval CLljka;C

Rljka

h iwhich varies randomly,

as shown in Fig. 4.Hence, (5) can be rewritten as:

maximizefor DM1

Prfl1 CL1ax

� �P h1g

maximizefor DM2

Pr l2 CL2ax

� �P h2

n osubject to A1x1 þ A2x2 6 b

x1 P 0; x2 P 0

9>>>>>>=>>>>>>;: ð6Þ

Fig. 4. An example of the a-level set of a fuzzy random variable eCljk .

Since ll(�), l = 1,2 are monotone decreasing, (6) can be rewritten as:

maximizefor DM1

Pr CL1ax 6 l�1ðh1Þ

n omaximize

for DM2Pr CL

2ax 6 l�2ðh2Þn o

subject to A1x1 þ A2x2 6 bx1 P 0; x2 P 0

9>>>>>>=>>>>>>;; ð7Þ

where l�l ð�Þ is a pseudo-inverse function of ll(�) defined byl�l ðhlÞ ¼ supfyjllðyÞP hlg for hl 2 (0,1].

In view of a ¼ L �dljk � CLljka

� �=bljk

� �in (7), it holds that

CLljka ¼ �dljk � L�ðaÞ � bljk;

where L⁄(�) is a pseudo-inverse function of L(�) defined by L⁄(a) =sup{sjL(s) P a}. From this result, the left side of the first and secondconstraint in (7) can be expressed as:

Pr CLlax 6 l�l ðhlÞ

n o¼ Pr ð�dl � L�ðaÞ � blÞx 6 l�l ðhlÞ

� �:

From the assumption, �dl is a random variable vector defined as�dl ¼ d1

l þ �tl � d2l using a random variable �tl with mean Ml. Provided

that d2l x > 0; l ¼ 1;2,

Pr ð�dl�L�ðaÞ �blÞx6l�l ðhlÞ� �

¼Pr d1l þ�tl �d2

l �L�ðaÞ �bl

� �x6l�l ðhlÞ

n o

¼Pr �tl6

�d1l þL�ðaÞ �bl

� �xþl�l ðhlÞ

d2l x

8<:

9=;

¼ Tl

�d1l þL�ðaÞ �bl

� �xþl�l ðhlÞ

d2l x

0@

1A;ð8Þ

where Tl(�) is a probability distribution of �tl for DMl.In this way, (7) can be transformed into the following problem:

maximizefor DM1

ZP1aðxÞ ¼ T1

�d11þL�ðaÞ�b1ð Þxþl�1ðh1Þ

d21x

�

maximizefor DM2

ZP2aðxÞ ¼ T2

�d12þL�ðaÞ�b2ð Þxþl�2ðh2Þ

d22x

� subject to A1x1 þ A2x2 6 b

x1 P 0; x2 P 0

9>>>>>>>=>>>>>>>;: ð9Þ

5. Interactive fuzzy programming

Observing the transformed problem (9) is a deterministic two-level programming problem, we can now construct the interactivealgorithm to derive a satisfactory solution for the decision maker atthe upper level in consideration of the cooperative relationshipsbetween DM1 and DM2.

5.1. Interactive fuzzy programming

Step 1: In order to calculate the individual minimum of E{zl(x1,x2)},solve the following problems:

minimize Efzlðx1; x2Þg ¼ d1l þMl � d2

l

� �x

subject to A1x1 þ A2x2 6 bx1 P 0; x2 P 0

9>>=>>;; l ¼ 1;2:

ð10Þ

Let xlmin and zE

l;min be optimal solutions to (10) and minimal objectivefunction values to (10), respectively. Observing that (10) are linearprogramming problems, they can be easily solved by some linear

1404 M. Sakawa, T. Matsui / Expert Systems with Applications 40 (2013) 1400–1406

programming technique like the simplex method.Step 2: Ask DMs to specify the membership functions ll(�), l = 1,2

by considering the obtained values of zEl;min; l ¼ 1;2.

Step 3: Ask DM1 to determine the initial value of the degree ofrealization a 2 (0,1) and those of permissible levels hl,l = 1,2.

