interactive fuzzy programming for two-level linear fractional programming problems with fuzzy...

$: Interactive fuzzy programming for two-level linear fractional programming problems with fuzzy parameters$
Fuzzy Sets and Systems 115 (2000) 93–103www.elsevier.com/locate/fss

Interactive fuzzy programming for two-level linear fractionalprogramming problems with fuzzy parametersMasatoshi Sakawa a ;∗, Ichiro Nishizaki a, Yoshio Uemura b

a Department of Industrial and Systems Engineering, Faculty of Engineering, Hiroshima University, 1-4-1 Kagamiyama,Higashi-Hiroshima, 739-8527 Japan

b Products Projects Development Department, Juken Sangyou Co., Ltd., 1-1 Mokuzaikouminami, Hatsukaichi, Hiroshima,738-8502 Japan

Received November 1998

Abstract

In this paper, we present interactive fuzzy programming for two-level linear fractional programming problems with fuzzyparameters. Using the level sets of fuzzy parameters, the corresponding nonfuzzy two-level linear fractional programmingproblem is introduced. In our interactive method, after determining fuzzy goals of decision makers at both levels, a satisfactorysolution is derived e�ciently by updating a minimal satisfactory level of the decision maker at the upper level withconsiderations of overall satisfactory balance between both levels. The satisfactory solution well-balanced between bothlevels is easily computed by combined use of the bisection method, the phase one of the simplex method and the variabletransformation method by Charnes and Cooper. An illustrative numerical example for two-level linear fractional programmingproblems with fuzzy parameters is provided to demonstrate the feasibility of the proposed method. c© 2000 Elsevier ScienceB.V. All rights reserved.

Keywords: Two-level linear fractional programming problems; Fuzzy parameters; Fuzzy programming; Fuzzy goals;Interactive methods

1. Introduction

In this paper, we consider a two-level programmingproblem in which two decision makers (DMs) makedecisions successively. For example, in a decentral-ized �rm, top management, an executive board, orheadquarters makes a decision such as a budget of the

∗ Corresponding author.E-mail address: [email protected] (M.

Sakawa).

�rm, and then each division determines a productionplan in the full knowledge of the budget.The Stackelberg solution, which is a solution con-

cept in two-person noncooperative games, has beenemployed as a solution concept to two-level program-ming problems, and a considerable number of algo-rithms for obtaining the solution have been developed(e.g. [1,2,4,20]).Concerning hierarchical decision problems in a de-

centralized �rm, however, it is natural that decisionmakers are regarded as to be cooperative rather thanto be completely noncooperative.

0165-0114/00/$ - see front matter c© 2000 Elsevier Science B.V. All rights reserved.PII: S 0165 -0114(99)00027 -5

94 M. Sakawa et al. / Fuzzy Sets and Systems 115 (2000) 93–103

Recently, Lai [9] and Shih et al. [18] have proposeda solution concept which is di�erent from the con-cept of the Stackelberg solution for two-level linearprogramming problems such that decisions of DMs atboth levels are sequential and all of the DMs essen-tially cooperate with each other. Their method is basedon the idea that the DM at the lower level optimizes aobjective function, taking a goal or preference of theupper level into consideration. The DMs elicit mem-bership functions of fuzzy goals for their objectivefunctions, and especially, the DM at the upper levelalso speci�es those of fuzzy goals for decision vari-ables. The DM at the lower level solves a fuzzy pro-gramming problem with a constraint on a satisfactorydegree of the DM at the upper level. Unfortunately,however, there is a possibility that their method leadsa �nal solution to an undesirable one because of in-consistency between the fuzzy goals of the objectivefunction and the decision variables.To overcome the problem in the methods of Lai et

al., eliminating the fuzzy goals for decision variables,we have developed interactive fuzzy programmingfor two-level linear programming problems [12].Moreover, from the viewpoint of experts’ impreciseor fuzzy understanding of the nature of parametersin a problem-formulation process [11,16], we haveextended it to interactive fuzzy programming fortwo-level linear programming problems with fuzzyparameters [13].However, especially for hierarchical decision prob-

