interactive fuzzy programming for multi-level 0–1 programming problems with fuzzy parameters...

Fuzzy Sets and Systems 117 (2001) 95–111www.elsevier.com/locate/fss

Interactive fuzzy programming for multi-level 0–1 programmingproblems with fuzzy parameters through genetic algorithms

Masatoshi Sakawa∗, Ichiro Nishizaki, Masatoshi HitakaDepartment of Industrial and Systems Engineering, Faculty of Engineering, Hiroshima University, 1-4-1 Kagamiyama,

Higashi-Hiroshima, 739-8527 Japan

Received January 1998; received in revised form June 1998

Abstract

In this paper, we propose interactive fuzzy programming for multi-level 0–1 programming problems with fuzzy para-meters through genetic algorithms. Our method can be applied to hierarchical decision problems in which decision makershave their own objective functions but they can coordinate their decisions, that is, they are essentially cooperative. Afterdetermining the fuzzy goals of the decision makers at all levels, a satisfactory solution is derived e�ciently by updatingsatisfactory levels of the decision makers with considerations of overall satisfactory balance among all levels. An illustrativenumerical example for a three-level 0–1 programming problem is provided to demonstrate the feasibility of the proposedmethod. c© 2001 Elsevier Science B.V. All rights reserved.

Keywords: Multilevel 0–1 programming problem; Fuzzy parameter; Fuzzy programming; Fuzzy goals; Genetic algorithms;Interactive methods

1. Introduction

The Stackelberg solution has been usually employed as a solution concept to multi-level programmingproblems [1–6,19–21]. To describe the concept of the Stackelberg solution, consider a two-level program-ming problem. There are two decision makers (DMs); each DM completely knows objective functions andconstraints of the two DMs, and the DM at the upper level (leader) �rst make a decision and then the DMat the lower level (follower) speci�es a decision so as to optimize an objective function with full knowledgeof the decision of the leader. According to the rule, the leader also make a decision so as to optimize theleader’s objective function. Then a solution de�ned as the above-mentioned procedure is called the Stackelbergsolution.When the Stackelberg solution is employed, it is assumed that there is no communication between the two

DMs, or they do not make any binding agreement even if there exists such communication. However, theabove assumption is not always reasonable when we model decision making problems in a decentralized �rm

∗ Corresponding author.

0165-0114/01/$ - see front matter c© 2001 Elsevier Science B.V. All rights reserved.PII: S 0165 -0114(98)00276 -0

96 M. Sakawa et al. / Fuzzy Sets and Systems 117 (2001) 95–111

as a two-level programming problem in which top management is a leader and an operation division of the�rm is a follower because it is supposed that there exists cooperative relationship between them.Consider a computational aspect of the Stackelberg solution. Even if objective functions of both DMs and

common constraint functions are linear, it is known that the problem for obtaining the Stackelberg solutionis a non-convex programming problem with special structure. Although a large number of algorithms forobtaining the Stackelberg solution have been developed, it is known that the problem �nding the solution isstrongly NP-hard [18]. From such di�culties, a new solution concept which is easy to compute and re ectsstructure of multi-level programming problems is expected.To realize two demands, following the idea of Lai [9] or Shih et al. [17], we improve and extend it for

multi-level 0–1 programming problems with fuzzy parameters. That is, we develop an algorithm based on abasic idea that the leader claims a solution with at least a minimal satisfactory level speci�ed by the leader,taking overall satisfactory balance among all levels into consideration, and the follower optimizes an objectivefunction of the follower on a condition satisfying the minimal satisfactory level of the leader.In general, an optimal solution to a problem with the leader’s objective function and the common constraints

does not coincide with an optimal solution to a problem with the follower’s objective function and theconstraints. Therefore, taking structure of the problem and their objective functions into consideration, theyhave to derive a solution compromisable each other.On the other hand, Sakawa et al. formulated mathematical programming problems with fuzzy parameters

from the viewpoint of experts’ imprecise or fuzzy understanding of the nature of parameters in a problem-formulation process, and proposed a fuzzy programming for multiobjective programming problems [11,16].In this paper, we present interactive fuzzy programming for multi-level 0–1 programming problems with

fuzzy parameters. In order to overcome the problem in the methods of Lai [9] and Shih, Lai and Lee [17],multi-level 0–1 programming problems without the fuzzy goals for decision variables is formulated [13]. Inour interactive method, after determining the fuzzy goals of the DMs at all levels, a satisfactory solutionis derived e�ciently by updating the satisfactory levels of the DMs with considerations of the satisfactorybalance between any two levels adjacent to each other. An illustrative numerical example for a three-level0–1 programming problem is provided to demonstrate the feasibility of the proposed method.

2. Interactive fuzzy programming for multi-level 0–1 programming problems

In this paper we consider multi-level 0–1 programming problems in a situation that each of the DMs at alllevels takes overall satisfactory balance among all levels into consideration and tries to optimize an objectivefunction of the DM, paying serious attention to preferences of the others. Such a multi-level 0–1 programmingproblem is formulated as

DM 1 minimizes z1(x1; : : : ; xt)= c11x1 + · · ·+ c1txt ; (1a)

...