Step 4: For the specified values of a and hl, l = 1,2, the followingmaximin problem is solved:

maximize min ZP1aðxÞ; Z

P2aðxÞ

n osubject to A1x1 þ A2x2 6 b

x1 P 0; x2 P 0

9>>=>>; ð11Þ

equivalently,

maximize vsubject to ZP

1aðxÞP vZP

2aðxÞP vA1x1 þ A2x2 6 bx1 P 0; x2 P 0

9>>>>>>=>>>>>>;: ð12Þ

In view of (8), this problem is rewritten as:

maximize v

subject to T1�d1

1þL�ðaÞ�b1ð Þxþl�1ðh1Þd2

1x

� P v

T2�d1

2þL�ðaÞ�b2ð Þxþl�2ðh2Þd2

2x

� P v

A1x1 þ A2x2 6 bx1 P 0; x2 P 0

9>>>>>>>>>>=>>>>>>>>>>;

ð13Þ

equivalently,

maximize v

subject to�d1

1þL�ðaÞ�b1ð Þxþl�1ðh1Þd2

1xP T�1ðvÞ

�d12þL�ðaÞ�b2ð Þxþl�2ðh2Þ

d22x

P T�2ðvÞ

A1x1 þ A2x2 6 bx1 P 0; x2 P 0

9>>>>>>>>=>>>>>>>>;: ð14Þ

where T�l ð�Þ is a pseudo-inverse function of Tl(�).

Obtaining the optimal value of v to this problem is equivalent tofinding the maximum of v so that the set of feasible solutions to(14) is not empty. Although this problem is a nonlinear program-ming problem, we can easily find the maximum of v by the follow-ing algorithm based on the bisection method and the simplexmethod since the constraints of (14) are linear if v is fixed.

5.2. The combined use of the bisection method and the simplex method

4-1 Set l :¼ 0 and v :¼ 0. Test whether the set of feasible solu-tions to (14) for v = 0 is empty or not using the phase oneof the simplex method. If it is empty, DMs must reassessmembership functions, a or hl. Otherwise, let vfeasible :¼ vand go to 4-2.

4-2 Set v :¼ 1. Test whether the set of feasible solutions to (14)for v = 1 is empty or not using the phase one of the simplexmethod. If it is not empty, v = 1 is the optimal value v⁄ to (14)and the algorithm is terminated. Otherwise, the maximumof v so that the set of feasible solutions to (14) is not emptyexists between 0 and 1. Let v infeasible :¼ v and go to 4-3.

4-3 Set v:¼(vfeasible + vinfeasible)/2, l :¼ l + 1 and go to 4-4.4-4 Test whether the set of feasible solutions to (14) for v deter-

mined in 4-3 is empty or not using the phase one of the sim-plex method. Observe that the sensitivity analysis technique

can be applied for this test. If it is not empty and (1/2)l6 e,

the current value of v is regarded as the optimal value v⁄

to (14) and the algorithm is terminated. If it is not emptyand (1/2)l > e, let v feasible :¼ v and return to 4-3. On the otherhand, if it is empty, let v infeasible :¼ v and return to 4-3.

Then, for the obtained optimal value v⁄, we can determine thecorresponding optimal value x⁄ by solving the following linearfractional programming problem:

maximize�d1

1þL�ðaÞ�b1ð Þxd2

1x

subject to T�2ðv�Þ � d22 þ d1

2 � L�ðaÞ � b2

� �x 6 l�2ðh2Þ

A1x1 þ A2x2 6 bx1 P 0; x2 P 0

9>>>>>>=>>>>>>;: ð15Þ

It should be noted here that linear fractional programmingproblems can be reduced to equivalent linear programming prob-lems through the variable transformation method by Charnesand Cooper (1969). Therefore, we can obtain an optimal solutionto (15) by some linear programming technique.

Step 5: The DM1 is supplied with the current values of ZP1aðx�Þ and

ZP2aðx�Þ for the optimal solution x⁄ calculated in step 4. If

DM1 is satisfied with the current membership functionvalues, the interaction process is terminated. If DM1 isnot satisfied and desires to update a and/or hl, l = 1,2, askDM1 to update a and/or hl and return to step 4. Otherwise,ask DM1 to specify the minimal satisfactory level d̂ forZP

1aðxÞ and the permissible range [Dmin,Dmax] of the ratioof membership functions D ¼ ZP

2aðxÞ=ZP1aðxÞ.

Observe that the larger the minimal satisfactory level isassessed, the smaller the DM2’s satisfactory degreebecomes. Consequently, in order to take account of theoverall satisfactory balance between both decision makers,DM1 needs to compromise with DM2 on DM1’s own min-imal satisfactory level. To do so, the permissible range ofthe ratio of the satisfactory degree of DM2 to that ofDM1 is helpful.