lems such as in �rms, it is frequently appropriatenot only to take the cooperative relationship be-tween decision makers into consideration but alsoto examine linear fractional objectives rather thanlinear ones. Examples of objectives or criteria rep-resented as fractional functions can be shown inthe following [7,19]: for �nance or corporate plan-ning, debt-to-equity ratio, return on investment,current ratio, risk-assets to capital, actual capital torequired capital, foreign loans to total loans, resi-dential mortgages to total mortgages, etc; for pro-duction planning, inventory to sales, actual cost tostandard cost, output per employee, and so forth.For instance, by adopting a criterion with respectto �nance or corporate planning as an objectivefunction at the upper level and employing a cri-terion regarding production planning as an objec-tive function at the lower level, a two-level linear

fractional programming problem can be formulatedfor hierarchical decision problems in �rms.Under these circumstances, in this paper, we formu-

late two-level linear fractional programming problemsinvolving fuzzy parameters. These fuzzy parameters,re ecting the experts’ imprecise or fuzzy under-standing of the nature of parameters in the problem-formulation process, are assumed to be characterizedas fuzzy numbers [11]. Using the �-level sets of fuzzynumbers, the corresponding nonfuzzy �-two-levellinear fractional programming problem is introduced.The fuzzy goals of the DMs for the linear fractionalobjective functions at both levels are quanti�ed byeliciting the corresponding linear membership func-tions. In our interactive fuzzy programming method,a satisfactory solution can be derived e�ciently byupdating the satisfactory degrees of the DM at theupper level with considerations of overall satisfactorybalance between both levels. The satisfactory solutionwell-balanced between both levels is easily computedby using the bisection method, the phase one of thesimplex method [17] and the variable transformationmethod by Charnes and Cooper [6]. An illustrativenumerical example for two-level linear fractional pro-gramming problems with fuzzy parameters demon-strates the feasibility of the proposed method.

2. Interactive fuzzy programming

In this paper, we consider two-level linear fractionalprogramming problems in a situation where each ofthe DMs at both levels takes overall satisfactory bal-ance between both levels into consideration and triesto minimize an objective function of the DM, payingserious attention to preferences of the other DM. Sucha two-level linear fractional programming problem isformulated as:

minimizeupper level

z1(x1; x2)

minimizelower level

z2(x1; x2)

subject to A1x1 + A2x26b;

x1¿0; x2¿0;

(1)

where the objective function zi(x1; x2), i=1; 2 is rep-resented by the following linear fractional function,

M. Sakawa et al. / Fuzzy Sets and Systems 115 (2000) 93–103 95

and it is assumed that a denominator is positive, i.e.,qi(x1; x2)¿ 0, i=1; 2:

zi(x1; x2)=pi(x1; x2)qi(x1; x2)

=ci1x1 + ci2x2 + ci3di1x1 + di2x2 + di3

; (2)

where xi ; i=1; 2 is an ni-dimensional decision vari-able; ci1 and di1; i=1; 2 are n1-dimensional constantrow vectors; ci2 and di2; i=1; 2 are n2-dimensionalconstant row vectors; ci3 and di3; i=1; 2 are con-stants; b is an m-dimensional constant column vec-tor; and Ai, i=1; 2 is an m × ni constant matrix. Forthe sake of simplicity, we use the following nota-tions: x=(x1; x2)T ∈Rn1+n2 ; ci= [ci1 ci2 ci3], i=1; 2,di= [di1 di2 di3], i=1; 2, where T denotes transposi-tion. In the two-level linear fractional programmingproblem (1), z1(x1; x2) and z2(x1; x2), respectively,represent objective functions of the upper and thelower levels and x1 and x2, respectively, representdecision variables of the upper and the lower levels.Let DM1 denote the DM at the upper level and DM2denote the DM at the lower level.When formulating a mathematical programming