DM t minimizes zt(x1; : : : ; xt)= ct1x1 + · · ·+ cttxt (1b)

subject to A1x1 + · · ·+ Atxt6b; (1c)

x1 ∈{0; 1}n1 ; : : : ; xt ∈{0; 1}nt ; (1d)

where xi, i=1; : : : ; t, is an ni-dimensional 0–1 decision variable; cij, i=1; : : : ; t, j=1; : : : ; t, is annj-dimensional constant row vector; b is an m-dimensional constant column vector, and Ai, i=1; : : : ; t, is an

M. Sakawa et al. / Fuzzy Sets and Systems 117 (2001) 95–111 97

m× ni constant matrix. For the sake of simplicity, we use the following notations:

x = (x1; : : : ; xt)T ∈{0; 1}n1+···+nt ; c=

c11 · · · c1t

...ct1 · · · ctt

;

ci· = (ci1; : : : ; cit); i=1; : : : ; t and A= [A1 : : : At];

and let DMi denote the DM at the ith level. The superscript T means transposition. In the multi-level 0–1programming problem (1), zi(x1; : : : ; xt), i=1; : : : ; t, represents the objective function of the ith level and isminimized by DMi, and xi, i=1; : : : ; t, represents a vector of 0–1 decision variables of the ith level.When formulating a mathematical programming problem which closely describes and represents the real-

world decision situation, various factors of the real-world system should be re ected in the description ofobjective functions and constraints. Naturally, these objective functions and constraints involve many para-meters whose possible values may be assigned by the experts. In the conventional approaches, such parametersare required to be �xed at some values in an experimental and=or subjective manner through the experts’understanding of the nature of the parameters in the problem-formulation process.It must be observed here that, in most real-world situations, the possible values of these parameters are

often only imprecisely or ambiguously known to the experts. With this observation in mind, it would becertainly more appropriate to interpret the experts’ understanding of the parameters as fuzzy numerical datawhich can be represented by means of fuzzy sets of the real line known as fuzzy numbers. The resultingmathematical programming problem involving fuzzy parameters would be viewed as a more realistic versionthan the conventional one [11,16].From this viewpoint, we assume that parameters involved in the objective functions and the constraints of

the multi-level 0–1 programming problem are characterized by fuzzy numbers. As a result, a multi-level 0–1programming problem with fuzzy parameters is formulated as

DM 1 minimizes z1(x1; : : : ; xt)= c11x1 + · · ·+ c1txt ; (2a)...



x1 ∈{0; 1}n1 ; : : : ; xt ∈{0; 1}nt ; (2d)

where cij =(cij;1; : : : ; cij; nj), i; j=1; : : : ; t, b=(b1; : : : ; bm)T, Ai=(ai; jk), i=1; : : : ; t, j=1; : : : ; m, k =1; : : : ; ni

are fuzzy parameters. For the sake of simplicity, we use the following notations:

c=

c11 · · · c1t

...ct1 · · · ctt

and A= [A1 : : : At]:

Assuming that these fuzzy parameters c, b, A are characterized by fuzzy numbers, let correspondingmembership functions be: �cij; k (cij; k), i; j=1; : : : ; t, k =1; : : : ; nj, �bi(bi), i=1; : : : ; m, �ai; jk (ai; jk), i=1; : : : ; t,j=1; : : : ; m, k =1; : : : ; ni.We introduce the �-level set of the fuzzy numbers c, b and A de�ned as the ordinary set (c; b; A)� in which

the degree of their membership functions exceeds the level �:

(c; b; A)� ={(c; b;A) | �cij; k (cij; k)¿�; i; j=1; : : : ; t; k =1; : : : ; nj; �bi(bi)¿�; i=1; : : : ; m;�ai; jk (ai; jk)¿�; i=1; : : : ; t; j=1; : : : ; m; k =1; : : : ; ni}: (3)


Now, suppose that DM1 considers that the degree of all the membership functions of the fuzzy numbersinvolved in the multi-level 0–1 programming problem should be greater than or equal to a certain value�. Then, for such a degree �, the problem can be interpreted as the following nonfuzzy multi-level 0–1programming problem which depends on the coe�cient vector (c; b;A)∈ (c; b; A)� [11,16]:

DM 1 minimizes z1(x1; : : : ; xt)= c11x1 + · · ·+ c1txt ; (4a)

...



x1 ∈{0; 1}n1 ; : : : ; xt ∈{0; 1}nt : (4d)

Observe that there exists an in�nite number of such a problem (4) depending on the coe�cient vector(c; b;A)∈ (c; b; A)� and the values of (c; b;A) are arbitrary for any (c; b;A)∈ (c; b; A)� in the sense that thedegree of all of the membership functions for the fuzzy numbers in problem (4) exceeds the level �. However,if possible, it would be desirable for the DMs to choose (c; b;A)∈ (c; b; A)� in problem (4) so as to minimizethe objective functions under the constraints. From such a point of view, for a certain degree �, it seems tobe quite natural to have understood the multi-level 0–1 programming problem with fuzzy parameters as thefollowing nonfuzzy �-multi-level 0–1 programming problem [11,16]:

DM 1 minimizesx; c;b;A

z1(x; c1·)= c11x1 + · · ·+ c1txt ; (5a)

...