Step 6: For the specified value of d̂, solve the following problem tomaximize the membership function ZP

2aðxÞ of DM2 consid-ering the constraint that the membership function ZP

1aðxÞof DM1 must be greater than or equal to d̂.

maximize ZP2aðxÞ

subject to ZP1aðxÞP d̂

A1x1 þ A2x2 6 bx1 P 0; x2 P 0

9>>>>=>>>>;: ð16Þ

This problem can be rewritten as:

maximize T2�d1

2þL�ðaÞ�b2ð Þxþl�2ðh2Þd2

2x

�

subject to T1�d1

1þL�ðaÞ�b1ð Þxþl�1ðh1Þd2

1x

� P d̂

A1x1 þ A2x2 6 bx1 P 0; x2 P 0

9>>>>>>>=>>>>>>>;

ð17Þ

equivalently,

maximize�d1

2þL�ðaÞ�b2ð Þxd2

2x

subject to T�1ðd̂Þ � d21 þ d1

1 � L�ðaÞ � b1

� �x 6 l�1ðh1Þ

A1x1 þ A2x2 6 bx1 P 0; x2 P 0

9>>>>>>=>>>>>>;:

ð18Þ

Table 2Value of each element of ai, i = 1,2,3,4.

x11 x12 x13 x14 x15 x21 x22 x23 x24 x25

a1 3.0 2.0 1.0 4.0 5.0 3.0 2.0 6.0 2.0 1.0a2 2.0 1.0 2.0 3.0 5.0 2.0 4.0 4.0 1.0 3.0a3 3.0 4.0 3.0 5.0 2.0 4.0 1.0 3.0 6.0 1.0a4 1.0 3.0 2.0 2.0 5.0 1.0 3.0 2.0 1.0 4.0

M. Sakawa, T. Matsui / Expert Systems with Applications 40 (2013) 1400–1406 1405

Observing that problem (18) is a linear fractional programmingproblem, it can be easily solved by some linear programming tech-nique through the variable transformation method by Charnes andCooper (1969) like (15). For the optimal solution x⁄ to (16), calculateZP

1aðx�Þ; ZP2aðx�Þ and D.

Step 7: DM1 is supplied with the current values of ZP1aðx�Þ; ZP

2aðx�Þand D calculated in step 6. If D 2[Dmin,Dmax] and DM1 issatisfied with the current membership function valuesfor the optimal solution x⁄, the interaction process is ter-minated. Otherwise, ask DM1 to update the degree of real-ization a, the permissible level hl, l = 1,2 or the minimalsatisfactory level d̂, and return to step 6.

In the proposed algorithm, Dmin and Dmax are usually set to beless than 1 since ZP

1aðx�Þ should be greater than ZP2aðx�Þ because of

the priority of DM1. In Step 6, if D < Dmin, i.e., ZP1aðx�Þ is much great-

er than ZP2aðx�Þ, DM1 will decrease d̂ to improve ZP

2aðx�Þ and in-crease D. Conversely, if Dmax < D, i.e., ZP

1aðx�Þ is slightly greater orless than ZP

2aðx�Þ, DM1 will increase d̂ to improve ZP1aðx�Þ and de-

crease D. On the other hand, if DM1 decreases (increases) a and/or hl, l = 1,2, both ZP

1aðx�Þ and ZP2aðx�Þ would increase (decrease).

With this observation, it can be expected that desirable values ofZP

1aðx�Þ; ZP2aðx�Þ and D will be obtained through a series of update

procedures of d̂; a and/or hl, l = 1,2 with DM1.

6. Numerical example

To demonstrate the feasibility and efficiency of the proposedmethod, consider the following two-level linear programmingproblem involving fuzzy random variable coefficients:

minimizefor DM1

z1ðx1; x2Þ ¼ eC 11x1 þ eC 12x2

minimizefor DM2

z2ðx1; x2Þ ¼ eC 21x1 þ eC 22x2

subject to a11x1 þ a12x2 6 100a21x1 þ a22x2 6 115a31x1 þ a32x2 6 155a41x1 þ a42x2 6 110x1 ¼ ðx11; x12; x13; x14; x15ÞT P 0

x2 ¼ ðx21; x22; x23; x24; x25ÞT P 0

9>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>;

; ð19Þ

where k(�) and q(�) are defined as kðtÞ ¼ qðtÞ ¼ 1� t; �t1 is a Gauss-ian random variable with mean M1 = 0.0 and variance r2

1 ¼ 4 and �t2

is also a Gaussian random variable with mean M2 = 0.0 and variancer2

2 ¼ 4.Table 1 shows values of parameter vectors of fuzzy random

variables d1l ; d2

l ; bl and cl l = 1,2, and Table 2 shows values of coef-ficients of constraints ai, i = 1,2,3,4.