problem which closely describes and represents thereal-world decision situation, various factors of thereal-world system should be re ected in the descrip-tion of objective functions and constraints. Natu-rally, these objective functions and constraints in-volve many parameters whose possible values maybe assigned by the experts. In the conventional ap-proaches, such parameters are required to be �xedat some values in an experimental and=or subjectivemanner through the experts’ understanding of thenature of the parameters in the problem-formulationprocess.It must be observed here that, in most real-world

situations, the possible values of these parameters areoften only imprecisely or ambiguously known to theexperts. With this observation in mind, it would cer-tainly be more appropriate to interpret the experts’ un-derstanding of the parameters as fuzzy numerical datawhich can be represented by means of fuzzy sets ofthe real line known as fuzzy numbers. The resultingmathematical programming problem involving fuzzyparameters would be viewed as a more realistic ver-sion than the conventional one [11,14,16].From this viewpoint, we assume that parameters

involving the objective functions and the constraintsof the two-level linear fractional programming prob-

lem are characterized by fuzzy numbers. As a result,the two-level linear fractional programming problemwith fuzzy parameters is formulated as:

minimizeupper level

z1(x1; x2)

minimizelower level

z2(x1; x2)


x1¿0; x2¿0;

(3)

zi(x1; x2) =pi(x1; x2)qi(x1; x2)

=ci1x1 + ci2x2 + ci3d i1x1 + d i2x2 + di3

;

i = 1; 2; (4)

where c=(cij); d =(dij); b; A=(Ai); i=1; 2; j=1; 2 represent fuzzy parameters. Assuming that thesefuzzy parameters c; b; A are characterized by fuzzynumbers, let corresponding membership functionsbe: �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; : : : ; ni;k =1; : : : ; m.We introduce the �-level set of the fuzzy numbers

c, b and A de�ned as the ordinary set (c; d ; b; A)� inwhich the degree of the membership functions exceedsthe level �:

(c; d ; b; A)�

= { (c; d ; b;A) | �cij; r (cij; r)¿�;i; j=1; 2; r=1; : : : ; nj;

�dij; r (dij; r)¿�; i; j=1; 2; r=1; : : : ; nj;

�bj (bj)¿�; j=1; : : : ; m;

�ai; jk (ai; jk)¿�;

i=1; 2; j=1; : : : ; m; k =1; : : : ; ni}: (5)

Now suppose that DM1 considers that the degree ofall of the membership functions of the fuzzy numbersinvolved in the two-level linear fractional program-ming problem should be greater than or equal to a cer-tain value �. Then, for such a degree �, the problemcan be interpreted as the following nonfuzzy two-levellinear fractional programming problem which depends


on the coe�cient vector (c; d ; b;A)∈ (c; d ; b; A)�[11,16]:

minimizeupper level

c11x1 + c12x2 + c13d11x1 + d12x2 + d13

minimizelower level

c21x1 + c22x2 + c23d21x1 + d22x2 + d23


x1¿0; x2¿0:

(6)

Observe that there exists an in�nite number of sucha problem (6) depending on the coe�cient vector(c; d ; b;A)∈ (c; d ; b; A)� and the values of (c; d ; b;A)are arbitrary for any (c; d ; b;A)∈ (c; d ; b; A)� in thesense that the degree of all of the membership func-tions for the fuzzy numbers in the problem (6) exceedsthe level �. However, if possible, it would be desir-able for the DMs to choose (c; d ; b;A)∈ (c; d ; b; A)�in the problem (6) so as to minimize the objectivefunctions under the constraints. From such a point ofview, for a certain degree �, it seems to be quite nat-ural to have understood the two-level linear fractionalprogramming problem with fuzzy parameters as thefollowing nonfuzzy �-two-level linear fractional pro-gramming problem [11,16]:

minimizeupper levelx; c;d ;b;A

z1(x; c1·; d1·)

minimizelower levelx; c;d ;b;A

z2(x; c2·; d2·)


x1¿0; x2¿0;

(c; d ; b;A)∈ (c; d ; b; A)�;

(7)

where ci·=(ci1; ci2), di·=(di1; di2), i=1; 2. We as-sume that a degree of the �-level is given by DM1.It should be noted that the parameters (c; d ; b;A) aretreated as decision variables rather than constants inthe problem (7).It is natural that the DMs have fuzzy goals for their

objective functions when they take fuzziness of humanjudgments into consideration. For each of the objectivefunctions zi(x; ci·; di·), i=1; 2 of (7), assume that theDMs have fuzzy goals such as “the objective functionzi(x; ci·; di·) should be substantially less than or equalto some value pi.”