DM t minimizesx; c;b;A

zt(x; ct·)= ct1x1 + · · ·+ cttxt (5b)


x1 ∈{0; 1}n1 ; : : : ; xt ∈{0; 1}nt ; (5d)

(c; b;A)∈ (c; b; A)�: (5e)

It is assumed that DM1 chooses a degree of the �-level. It should be noted that the parameters (c; b;A) aretreated as decision variables rather than constants in problem (5).It is natural that the DMs have fuzzy goals for their objective functions when they take fuzziness of human

judgments into consideration. For each of the objective functions zi(x; ci·), i=1; : : : ; t of (5), assume that theDMs have fuzzy goals such as “the objective function zi(x; ci·) should be substantially less than or equal tosome value pi”.Let �=0, and then the individual minimum

zmini = zi(xio; coi·)=min{zi(x; ci·) |A1x1 + · · ·+ Atxt6b; xi ∈{0; 1}ni ; i=1; : : : ; t; (c; b;A)∈ (c; b; A)0}(6)

and the individual maximum

zmaxi =max{zi(x; ci·) |A1x1 + · · ·+ Atxt6b; xi ∈{0; 1}ni ; i=1; : : : ; t; (c; b;A)∈ (c; b; A)0} (7)

of the objective function zi(x; ci·) are referred to when DMi elicits a membership function prescribing a fuzzygoal for the objective functions zi(x; ci·). DMi determines the membership functions �i(zi(x; ci·)), which are


Fig. 1. Linear membership function.

strictly monotone decreasing for zi(x; ci·), consulting the variation ratio of degree of satisfaction in the intervalbetween individual minimum (6) and individual maximum (7). The domain of the membership function isthe interval [zmini ; zmaxi ], and DMi speci�es the value z0i of the objective function for which the degree ofsatisfaction is 0 and the value z1i of the objective function for which the degree of satisfaction is 1. For thevalue undesired (larger) than z0i , it is de�ned that �i(zi(x; ci·))= 0, and for the value desired (smaller) thanz1i , it is de�ned that �i(zi(x; ci·))= 1.For the sake of simplicity, in this paper, we adopt a linear membership function, which characterizes the

fuzzy goal of DMi for all i=1; : : : ; t. The corresponding linear membership function �i(zi) is de�ned as

�i(zi(x; ci·))=

0; zi(x; ci·)¿z0i ;

zi(x; ci·)− z0iz1i − z0i

; z1i ¡zi(x; ci·)6z0i ;

1; zi(x; ci·)6z1i ;

(8)

where z0i and z1i denote the values of the objective function zi(x; ci·) such that the degree of membership

function is 0 and 1, respectively, and it is assumed that DMi subjectively assesses z0i and z1i . The membership

function (8) is depicted in Fig. 1.For all i=1; : : : ; t, suppose that applying the way suggested by Zimmermann [22] and setting �=0, DMi

speci�es z0i and z1i in the following. That is, using the individual minimum (6) together with

zmi =max(zi(x1o; coi·); : : : ; zi(x

i−1; o; coi·); zi(xi+1; o; coi·); : : : ; zi(x

to; coi·)); (9)

DMi determines the linear membership function as in (8) by choosing z1i = zmini , z0i = z

mi .

After eliciting a membership function, DMi, i=1; : : : ; t − 1, subjectively speci�es a minimal satisfactorylevel �i ∈ [0; 1] for the membership function �i(zi(x; ci·)). Then, DMt also elicits a membership function andmaximizes the membership function subject to a condition that the membership functions �1(z1(x; c1·)); : : : ;�t−1(zt−1(x; ct−1·)) of the DMs at upper levels are larger than or equal to �1; : : : ; �t−1 under the givenconstraints, that is, DMt solves the following problem:

maximizex; c;b;A

�t(zt(x; ct·)) (10a)

subject to A1x1 + · · ·+ Atxt6b; (10b)

�i(zi(x; ci·))¿�i; i=1; : : : ; t − 1; (10c)

x1 ∈{0; 1}n1 ; : : : ; xt ∈{0; 1}nt ; (10d)

(c; b;A)∈ (c; b; A)�: (10e)

In our formulation (10), the constraints on the fuzzy goals for decision variables of DMs are eliminatedwhile they are involved in the formulations by Lai [9] and Shih et al. [17]. The authors have pointed out that


undesired solutions are produced when fuzzy goals are introduced for both an objective function and decisionvariables at the upper level [13].If an optimal solution to problem (10) exists, it follows that DMi, i=1; : : : ; t, obtains a satisfactory solution

having a satisfactory degree larger than or equal to the minimal satisfactory level speci�ed by DMi’s ownself. However, if some DMs specify larger values to their minimal satisfactory levels, it follows that thesatisfactory degree of DMt becomes fairly small or there exists no feasible solution. Consequently, it is fearedthat overall satisfactory balance among all the levels cannot be maintained.To take account of overall satisfactory balance among all the levels, DMi needs to compromise with DMs

at lower levels on the minimal satisfactory level of DMi for all i=1; : : : ; t−1. To do so, a satisfactory degreeof the DMs at all levels is de�ned as

�=min(�1(z1(x; c1·)); : : : ; �t(zt(x; ct·))); (11)

and the following problem is substituted for problem (10):

maximizex; �; c;b;A

� (12a)


�i(zi(x; ci·))¿�i¿�; i=1; : : : ; t − 1; (12c)

�t(zt(x; ct·))¿�; (12d)

06�61; (12e)

x∈{0; 1}n1+···+nt ; (12f)

(c; b;A)∈ (c; b; A)�: (12g)

For problem (12), introduce the auxiliary problem


� (13a)


�i(zi(x; ci·))¿�; i=1; : : : ; t; (13c)

06�61; (13d)

x∈{0; 1}n1+···+nt ; (13e)

(c; b;A)∈ (c; b; A)�: (13f)

By solving problem (13), we obtain a solution maximizing a minimal satisfactory degree among the DMs.Unfortunately, problem (13) is not linear 0–1 programming problem even if all the membership functions�i(zi(x; ci·)); i=1; : : : ; t are linear. To solve problem (13), we introduce the set-valued functions:

Si(ci·)= {(x; �) | �i(zi(x; ci·))¿�}; i=1; : : : ; t;

Tj(bj; A1; j ; : : : ; At; j)= {x |A1; j x1 + · · ·+ At; j xt6bj}; j=1; : : : ; m;(14)

where Ai; j is a row vector corresponding to the jth row of the m×ni matrix Ai. Then it can be easily veri�edthat the following relations hold for Si(ci·) and Tj(bj; A1; j ; : : : ; At; j) when x∈{0; 1}n1+···+nt [16,11].