Through the use of this numerical example, it is now appropri-ate to illustrate the proposed interactive fuzzy programming.

Table 1Value of each element of d1

l ; d2l ; bl ; cl ; l ¼ 1;2.

x11 x12 x13 x14 x15 x21 x22 x23 x24 x25

d11�5.0 �3.0 �3.0 �4.0 �4.0 �2.0 �3.0 �3.0 �2.0 �2.0

d12�1.0 �2.0 �2.0 �3.0 �3.0 �4.0 �3.0 �4.0 �4.0 �3.0

d21

2.0 1.0 2.0 3.0 2.0 1.0 2.0 2.0 1.0 2.0

d22

1.0 1.0 1.0 2.0 2.0 2.0 2.0 3.0 3.0 2.0

b1 0.3 0.4 0.2 0.5 0.2 0.4 0.3 0.2 0.5 0.3b2 0.3 0.1 0.2 0.3 0.5 0.3 0.1 0.5 0.4 0.1c1 0.3 0.1 0.2 0.3 0.5 0.3 0.1 0.5 0.4 0.1c2 0.3 0.4 0.2 0.5 0.2 0.4 0.3 0.2 0.5 0.3

Expectation optimization problems are solved by the simplexmethod and the individual minimum z1

min ¼ �205:000; z2min ¼ �

178:243 are obtained.Although the membership function does not always need to be

linear, for the sake of simplicity, we adopt the linear membershipfunction defined as:

llðzlðx1; x2ÞÞ ¼1; if zlðx1; x2Þ < zl;1;zlðx1 ;x2Þ�zl;0

zl;1�zl;0; if zl;1 6 zlðx1; x2Þ 6 zl;0;

0; if zlðx1; x2Þ > zl;0;

8>><>>:

Following the Zimmermann’s method (Zimmermann, 1978), theparameter values characterizing the linear membership functionsare determined as z1,1 = �205.000, z1,0 = �99.189, z2,1 = �178.243and z2,0 = �92.500.

Initial values of the degree of realization of the problem a andthe permissible levels hl, l = 1,2 are set as a = 0.7, h1 = 0.3 andh2 = 0.3. For these initial values, (13) is solved through the com-bined use of the bisection method and the simplex method. Theobtained result is shown at the column labeled ‘‘1st’’ in Table 3.DM1 is not satisfied with this solution, but he does not desire toupdate a and hl, l = 1,2. Thus, DM1 determines the minimal satis-factory level d̂ ¼ 0:65 to improve ZP

1aðxÞ at the expense of ZP2aðxÞ.

Furthermore, DM1 specifies the upper bound Dmax = 0.80 and thelower bound Dmin = 0.60 for the ratio of objective functionsD ¼ ZP

2aðxÞ=ZF1aðxÞ.

For the updated value of d̂, (18) is solved by the simplex meth-od. The obtained result is shown at the column labeled ‘‘2nd’’ inTable 3. DM1 considers that ZP

1aðxÞ is improved but ZP2aðxÞ is too

bad, and D is less than Dmin. Hence, DM1 is not satisfied with thissolution and updates the minimal satisfactory level d̂ from 0.65 to0.55. (18) is solved for the updated value of d̂, and the obtained re-sult is shown at the column labeled ‘‘3rd’’ in Table 3. Since ZP

2aðxÞ isimproved but D is greater than Dmax, DM1 is not satisfied with thissolution and updates the minimal satisfactory level d̂ from 0.55 to0.60. For the updated d̂, (18) is solved and the obtained result isshown at the column labeled ‘‘4th’’ in Table 3. Since D exists inthe interval [Dmin,Dmax] and DM1 is satisfied with the balance be-tween ZP

1aðxÞ and ZP2aðxÞ, the interactive algorithm is terminated.

In the proposed interactive fuzzy programming, through a ser-ies of update procedures of the minimal satisfactory level d̂, thedegree of realization a and the probability level hl, l = 1,2, it canbe possible to obtain a satisfactory solution where the satisfactorydegree of DM1 is guaranteed to be greater than or equal to theminimal satisfactory level d̂ and is well balanced with that of DM2.

Table 3Interaction process.