Let �=0, and then the individual minimum

zmini =zi(xio; coi·; doi·)

=min{zi(x; ci·; di·) |A1x1 + A2x26b; xi¿0;i=1; 2; (c; d ; b;A)∈ (c; d ; b; A)0} (8)

and the individual maximum

zmaxi =max{zi(x; ci·; di·) |A1x1 + A2x26b; xi¿0;i=1; 2; (c; d ; b;A)∈ (c; d ; b; A)0} (9)

of the objective function zi(x; ci·; di·) are referredto when DMi elicits a membership function pre-scribing a fuzzy goal for the objective functionszi(x; ci·; di·). DMi determines the membership func-tions �i(zi(x; ci·; di·)), which are strictly monotonedecreasing for zi(x; ci·; di·), consulting the variationratio of degree of satisfaction in the interval be-tween the individual minimum (8) and the individualmaximum (9). The domain of the membership func-tion is the interval [zmini ; zmaxi ], and DMi speci�esthe value z0i of the objective function for which thedegree of satisfaction is 0 and the value z1i of theobjective function for which the degree of satisfac-tion is 1. For the value undesired (larger) than z0i ,it is de�ned that �i(zi(x; ci·; di·))= 0, and for thevalue desired (smaller) than z1i , it is de�ned that�i(zi(x; ci·; di·))= 1.For the sake of simplicity, in this paper, we adopt

a linear membership function, which characterizes thefuzzy goal of DMi for all i=1; 2. The correspondinglinear membership function �i(zi) is de�ned as

�i(zi(x; ci·; di·))

=

0; zi(x; ci·; di·)¿ z0i ;

zi(x; ci·; di·)− z0iz1i − z0i

; z1i ¡ zi(x; ci·; di·)6z0i ;

1; zi(x; ci·; di·)6z1i ;

(10)

where z0i and z1i denote the values of the objective

function zi(x; ci·; di·) such that the degree of member-ship function is 0 and 1, respectively, and it is assumedthat DMi subjectively assesses z0i and z

1i . See Fig. 1.


Fig. 1. Linear membership function.

For all i=1; 2, suppose that applying the way sug-gested by Zimmermann [21] and setting �=0, DMispeci�es z0i and z

1i in the following. That is, using the

individual minimum (8) together with

zmi = zi(x�io; co�i·; d

o�i·); (11)

DMi determines the linear membership function as in(10) by choosing z1i = z

mini , z0i = z

mi , where �i=2 when

i=1 and �i=1 when i=2.After eliciting a membership function, DM1 subjec-

tively speci�es a minimal satisfactory level �∈ [0; 1]for the membership function �1(z1(x; c1·; d1·)). Then,DM2 also elicits a membership function and maxi-mizes the membership function subject to a condi-tion that the membership function �1(z1(x; c1·; d1·)) ofDM1 is larger than or equal to � under the given con-straints, that is, the following problem is solved:

maximizex; c;d ;b;A

�2(z2(x; c2·; d2·))


�1(z1(x; c1·; d1·))¿�;

x1¿0; x2¿0;

(c; d ; b;A)∈ (c; d ; b; A)�:

(12)

In our formulation (12), the constraint on the fuzzygoal for decision variables of DM1 is eliminatedwhile it is involved in the formulations by Lai [9] andShih et al. [18].If an optimal solution to the problem (12) exists, it

follows that DMi, i=1; 2, obtains a satisfactory solu-tion having a satisfactory degree larger than or equalto the minimal satisfactory level speci�ed by DM1.However, if DM1 speci�es larger value to the mini-mal satisfactory level, it follows that the satisfactorydegree of DM2 becomes fairly small or there exists no

feasible solution. Consequently, it is feared that over-all satisfactory balance between both levels cannot bemaintained.To take account of overall satisfactory balance be-

tween both levels, DM1 needs to compromise withDM2 on the minimal satisfactory level of DM1.To do so, a satisfactory degree of the DMs at bothlevels is de�ned as