Proposition 1. (1) For j=1; : : : ; t, if c1ij6c2ij, then Si(: : : ; c

1ij ; : : :)⊇ Si(: : : ; c2ij ; : : :).

(2) If b1j6b2j , then Tj(b

1j ; : : :) ⊆ Tj(b2j ; : : :).

(3) For i=1; : : : ; t, if A1i; j6A2i; j, then Tj(: : : ; A

1i; j ; : : :)⊇Tj(: : : ; A2i; j ; : : :).

From the properties of the �-level set for the vectors of fuzzy numbers cij, b and the matrices of fuzzynumbers Ai, it should be noted that the feasible regions for cij ; bj and Ai can be denoted, respectively, bythe closed intervals [cLij ; c

Rij ], [b

Lj ; b

Rj ], and [A

Li ; A

Ri ].

Therefore, through the use of Proposition 1, we can obtain an optimal solution of problem (13) by solvingthe following 0–1 programming problem:

maximizex; �

� (15a)

subject to AL1 x1 + · · ·+ ALt xt6bR ; (15b)

�i(zi(x; cLi·))¿�; i=1; : : : ; t; (15c)

06�61; (15d)

x∈{0; 1}n1+···+nt : (15e)

By solving problem (15), we obtain a solution maximizing the minimal satisfactory degree among those ofall the DMs. If the optimal solution (x∗; �∗; cL; bR ;AL) to problem (13) satis�es the condition �i(zi(x∗; cLi·))¿�i; i=1; : : : ; t − 1, it follows that DMi, i=1; : : : ; t, obtains a satisfactory solution. However, the solution(x∗; �∗; cL; bR ;AL) does not always satisfy the conditions. Then the ratio of satisfactory degree betweenadjacent two levels

�i=�i+1(zi+1(x∗; cLi+1·))�i(zi(x∗; cLi·))

; i=1; : : : ; t − 1; (16)

which is de�ned by Lai [9], is useful. Let �iL and �iU denote the lower bound and the upper bound of �ispeci�ed by DMi, respectively. If �i¿�iU, i.e., �i+1(zi+1(x∗); cL1·; c

L2·)¿�iU�i(zi(x

∗); cL1·; cL2·), then DMi updates

the minimal satisfactory level �i by increasing �i. Then DMi obtains a larger satisfactory degree and DM(i+1)accepts a smaller satisfactory degree. Conversely, if �i¡�iL, i.e., �i+1(zi+1(x∗); cL1·; c

L2·)¡�iL�i(zi(x

∗; cL1·; cL2·)),

then DMi updates the minimal satisfactory level �i by decreasing �i, and DMi accepts a smaller satisfactorydegree and DM(i + 1) obtains a larger satisfactory degree.At an iteration ‘, let �i(z‘i ), i=1; : : : ; t, and �

‘ denote satisfactory degrees of DMi, i=1; : : : ; t, and asatisfactory degree of all the levels, respectively, and let �‘i = �i+1(z

‘i+1)=�i(z

‘i )‘ denote a ratio of satisfactory

degrees of the ith and the (i+1)th levels. Let a corresponding solution be x‘. For all i=1; : : : ; t− 1, DMi isproposed a solution by DM(i+1). Then the DMs at all the levels except for the tth level obtain the satisfactorysolution and the interactive process terminates if the following two conditions are satis�ed simultaneously.

Termination conditions of the interactive process for multi-level 0–1 programming problems(1) For all i=1; : : : ; t − 1, DMi’s satisfactory degree is larger than or equal to the minimal satisfactory

level �i speci�ed by DMi, i.e., �i(z‘i )¿�i, i=1; : : : ; t − 1.(2) For all i=1; : : : ; t − 1, the ratio �‘i of satisfactory degrees is in the closed interval the lower and the

upper bounds of which are speci�ed by DMi.Condition (1) means DMi’s required condition for solutions proposed by DM(i + 1). Condition (2) is

provided in order to keep overall satisfactory balance among all the levels.Unless the conditions are satis�ed simultaneously, DMi, i=1; : : : ; t, needs to update the minimal satisfactory

level �i. Suppose that the DMs from at the (q+1)th level to at the (t−1)th level, i.e., DM(q+1), DM(q+2); : : : ;


and DM(t − 1), satisfy the proposed solution but DMq does not satisfy it. Then DMq, DM(q + 1); : : : ; andDM(t−1) need to update their minimal satisfactory levels �i, i= q; q+1; : : : ; t−1. For any two levels adjacentto each other, giving a DM at an upper level serious consideration, a DM at a lower level should update theminimal satisfactory level.