Interaction 1st 2nd 3rd 4th

d̂ – 0.65 0.55 0.60

a 0.70 0.70 0.70 0.70h1 0.30 0.30 0.30 0.30h2 0.30 0.30 0.30 0.30

ZP1aðxÞ 0.524 0.650 0.550 0.600

ZP2aðxÞ 0.524 0.244 0.503 0.445

D 1.0 0.375 0.916 0.741

1406 M. Sakawa, T. Matsui / Expert Systems with Applications 40 (2013) 1400–1406

7. Conclusions

In this paper, assuming cooperative behavior of the decisionmakers, interactive decision making methods in hierarchical orga-nizations under fuzzy random environments were considered. Forthe formulated fuzzy random two-level linear programming prob-lems, a-level sets of fuzzy random variables were introduced andan a-stochastic two-level linear programming problem was de-fined for guaranteeing the degree of realization of the problem.Considering the vague natures of decision makers’ judgments, fuz-zy goals were introduced and the a-stochastic two-level linear pro-gramming problem was transformed into the problem to maximizethe satisfaction degree for each fuzzy goal. Through the probabilitymaximization model, the transformed stochastic two-level pro-gramming problem was reduced to a deterministic one. Interactivefuzzy programming to obtain a satisfactory solution for the deci-sion maker at the upper level in consideration of the cooperativerelation between decision makers was presented. It should beemphasized here that all problems to be solved in the proposedinteractive fuzzy programming can be easily solved by the simplexmethod or the combined use of the bisection method. An illustra-tive numerical example demonstrated the feasibility and efficiencyof the proposed method. However, as a subject of future work,applications of the proposed method to the real world decisionmaking situations will be required in the near future. Extensionsto other stochastic programming models will be considered else-where. Considerations from the view point of fuzzy random two-level linear programming problems with two decision makersunder noncooperative environments will be reported elsewhere.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.eswa.2012.08.048.

References

Bialas, W. F., & Karwan, M. H. (1984). Two-level linear programming. ManagementScience, 30(8), 1004–1020.

Birge, J. R., & Louveaux, F. (1997). Introduction to stochastic programming. London:Springer.

Charnes, A., & Cooper, W. W. (1959). Chance constrained programming.Management Science, 6(1), 73–79.

Charnes, A., & Cooper, W. W. (1963). Deterministic equivalents for optimizing andsatisficing under chance constraints. Operations Research, 11(1), 18–39.

Charnes, A., & Cooper, W. W. (1969). Programming with linear fractionalfunctionals. Naval Research Logistics Quarterly, 9(3–4), 181–186.

Gil, M. A., Lopez-Diaz, M., & Ralescu, D. A. (2006). Overview on the development offuzzy random variables. Fuzzy Sets and Systems, 157(19), 2546–2557.

Kataoka, S. (1963). A stochastic programming model. Econometorica, 31(1–2),181–196.

Kruse, R., & Meyer, K. D. (1987). Statistics with vague data. Dordrecht: D. RiedelPublishing Company.

Kwakernaak, H. (1978). Fuzzy random variables – I. Definitions and theorems.Information Sciences, 15(1), 1–29.

Lai, Y. J. (1996). Hierarchical optimization: A satisfactory solution. Fuzzy Sets andSystems, 77(3), 321–325.

Liu, B. (2001a). Fuzzy random chance-constrained programming. IEEE Transactionson Fuzzy Systems, 9(5), 713–720.

Liu, B. (2001b). Fuzzy random dependent-chance programming. IEEE Transactionson Fuzzy Systems, 9(5), 721–726.

Liu, Y.-K., & Liu, B. (2003). Fuzzy random variables: A scalar expected valueoperator. Fuzzy Optimization and Decision Making, 2(2), 143–160.

Luhandjula, M. K. (1996). Fuzziness and randomness in an optimization framework.Fuzzy Sets and Systems, 77(3), 291–297.

Luhandjula, M. K. (2006). Fuzzy stochastic linear programming: Survey and futureresearch directions. European Journal of Operational Research, 174(3),1353–1367.

Luhandjula, M. K., & Gupta, M. M. (1996). On fuzzy stochastic optimization. FuzzySets and Systems, 81(1), 47–55.

Nishizaki, I., & Sakawa, M. (2000). Computational methods through geneticalgorithms for obtaining Stackelberg solutions to two-level mixed zero-oneprogramming problems. Cybernetics and Systems: An International Journal, 31(2),203–221.

Puri, M. L., & Ralescu, D. A. (1986). Fuzzy random variables. Journal of MathematicalAnalysis and Applications, 114(2), 409–422.

Qiao, Z., Zhang, Y., & Wang, G.-Y. (1994). On fuzzy random linear programming.Fuzzy Sets and Systems, 65(1), 31–49.