�=min(�1(z1(x; c1·; d1·)); �2(z2(x; c2·; d2·))) (13)

and the following problem is substituted for the prob-lem (12):

maximizex; �; c;d ;b;A

�


�1(z1(x; c1·; d1·))¿�¿�;�2(z2(x; c2·; d2·))¿�;06�61;

x1¿0; x2¿0;

(c; d ; b;A)∈ (c; d ; b; A)�:

(14)

For the problem (14), introduce an auxiliary problem


�


�1(z1(x))¿�;

�2(z2(x))¿�;

06�61;

x1¿0; x2¿0;

(c; d ; b;A)∈ (c; d ; b; A)�:

(15)

Let �−1i (·), i=1; 2 be an inverse function of the con-tinuous and strictly monotone decreasing membershipfunction �i(·); i=1; 2, and then the problem (15) canbe transformed into the following equivalent problem:


�


p1(x; c1·)6�−11 (�)q1(x; d1·);

p2(x; c2·)6�−12 (�)q2(x; d2·);

06�61;

x1¿0; x2¿0;

(c; d ; b;A) ∈ (c; d ; b; A)�:

(16)


By solving the problem (16), we obtain a solutionmaximizing a minimal satisfactory degree betweenthe DMs. The problem (16) is not linear program-ming problem even if all the membership functions�i(zi(x; ci·; di·)); i=1; 2 are linear. From the prop-erties of the �-level set for the vectors of fuzzynumbers c; d ; b and the matrices of fuzzy num-bers A, it should be noted that the feasible regionsfor c; b and A can be denoted respectively by theclosed intervals [cL; cR]; [dL; dR]; [bL; bR], and[AL;AR].Therefore, we can obtain an optimal solution to the

problem (16) by solving the following mathematicalprogramming problem:

maximizex; �

�

subject to AL1x1 + AL2x26b

R ;

p1(x; cL1·)6�−11 (�)q1(x; d

R1·);

p2(x; cL2·)6�−12 (�)q2(x; d

R2·);

06�61;

x1¿0; x2¿0:

(17)

By solving the problem (17), we can obtain a solu-tion maximizing a smaller satisfactory degree betweenthose of both DMs. In the problem (17), however,even if the membership function �i(·) is linear, theproblem (17) is not a linear programming problem be-cause �−1i (�)qi(x); i=1; 2 is nonlinear. Thus, we can-not directly apply the linear programming technique,but from the following fact, we can solve the problem(17) by combined use of the bisection method andphase one of linear programming technique [11,17].In the problem (17), if the value of �, which sat-is�es 06�61, is �xed, it can be reduced to a setof linear inequalities. Obtaining the optimal solution�∗ to the above problem is equivalent to determiningthe maximum value of � so that there exists an ad-missible set satisfying the constraints of the problem(17).After �∗ has been determined, we minimize the ob-

jective function z2(x) at the lower level subject to theconstraints of the problem (17) for �= �∗, i.e., wesolve the following problem:

minimizex; c;d ;b;A

z2(x)=c21x1 + c22x2 + c23d21x1 + d22x2 + d23


p1(x)6�−11 (�∗)q1(x);

x1¿0; x2¿0;

(c; d ; b;A) ∈ (c; d ; b;A)�;

(18)

where the constraint with respect to the fuzzy goal ofthe lower level is eliminated because �∗ is a solutionto the problem (17).In order to solve this linear fractional programming

problem, using the variable transformation by Charnesand Cooper [6]

t=1=q2(x); (y1; y2; t)T = (x1; x2; 1)Tt (19)

and the properties of the �-level set for the vectors offuzzy numbers, we can transform the problem (18)into:

minimizey

cL2 y

subject to [AL1 AL2 −bR]y60;

cL1 y6�−11 (�

∗)dR1 y;

dR2 y=1;

y¿0;