Procedure for updating the minimal satisfactory level �i(1) A DM, say DMi, who does not satisfy the �rst termination condition decreases the minimal satisfactory

level �i.(2) If the ratio �‘i exceeds its upper bound, then DMi increases the minimal satisfactory level �i. Conversely,

if the ratio �‘i is below its lower bound, then DMi decreases the minimal satisfactory level �i.(3) If the ratio �i+1=�i of the minimal satisfactory levels also is not in the valid interval of �i, then the

minimal satisfactory level �i is updated in a way similar to the update in (2).Let �′i ; i= q; : : : ; t−1, denote the updated minimal satisfactory level. DMt solves the following maximization

problem with the updated minimal satisfactory levels �′i :


� (17a)

subject to A1 x1 + · · ·+ At xt6b; (17b)

�i(zi(x; ci·))¿�; i=1; : : : ; q− 1; (17c)

�i(zi(x; ci·))¿�′i ; i= q; : : : ; t − 1; (17d)

�t(zt(x; ct·))¿�; (17e)

06�61; (17f)

x∈{0; 1}n1+···+nt ; (17g)

(c; b;A)∈ (c; b; A)�: (17h)

We can obtain an optimal solution to problem (17) by solving the following problem through a transformationsimilar to the transformation from problem (13) to problem (15).

maximizex; �

� (18a)

subject to AL1 x1 + · · ·+ ALt xt6bR ; (18b)

�i(zi(x; cLi·))¿�; i=1; : : : ; q− 1; (18c)

�i(zi(x; cLi·))¿�′i ; i= q; : : : ; t − 1; (18d)

�t(zt(x; cLt·))¿�; (18e)

06�61; (18f)

x∈{0; 1}n1+···+nt : (18g)

The above-mentioned algorithm is summarized as follows:


Algorithm of the interactive fuzzy programming for solving multi-level 0–1 programming problemsStep 1: Set ‘=1. DM1 determines a degree � of the �-level set (c; b; A)�. For all i=1; : : : ; t − 1, DMi

elicits a membership function �i(zi) of a fuzzy goal, and speci�es a minimal satisfactory level �‘i and a lower

bound and an upper bound of a ratio �i of satisfactory degrees.Step 2: DMt elicits a membership function �t(zt) of a fuzzy goal.Step 3: DMt solves the auxiliary problem (13) through problem (15). For all i= t; t−1; : : : ; 2, DMi proposes

a solution x‘, (z‘1 ; : : : ; z‘t )�

‘; �i(z‘i ), and �‘i to DM(i − 1) successively.

Step 4: If the solution satis�es the termination conditions of all levels, it follows that the DMs obtain asatisfactory solution, and the algorithm stops. Otherwise ‘= ‘ + 1.Step 5: If DM(q+1);DM(q+2); : : : ; and DM(t−1) satisfy the proposed solution but DMq does not satisfy

it, DMq; : : : ;DM(t − 1) update the minimal satisfactory levels �q; : : : ; �t−1, in accordance with the procedurefor updating minimal satisfactory level.Step 6: DMt solves problem (17) through problem (18). For all i= t; t−1; : : : ; 2;DMi proposes an optimal

solution to problem (17) to DM(i − 1) successively. Return to Step 4. If there does not exist any feasiblesolution to problem (17), go to Step 7.Step 7: For all i= q; : : : ; t − 1, DMi updates the minimal satisfactory levels �i by decreasing the values �i,

and return to Step 6.To illustrate the use of multi-level programming, consider a transportation �rm with special skills which

has received n inquiries (projects). Its headquarters, which is thought of as a DM at upper level, receivessome orders and must decline the others because of constraints of resources. For the undertaken transportationprojects, its operation division, which is thought of as a DM at the lower level, makes a plan of transportationso as to minimize costs of transportation on condition that, for Project ‘, there are m‘ depots and o‘ demandpoints. Such a problem can be formulated as a two-level 0–1 programming problem:

DM 1 maximizesn∑‘=1

c‘ x‘; (19a)

DM 2 minimizesn∑‘=1

m‘∑j=1

o‘∑k=1

d‘jky‘jk x‘ (19b)

subject ton∑‘=1

a‘s x‘6bs; s=1; : : : ; r; (19c)

m‘∑j=1

y‘jk = �‘k x‘; k =1; : : : ; o‘; ‘=1; : : : ; n; (19d)

o‘∑k=1

y‘jk = �‘j x‘; j=1; : : : ; m‘; ‘=1; : : : ; n; (19e)

x‘ ∈{0; 1}; ‘=1; : : : ; n; (19f)

y‘jk ∈{0; 1}; ‘=1; : : : ; n; j=1; : : : ; m‘; k =1; : : : ; o‘; (19g)

wherex 0–1 variable for choosing transportation projectsc pro�t coe�cients on the transportation projectsa technology coe�cients on the transportation projects


b resources of the transportation �rmyl·· transportation variables of Project ‘d cost coe�cients on the transportation projects� supply coe�cients� demand coe�cients

Problem (19) may be formulated as a problem with fuzzy parameters like problem (2) by taking into accountthe experts’ understanding of the nature of the parameters in the problem-formulation process. Employing theproposed algorithm of the interactive fuzzy programming, we can interpret the parameter �i as a degree ofsatisfaction on the total pro�t which the headquarters perceive and the parameter �i as a ratio of the degree ofsatisfaction of the headquarters and a degree of satisfaction on the transportation cost on which the operationdivision spends.

3. Genetic algorithms with double strings

We can apply the genetic algorithms with double strings proposed by Sakawa et al. [14,12,15] for solvingproblems (15) and (18) if the constraints of the problems are linear and all of the technology coe�cients arepositive.