Roghanian, E., Sadjadi, S. J., & Aryanezhad, M. B. (2007). A probabilistic bi-levellinear multi-objective programming problem to supply chain planning. AppliedMathematics and Computation, 188(1), 786–800.

Rommelfanger, H. (2007). A general concept for solving linear multicriteriaprogramming problems with crisp, fuzzy or stochastic values. Fuzzy Sets andSystems, 156(17), 1892–1904.

Sakawa, M. (1993). Fuzzy sets and interactive multiobjective optimization. New York:Plenum Press.

Sakawa, M. (2000). Large scale interactive fuzzy multiobjective programming.Heidelberg: Physica-Verlag.

Sakawa, M. (2001). Genetic algorithms and fuzzy multiobjective optimization. Boston:Kluwer Academic Publishers.

Sakawa, M., & Katagiri, H. (2010). Interactive fuzzy programming based on fractilecriterion optimization model for two-level stochastic linear programmingproblems. Cybernetics and Systems: An International Journal, 41(7), 508–521.

Sakawa, M., Katagiri, H., & Matsui, T. (2012). Interactive fuzzy stochastic two-levelinteger programming through fractile criterion optimization. OperationalResearch: An International Journal, 12(2), 209–227.

Sakawa, M., & Kato, K. (2008). Interactive fuzzy multi-objective stochastic linearprogramming. In C. Kahraman (Ed.), Fuzzy multi-criteria decision making –Theory and applications with recent developments (pp. 375–408). New York:Springer.

Sakawa, M., & Nishizaki, I. (2002). Interactive fuzzy programming for decentralizedtwo-level linear programming problems. Fuzzy Sets and Systems, 125(3),301–315.

Sakawa, M., & Nishizaki, I. (2002). Interactive fuzzy programming for two-levelnonconvex programming problems with fuzzy parameters through geneticalgorithms. Fuzzy Sets and Systems, 127(2), 185–197.

Sakawa, M., & Nishizaki, I. (2009). Cooperative and noncooperative multi-levelprogramming. New York: Springer.

Sakawa, M., & Nishizaki, I. (2012). Interactive fuzzy programming for multi-levelprogramming problems: A review. International Journal of Multicriteria DecisionMaking, 2(3), 241–266.

Sakawa, M., Nishizaki, I., & Uemura, Y. (1998). Interactive fuzzy programming formulti-level linear programming problems. Computers & Mathematics withApplications, 36(2), 71–86.

Sakawa, M., Nishizaki, I., & Uemura, Y. (2000). Interactive fuzzy programming fortwo-level linear fractional programming problems with fuzzy parameters. FuzzySets and Systems, 115(1), 93–103.

Sakawa, M., Nishizaki, I., & Uemura, Y. (2001). Interactive fuzzy programming fortwo-level linear and linear fractional production and assignment problems: Acase study. European Journal of Operational Research, 135(1), 142–157.

Sakawa, M., Nishizaki, I., & Uemura, Y. (2002). A decentralized two-leveltransportation problem in a housing material manufacturer – Interactivefuzzy programming approach –. European Journal of Operational Research,141(1), 167–185.

Shih, H. S., Lai, Y. J., & Lee, E. S. (1996). Fuzzy approach for multi-level programmingproblems. Computers and Operations Research, 23(1), 73–91.

Shimizu, K., Ishizuka, Y., & Bard, J. F. (1997). Nondifferentiable and two-levelmathematical programming. Boston: Kluwer Academic Publishers..

Simaan, M., & Cruz, J. B. Jr., (1973). On the Stackelberg strategy in nonzero-sumgames. Journal of Optimization Theory and Applications, 11(5), 533–555.

Stancu-Minasian, I. M. (1984). Stochastic programming with multiple objectivefunctions. Dordrecht: D. Reidel Publishing Co..

Stancu-Minasian, I. M. (1990). Overview of different approaches for solvingstochastic programming problems with multiple objective functions. In R.Slowinski & J. Teghem (Eds.), Stochastic versus fuzzy approaches to multiobjectivemathematical programming under uncertainty (pp. 71–101). Dordrecht/Boston/London: Kluwer Academic Publishers.

Wang, G.-Y., & Qiao, Z. (1993). Linear programming with fuzzy random variablecoefficients. Fuzzy Sets and Systems, 57(3), 295–311.

Zimmermann, H.-J. (1978). Fuzzy programming and linear programming withseveral objective functions. Fuzzy Sets and Systems, 1(1), 45–55.