(20)

where, for the sake of simplicity, we use the notationy=(y1; y2; t)T ∈ Rn1+n2+1.If an optimal solution x∗ to the problem (15),

which is obtained by solving the problems (17) and(20), satis�es the condition �1(z1(x∗))¿�, it followsthat DM1 obtains a satisfactory solution. However,the solution x∗ does not always satisfy the condition.Then the ratio of satisfactory degree between bothlevels

�=�2(z2(x∗; cL2·; d

R2·))

�1(z1(x∗; cL1·; dR1·))

(21)

which is de�ned by Lai [9], is useful. If �¿1, i.e.,�2(z2(x∗; cL2·; d

R2·))¿�1(z1(x∗; cL1·; d

R1·)), then DM1

updates the minimal satisfactory level � by increas-ing the value �. Receiving the updated level �′, DM2solves the problem (12) with �′, and then DM1 ob-tains a larger satisfactory degree and DM2 accepts a


smaller satisfactory degree. Conversely, if �¡1, i.e.,�2(z2(x∗; cL2·; d

R2·))¡�1(z1(x∗; cL1·; d

R1·)), then DM1

updates the minimal satisfactory level � by decreasingthe value �, and DM1 accepts a smaller satisfac-tory degree and DM2 obtains a larger satisfactorydegree.At an iteration ‘, let �i(z‘i ); i=1; 2, and �

‘ de-note satisfactory degrees of DMi; i=1; 2, and a sat-isfactory degree of both levels, respectively, and let�‘= �2(z‘2)=�1(z

‘1) denote a ratio of satisfactory de-

grees of the upper and the lower levels. Let a cor-responding solution be x‘. When DM1 is proposeda solution by DM2 and the following two conditionsare satis�ed, DM1 concludes the solution as a satis-factory solution and the iterated interactive processterminates.

Termination conditions of the interactive process fortwo-level linear fractional programming problemswith fuzzy parameters

1. DM1’s satisfactory degree is larger than or equal tothe minimal satisfactory level � speci�ed by DM1,i.e., �1(z‘1)¿�.

2. The ratio �‘ of satisfactory degrees is in the closedinterval, the lower and the upper bounds speci�edby DM1.The condition 1. means DM1s required condition

for solutions proposed by DM2. The condition 2. isprovided in order to keep overall satisfactory balancebetween both levels.Unless the conditions are satis�ed simultaneously,

DM1 needs to update the minimal satisfactory level �.

Procedure for updating the minimal satisfactorylevel �

1. If the condition 1. is not satis�ed, then DM1 de-creases the minimal satisfactory level �.

2. If the ratio �‘ exceeds its upper bound, then DM1increases the minimal satisfactory level �. Con-versely, if the ratio �‘ is below its lower bound,then DM1 decreases the minimal satisfactorylevel �.Let �′ denote the updatedminimal satisfactory level.

DM2 solves the following maximization problem with

the updated minimal satisfactory level �′.

maximizex; c;d ;b;A

�2(z2(x; c2·; d2·))


�1(z1(x; c1·; d1·))¿�′;

x1¿0; x2¿0;

(c; d ; b;A) ∈ (c; d ; b; A)�:

(22)

Because the membership function �2(·) is strictlymonotone decreasing, the problem (22) is equivalentto the linear fractional problem:

minimizex; c;d ;b;A

z2(x; c2·; d2·)


p1(x; c1·)6�−11 (�′)q1(x; d1·);

x1¿0; x2¿0;

(c; d ; b;A) ∈ (c; d ; b; A)�:

(23)

Using the variable transformation and the properties ofthe �-level set for the vectors of fuzzy numbers again,we can formulate the linear programming problem:

minimizex

cL2 y

subject to [AL1 AL2 −bR] y60;

cL1 y6�−11 (�′)d

R1 y;

dR2 y=1;

y¿0:

(24)

The above mentioned algorithm is summarized asfollows:

Algorithm of the interactive fuzzy programmingfor solving two-level linear fractional programmingproblems with fuzzy parameters

Step 0: DM1 determines the value of � (06�61) forthe degree of all of the membership functionsof the fuzzy parameters.