3.1. Coding and decoding

For solving problems (15) and (18) through genetic algorithms, an individual is usually represented by abinary 0–1 string [7,10]. This representation, however, may weaken ability of genetic algorithms since anindividual whose phenotype is feasible is scarcely generated under this representation. In this paper, as onepossible approach to generate only feasible solutions, a double string as is shown in Fig. 2 is adopted forrepresenting an individual, where si( j) ∈{1; 0}, i(j)∈{1; : : : ; n1 + · · ·+ nt}, and i(j) 6= i(j′) for j 6= j′.In a double string, regarding i(j) and si( j) as the index of an element in a solution vector and the value of

the element respectively, a string S can be transformed into a solution x=(x11; : : : ; x1n1 ; : : : ; xt1; : : : ; xtnt )T as

xi( j) = si( j); j=1; : : : ; n1 + · · ·+ nt :

Unfortunately, however, since this mapping may generate infeasible solutions, we propose the followingdecoding algorithm for eliminating infeasible solutions. It should be noted that the following algorithm gen-erates only feasible solutions for problem (15) but it does not always generate only feasible solutions forproblem (18). In the algorithm, n1 + · · · + nt; j; i(j); xi( j) and ai( j) denote, respectively, length of a string,a position in a string, an index of a variable, 0–1 value of a variable with index i(j) decoded from a stringand an i(j)th column vector of the coe�cient matrices [A1 : : : At].Step 1: Set j=1; �= 0.Step 2: If si( j) = 1, set j= j+1 and go to step 3. Otherwise, i.e., if si( j) = 0, set j= j+1 and go to step 4.Step 3: If �+as(i)6b, set xs(i) = 1, �=�+ai( j) and go to step 4. Otherwise, set xi( j) = 0 and go to step 4.Step 4: If j¿n1+ · · ·+nt , stop and regard x=(x1; : : : ; xn1+···+nt )T as phenotype of the individual represented

by the double string. Otherwise, return to step 2.

index of variable:0–1 value:

(i(1) i(2) : : : i(n1 + · · ·+ nt)si(1) si(2) : : : si(n1+···+nt)

)

Fig. 2. Double string.


3.2. Fitness and scaling

An individual reproduced by the genetic algorithms with double strings does not always satisfy all theconstraints of problem (18) because of constraint (18d). To overcome this di�culty, we adopt the following�tness function, which becomes −1 as a penalty in case of violating the constraint. We de�ne the �tnessfunction of each individual S by

f(S)=min{mini=1;:::; t

�i(zi(x; cLi·)); �(x)}; (20)

�(x)={1 if �i¿�′i ; ∀i∈{q; : : : ; t − 1};−1 otherwise;

(21)

where S and x, respectively, denote an individual represented by double string and phenotype of S. Forproblem (15), we de�ne �(x)= 1 for all x because the decoding algorithm for an individual represented bydouble string always yields a feasible solution.

3.3. Reproduction

Up to now, various reproduction methods have been proposed and considered [7,10]. As a reproductionoperator, elitist roulette wheel selection is adopted here. Elitist roulette wheel selection is a combination ofelitism and roulette wheel selection as mentioned below.Elitism: If the �tness of a string in the past populations is larger than that of every string in the current

population, preserve this string into the current generation.Roulette wheel selection: The roulette wheel selection is the most popular way of selection. This reproduc-

tion allocates o�springs using a roulette wheel with slots sized according to �tness values. The size of slot isgiven by the probability, f(Si)=

∑f(Si).

3.4. Crossover and mutation

If a single-point or multi-point crossover operator is applied to individuals represented by double strings, anindex i(j) in an o�spring may take the same number that an index i(j′) (j 6= j′) takes. Recall that the sameviolation occurs in solving traveling salesman problems or scheduling problems through genetic algorithms.One possible approach to circumvent such violation, a crossover method called partially matched crossover(PMX) is useful. The PMX was �rst proposed by Goldberg and Lingle [8] for tackling a blind travelingsalesman problem. It enables us to generate desirable o�springs without changing the double string structureunlike the ordinal representation. However, in order to process each element si( j) in the double string structuree�ciently, it is necessary to revise some points of the procedures. Our revised procedures of the PMX canbe illustrated as follows:Step 1: For two individuals S1 and S2 represented by double strings, choose two crossover points.Step 2: According to the PMX, reorder upper strings of S1 and S2 together with the corresponding lower

strings which yields S′1 and S′2.

Step 3: Exchange lower substrings between two crossover points of S′1 and S′2 for obtaining the resulting

o�springs S′′1 and S′′2 after the revised PMX for double strings.

It is well recognized that a mutation operator plays a role of local random search in genetic algorithms. Inthis paper, for the lower string of a double string, mutation of bit-reverse type is adopted, and we introduceanother genetic operator, an inversion, together with PMX operator. The inversion proceeds as follows:


Step 1: For an individual S, choose two inversion points at random, i.e.,

S=

(i(1) · · · | i(l) i(l+ 1) · · · i(m) | · · · i(n1 + · · ·+ nt)si(1) · · · | si(l) si(l+1) · · · si(m) | · · · si(n1+···+nt)

):

Step 2: Invert both upper and lower substrings between two inversion points, i.e.,

S′=

(i(1) · · · | i(m) i(m− 1) · · · i(l) | · · · i(n1 + · · ·+ nt)si(1) · · · | si(m) si(m−1) · · · si(l) | · · · si(n1+···+nt)

):

4. A numerical example for a multi-level 0–1 programming problem

As an example for a multi-level 0–1 programming problem, consider the following three-level problem:

DM 1 minimizes z1 = c11x1 + c12x2 + c13x3; (22a)