Step 1: Set ‘=1. DM1 elicits the membership func-tion �1(z1) of the fuzzy goal of DM1, andspeci�es the minimal satisfactory level � andthe lower and the upper bounds of the ratio ofsatisfactory degrees �.


Step 2: DM2 elicits the membership function �2(z2)of the fuzzy goal of DM2.

Step 3: The auxiliary problem (15) is solved throughthe problems (17) and (20), and then proposesa solution x‘; (z‘1 ; z

‘2), �

‘; �1(z‘1), �2(z‘2) and

�‘ to DM1.Step 4: If the solution proposed by DM2 to DM1 sat-

is�es the termination conditions, DM1 con-cludes the solution as a satisfactory solution,and the algorithm stops. Otherwise ‘= ‘+1.

Step 5: DM2 updates the minimal satisfactory level �in accordance with the procedure of updatingminimal satisfactory level.

Step 6: The problem (22) is solved through the prob-lem (24) and proposes an obtained solution toDM1. Return to Step 4.

3. A numerical example for two-level linearfractional programming problems with fuzzyparameters

Consider the following two-level linear fractionalprogramming problem:

minimizeupper level

z1 =c1x1 + c2x2 + 1

d1x1 + d2x2 + 1

minimizelower level

z2 =c3x1 + c4x2 + 1

d3x1 + d4x2 + 1

subject to A1x1 + A2x26 b;

xj¿0; j=1; 2; : : : ; 10;

(25)

where x1 = (x1; : : : ; x5)T; x2 = (x6; : : : ; x10)T; eachmean value of an entry of 5-dimensional coe�-cient vectors represented by fuzzy numbers ci ; d i,i=1; 2; 3; 4, and of 11 × 5 coe�cient matrices rep-resented by fuzzy numbers A1 and A2 is a randomvalue in the interval [−50; 50]; each mean value of anentry of the right hand side constant column vectorrepresented by fuzzy numbers b is a sum of meanvalues of entries of the corresponding row vectorof A1 and A2 multiplied by 0:6; 20% of the coe�cientsare set as fuzzy parameters (numbers). Coe�cientsare shown in Tables 1 and 2.Suppose that DM1 determines degree of the �-level

set as �=0:8, the initial minimal satisfactory level

as �=1:0, and the lower and the upper bounds of� as 0:6 and 1:0. The membership functions (10) ofthe fuzzy goals are assessed by using values (8) and(11). The individual minima and the correspondingoptimal solutions are shown in Table 3, and we havezm1 = − 0:900385 and zm2 =0:510708.Then the problem (15) for this numerical example

can be formulated as


�

subject to x ∈ X; 06�61;

(z1(x; c1·; d1·)+0:90)=(−1:61+0:90)¿�;

(z2(x; c2·; d2·)−0:51)=(−0:31−0:51)¿�;

(c; d ; b;A) ∈ (c; d ; b;A )�; (26)

where X denotes the feasible region of the problem(25). Data of the �rst iteration including an optimalsolution to the problem (26) are shown in Table 4.The �rst termination condition of the interactive

process is not satis�ed because the satisfactory de-gree �11(z

11)= 0:484551 of DM1 does not exceed

the minimal satisfactory level �=1:0. Consequently,DM1 must change the minimal satisfactory level, andsuppose that DM1 changes the minimal satisfactorylevel to �′=0:60 because, neither for �′=0:90, for�′=0:80 nor for �′=0:70, any solution satisfyingthe condition on the upper and lower bounds of � isnot obtained. Then a problem corresponding to theproblem (22) is formulated as

maximizex; c;d ;b;A

�2(z2(x; c2·; d2·))

subject to x ∈ X;(z1(x; c1·; d1·) + 0:90)=(−1:61 + 0:90)¿0:6;

(c; d ; b;A) ∈ (c; d ; b;A)�: (27)

Data of the second iteration including an optimal so-lution to the problem (27) are shown in Table 5.At the second iteration, the satisfactory degree

�21(z21)= 0:60 of DM1 becomes equal to the minimal

satisfactory level�′=0:60 and the ratio�2 = 0:650406of satisfactory degrees is in the valid interval [0:6; 1:0]


Table 1Coe�cients

c1 −39 c1 −21 −42 −46 d1 13 36 30 45 50c2 c2 − 1 −42 c3 −19 d2 29 42 21 23 48c3 −33 −22 −12 −33 −19 d3 38 31 d1 19 3c4 c4 1 42 35 19 d4 29 42 d2 23 48

A1 −23 −2 42 a1 −48 A2 39 −9 −44 −20 47a2 34 a3 5 −11 4 a20 −20 −22 −10−7 30 a4 41 a5 −33 a21 2 −14 −15

−19 a6 −33 −44 4 37 −48 26 −47 36−46 −48 a7 a8 −2 a22 −1 a23 a24 −19−25 a9 −9 −42 a10 1 3 −4 29 −27a11 −10 2 a12 10 7 44 5 46 2614 a13 4 34 a14 −37 −26 −11 8 −5a15 a16 11 −19 8 26 −11 −12 −11 −18a17 5 −32 a18 −30 47 −17 a25 8 32a19 33 12 8 43 a26 a27 −31 −43 −31

b −40 −44 b1 −48 −95 −16 131 4 −2 45 11

Table 2Fuzzy parameters

c1 −57 −48 −39 c2 −10 −9 −8 c3 −42 −35 −28c4 8 9 10 d1 31 38 45 d2 17 21 25

a1 −58 −49 −40 a2 −6 −5 −4 a3 −8 −7 −6a4 11 13 15 a5 −6 −5 −4 a6 6 7 8a7 −18 −15 −12 a8 −6 −5 −4 a9 26 32 38a10 12 14 16 a11 39 48 57 a12 34 42 50a13 −7 −6 −5 a14 27 33 39 a15 31 38 45a16 −20 −17 −14 a17 22 27 32 a18 4 5 6a19 36 45 54 a20 −51 −43 −35 a21 23 28 33a22 12 14 16 a23 −10 −9 −8 a24 −34 −29 −24a25 25 31 37 a26 13 16 19 a27 −38 −32 26

b1 19 23 27

Table 3Individual minima and the corresponding optimal solutions

zmin1 −1.610991x1 0.814301 0.000000 1.137982 0.127555 0.000000x2 0.000000 0.000000 2.202210 0.991914 0.000000

zmin2 −0.318552x1 1.075991 0.527692 0.000000 0.276268 0.258014x2 0.000000 0.866805 0.000000 0.195700 0.551630

of the ratio. Therefore this solution satis�es bothof the termination conditions of the interactiveprocess and becomes a satisfactory solution forboth DMs.

4. Conclusions

In this paper, we have proposed interactivefuzzy programming for two-level linear fractional


Table 4Iteration 1

�1 0.484551

x11 0.857028 0.380029 0.378030 0.289072 0.000000

x12 0.000000 0.402141 0.930456 0.580505 0.187229

z11 −1.244710 �1(z11) 0.484551

z12 0.108889 �2(z12) 0.484551

�1 1.000000

Table 5A satisfactory solution to the two-level linear fractional programming

x21 0.796402 0.363118 0.569424 0.272600 0.000000

x22 0.000000 0.277158 1.199084 0.568590 0.119415

z21 −1.326748 �1(z21) 0.600000

z22 0.187094 �2(z22) 0.390244

�2 0.650406

programming problems with fuzzy parameters. In ourinteractive method, once DM1 determines the value of�, the fuzzy goals of the decision makers at both levelsare determined, and a satisfactory solution is derivede�ciently by updating the minimal satisfactory levelsof decision makers at upper levels with considerationsof overall satisfactory balance between both levels.A solution at each iteration is computed by using thebisection method, the phase one of the simplexmethod and the variable transformation by Charnesand Cooper. An illustrative numerical example fortwo-level linear fractional programming problemswith fuzzy parameters has been provided to demon-strate the feasibility of the proposed method.

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