DM 2 minimizes z2 = c21x1 + c22x2 + c23x3; (22b)

DM 3 minimizes z3 = c31x1 + c32x2 + c33x3 (22c)

subject to A1x1 + A2x2 + A3x36b; (22d)

xj ∈{0; 1}; j=1; 2; : : : ; 30; (22e)

where x1 = (x1; : : : ; x10)T, x2 = (x11; : : : ; x20)T, x3 = (x21; : : : ; x30)T; each entry of 10-dimensional row constantvectors cij, i; j=1; 2; 3, and each entry of 3× 10 coe�cient matrices A1, A2 and A3 are random values in theinterval (0; 100), and c22 =−c12, c33 =−c13; each entry of the right-hand-side constant column vector b is asum of entries of the corresponding row vector of A1, A2 and A3 multiplied by 0:6; 90% of the coe�cientsare set as fuzzy parameters (numbers). Values shown in Tables 1 and 2 are means of the fuzzy numbersrepresenting the coe�cients, and their spread are given by the values of the means multiplied by randomnumbers in [1:0; 1:1].Suppose that DM1 and DM2 determine the initial minimal satisfactory levels as �1 = �2 = 1:0, and the lower

and the upper bounds of �1 and �2 as [0:6; 1:0]. The membership functions (8) of the fuzzy goals are assessedby using values (6) and (9). Because all the coe�cients of the constraints are positive, the individual minimaand the corresponding optimal solutions are obtained by applying the genetic algorithms with double stringsand are shown in Table 3, and we have zm1 =−972, zm2 =−85 and zm3 =166. In the computation for obtainingan approximate optimal solution to each problem, we set 0.7 to the probability of the crossover, 0.05 to theprobability of the mutation, 100 to the population size and 1000 to the number of the generations, and wemade this computational trial ten times. Then three membership functions for fuzzy goals are assessed as

�1(z1(x; c1·)), (z1(x; cL1·) + 972)=(−1439 + 972); (23)

�2(z2(x; c2·)), (z2(x; cL2·) + 85)=(−971 + 85); (24)

�3(z3(x; c3·)), (z3(x; cL3·)− 166)=(−1011− 166): (25)

Let �=0:9, and then problem (15) for this numerical example can be formulated as

maximizex; �

min(�1(z1(x; c1·)); �2(z2(x; c2·)); �3(z3(x; c3·))) (26a)

subject to AL1x1 + AL2x2 + A

L3x36b

R ; (26b)

xj ∈{0; 1}; j=1; 2; : : : ; 30; (26c)


Table 1Coe�cients of the three-level problem

c11 −67 −73 −93 −24 −84 −96 −78 −43 −67 −81c12 −16 −28 −14 −86 −75 −21 −14 −30 −80 −22c13 −56 −71 −20 −98 −25 −43 −75 −86 −89 −97c21 −40 −43 −13 −46 −24 −98 −65 −60 −24 −46c22 16 28 14 86 75 21 14 30 80 22c23 −79 −8 −48 −16 −25 −94 −61 −98 −48 −80c31 −74 −38 −48 −53 −10 −59 −35 −15 −78 −71c32 −45 −70 −10 −96 −55 −74 −58 −64 −78 −19c33 56 71 20 98 25 43 75 86 89 97

A1 51 18 31 53 94 18 70 23 49 1393 87 86 67 76 58 39 36 20 8244 91 57 12 57 25 50 24 48 41

A2 9 39 28 37 98 54 76 65 76 7842 46 97 13 22 95 74 41 78 7687 43 36 38 5 16 52 69 10 40

A3 82 16 62 32 35 91 52 40 61 7895 3 32 75 25 59 5 95 32 677 25 34 23 30 31 88 4 65 40

b 917 993 757

Note: The underlined values are nonfuzzy numbers.

Table 2Extreme points of fuzzy parameters (�=0:9)

cL11 −69 −75 −97 −25 −88 −104 −84 −45 −68 −86cL12 −16 −28 −14 −87 −76 −21 −14 −31 −84 −23cL13 −57 −75 −21 −105 −27 −46 −75 −87 −90 −105cL21 −40 −43 −13 −48 −25 −98 −69 −63 −24 −49cL22 15 25 13 83 73 19 12 27 78 21

cL23 −82 −8 −49 −17 −25 −94 −61 −107 −52 −80cL31 −75 −38 −48 −53 −10 −59 −35 −15 −83 −72cL32 −45 −75 −10 −102 −57 −74 −62 −65 −84 −20cL33 51 68 19 97 23 42 74 81 80 91

AL1 46 17 31 51 89 16 64 22 47 1293 78 85 61 70 53 36 33 18 7543 90 53 11 56 24 47 21 43 40

AL2 8 37 26 37 92 52 76 64 74 7340 43 89 11 20 85 69 39 78 7486 42 33 36 4 14 50 66 9 38

AL3 75 15 61 29 35 85 50 40 56 7295 2 31 69 24 54 4 87 31 574 22 31 21 28 29 83 3 60 39

bR 920 1035 785


Table 3The individual minima and the corresponding solutions

zmin1 −1439 x1o1 1 1 1 0 1 1 1 1 1 1

x1o2 0 0 0 1 1 0 0 0 1 0

x1o3 0 1 0 1 0 0 1 1 1 1

zmin2 −971 x2o1 1 1 0 1 0 1 1 1 1 1

x2o2 0 0 0 0 0 0 0 0 0 0

x2o3 1 0 1 1 1 1 1 1 1 1

zmin3 −1011 x3o1 1 1 1 1 0 1 1 1 1 1

x3o2 1 1 0 1 1 1 1 1 1 0

x3o3 0 0 0 0 0 0 0 0 0 0

Table 4 Table 5The �rst iteration of the three-level problem The second iteration for the three-level programming

�1 0.537473

x11 1 1 1 1 1 1 1 1 1 1

x12 0 1 0 1 1 1 0 0 0 0

x13 0 0 1 0 1 0 1 1 0 1

z11 −1223 �1(z11) 0.537473

z12 −564 �2(z12) 0.540632

z13 −476 �3(z13) 0.545455

�11 1.005877

�12 1.008920

z21 1095:9 �1(z21) 0.263383

z22 884:7 �2(z22) 0.901806

z23 99:9 �3(z23) 0.225149

�21 3.423930

�22 0.249664

where cLij, ALi and b

R are shown in Table 2. We apply the genetic algorithms with double strings with the�tness function (20) and �(x)= 1. Data of the �rst iteration including an approximate optimal solution toproblem (26) are shown in Table 4.The �rst termination condition of the interactive process is not satis�ed because the satisfactory degree

�12 = 0:540632 of DM2 does not exceed the minimal satisfactory level �2 = 1:0. Consequently, suppose thatDM2 updates the minimal satisfactory level from �2 = 1:0 to �′2 = 0:9. Then a problem corresponding toproblem (18) is formulated as

maximizex; �

min(�1(z1(x; c1·)); �3(z3(x; c3·))) (27a)

subject to �2(z2(x; c2·))¿0:9; (27b)

AL1x1 + AL2x2 + A

L3x36b

R ; (27c)

xj ∈{0; 1}; j=1; 2; : : : ; 30; (27d)

and solved by using the genetic algorithms with double strings with the �tness function (20) and (21). Dataof the second iteration including an optimal solution to problem (27) are shown in Table 5.The �rst termination condition of the interactive process is not satis�ed because the satisfactory de-

gree �21 = 0:263383 of DM1 does not exceed the minimal satisfactory level �1 = 1:0. Consequently, suppose


Table 6The third iteration for the three-level programming

z31 1253:9 �1(z31) 0.601713

z32 621:4 �2(z32) 0.604966

z33 233:2 �3(z33) 0.338997

�31 1.005406

�32 0.560358

that DM1 changes the minimal satisfactory level from �1 = 1:0 to �′1 = 0:9. Then the following problem isformulated.

maximizex; �

�3(z3(x; c3·)) (28a)

subject to �1(z1(x; c1·))¿0:9; (28b)

�2(z2(x; c2·))¿0:9; (28c)

AL1x1 + AL2x2 + A

L3x36b

R ; (28d)

xj ∈{0; 1}; j=1; 2; : : : ; 30: (28e)

Because problem (28) has no feasible solution, suppose that the minimal satisfactory levels �′1 and �′2 are

updated as 0:6, and a problem is formulated as

maximizex; �

�3(z3(x; c3·)) (29a)

subject to �1(z1(x; c1·))¿0:6; (29b)

�2(z2(x; c2·))¿0:6; (29c)

AL1x1 + AL2x2 + A

L3x36b

R ; (29d)

xj ∈{0; 1}; j=1; 2; : : : ; 30: (29e)

Data of the third iteration including an optimal solution to problem (29) are shown in Table 6.Because the ratio of minimal satisfactory degrees �32 = 1:125 and �

31 = 1:005406 are not in the valid interval

[0:6; 1:0] of �1 and �2, suppose that the minimal satisfactory levels �′1 and �′2 are updated as 0:7 and 0:5,

respectively.Then, the following problem is formulated.

maximizex; �

�3(z3(x; c3·)) (30a)

subject to �1(z1(x; c1·))¿0:7; (30b)

�2(z2(x; c2·))¿0:5; (30c)

AL1x1 + AL2x2 + A

L3x36b

R ; (30d)

xj ∈{0; 1}; j=1; 2; : : : ; 30: (30e)

Data of the fourth iteration including an approximate optimal solution to problem (30) are shown in Table 7.


Table 7A satisfactory solution to the three-level problem

x41 1 1 1 0 1 1 1 1 1 1

x42 0 1 0 1 0 1 0 0 1 0

x43 0 1 0 0 1 1 1 1 0 1

z41 −1299:7 �1(z41) 0.700214

z42 −567:0 �2(z42) 0.542889

z43 −353:2 �3(z43) 0.440952

�41 0.775319

�42 0.812231

At the fourth iteration, the satisfactory degree �1(z41)= 0:700214 of DM1 becomes larger than the minimalsatisfactory level �′1 = 0:7 and the satisfactory degree �2(z

42)= 0:542889 of DM2 becomes larger than the

minimal satisfactory level �′2 = 0:5. The ratios �41 = 0:775319 and �

42 = 0:812231 of satisfactory degrees are in

the valid interval [0:6; 1:0] of the ratios �1 and �2. Therefore this solution satis�es the termination conditionsof the interactive process and then becomes a satisfactory solution for all the DMs.

5. Conclusions

In this paper, we have proposed interactive fuzzy programming for multi-level 0–1 programming problemswith fuzzy parameters. In our interactive method, after determining the fuzzy goals of the DMs at all levels,a satisfactory solution is derived e�ciently by updating the minimal satisfactory levels with considerationsof overall satisfactory balance. An illustrative numerical example for a three-level 0–1 programming problemhas been provided to demonstrate the feasibility of the proposed method.

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