Asia-Pacific Journal of Operational ResearchVol. 24, No. 4 (2007) 557-573 World Scientific Publishing Co. k Operational Research Society of Singapore

A METHOD FOR SOLVING LINEAR PROGRAMMINGPROBLEMS WITH FUZZY PARAMETERS BASED ON

MULTIOBJECTIVE LINEAR PROGRAMMING TECHNIQUE

M. ZANGIABADIFaculty of Mathematics and Computer Sciences

Shahid Bahonar University of Kerman, Kerman, IranZangiabadi^raail.uk. ac. ir

H. R. MALEKI*Department of Basic Sciences

Shiraz Univemity of Technology, Shiraz, Iranmaleki@sutech. ac. ir

Received 26 October 2005Accepted 11 November 2006

In the real-world optimization problems, coefficients of the objective function are notknown precisely and can be interpreUid as fuzzy numbers. In this paper we define theconcepts of optiniality for linear programming problems with fuzzy parameters basedon those for miiltiobjective line.ar programniirig problems. Then by using the concept ofrompiiriaon of fuzzy numbers, we transform a linear programming problem with fuzzyparameters to a multiobjective linear programming problem. To this end, we proposeseveral theorems which are used to obtain optimal solutions of linc-ar programmitig prob-lems with fuzzy parameters. Finally some examples are given for illustrating the proposedmethod of solving linear programming problem with fuzzy parameters.

Kejjwords: Linear optimization: fuzzy number; ranking function; multiobjective linearprogramming.

1. IntroductionFuzzy set theory presented by Zadeh (1965) has been found extensive applica-tions in various fields. Following tbe fuzzy decision-mak ing concept proposed byBellman and Zadeh (1D7U), Zimmermaim (1974, lU7(j) and Tauaka et al. (1974)introduced fuzzy sets into an ordinary LP problem with fuzzy objective and con-straints. Subsequently, several kinds of fuzzy linear programming (FLP) problemshave appeared in the literature (Cadenas and Verdegay, 1997; Delgado et at., 1985,1987, 1989; Ishibuchi aud Tanaka, 199(J; Luhandjula, 1983: Oh'Eigeartaigh, 1982;

* Corresponding author.

557

558 M. Zangiabadi & H. R. Maleht

Rommelfanger et al, 1989; Tanaka et a/., 1984; Verdegay, 1982; Zimmermann, 1985)and different approaches of resolution bave been proposed. Delgado ei at. (1989)described the more important problems in fuzzy linear programming. In continue,they proposed a resolution method for a general model of FLP problems involvingall of the above items.

To mathematically model the FLP problem, perhaps a crisp version of it hasadvantageous because of necessary tools for solving it whicJi will be conventional(Delgado et a/., 1987). For this means Dolgado et al. (1987) coasidered FLP problemin which coefficients of the objective function are fuzzy numbers. To give a resolutionmethod for this problem they introduced an auxihary multiobjective parametric lin-ear programming (MPLP) problem. They showed that, the parametric solution ofMPLP is a fuzzy solution for FLP problem. Rommelfanger et al. (1989) presenteda method for solving linear prograumiing problems with fuzzy parameter in theobjective function. In their method, FLP problem was reduced to a few extremeobjective functions. Ishibuchi e( al. (1990) converted the mathematical program-ming problem whose objective function lias interval coefficient into a multiobjectiveproblem using the order relations. Luhandjura (1987) formulated FLP problem as asemi-infinite linear prograimning probleirLs with infinitely many objective functions.Mae(ia (2001) formulated the FLP problem as a two-objective linear programmingproblem and Zhang et al. (2003) formulated it as a four-objective linear program-ming problem.

On the other hand, one of the most convenient methods is based on the conceptof comparison of fuzzy numbers by nsing ranking functions (Bortolan and Degani.1985; Maleki et ai, 2000, 2002; Mishmast et at., 2004: Yazdani Peraei et /., 2001;Yoon, 1996). However, it is clear that using a single ranking function will producetoo broad a surmnary of the aforementioned information (as in probability theorywhen one uses only the average value to represent a certain j)robability distribu-tion). Therefore, a description based on more than one characteristic seems moreappropriate (Delgado, 1998; Mishmast, 2004). In this paper, to remove the short-coming in applying ranking functions we associate a A'-dimcntional vector rankingfunction to a fuzzy number, where the components of that are detcriiiincd on thebasis of the decision maker's preferences.

The aim of this paper is to extend the Zhang et al. method by using a vectorranking function. In fact, we solve linear programming problem with fuzzy param-eters based on multiobjective linear programming techniques.

The paper has the following structure. In Sec. 2, we present comparison offuzzy numbers by using ranking functions and review the concept of optimality formultiobjective linear prograimning problem. In Sec. 3, we apply a vector rankingfunction to convert FLP problem to a multiobjective linear programming problem.In Sec. 4, we briefly describe Delgado and Verdegays' method. In Sec. 5, we applylexicographic ranking function for solving FLP problem. Also some examples arepresented.

Solving Linear Programming Problems with Fuzzy Parameters .559

2. Preliminaries2.1. Vector ranking functionLetx - {xi,X2,.-.^XnY. y = (yi,y2,---.yn)' G K " be two vectors, where ' denotestranspose of the vector. Then we write x > y if and only if Xi > yi, for all i belongto iV = {1 ,2 , . . . , n}; x > y if and only if a:, > yi, for all i iV; x ^ y if and onlyif Xi ^ yi, for some i N.

Definition 2.1. (Cadenas and Verdegay, 1997) A fuzzy set a on M is called afuzzy number if it holds:

(1) Its membership function is upper semi continuous.(2) There exist three intervals [a, 6], [6,c], [c,d\ such that a is increasing on [a,6],

equal to 1 on [6, c], decreasing on [c, d] and equal to 0 anywhere else.We denote the set of all fuzzy numbers by F(E). A simple method for ordering

the elements of F(R) consists in the defining of a ranking function R : F(R) v Mwhich maps each fuzzy number into a point of the real line, where a natural orderexists. It is obvious tbat more than one ranking function can be defined (Bortolanand Degani, 1985; Cadenas and Verdegay, 1997).

Based on the decision maker's preferences, assume there exist k importantattributes associated to fuzzy number a such tbat the ith of them can be char-acterized by the ranking function Ri : F{R) -> R. In tbis (-ase, we associate a crisp/c-dimensional vector, R(a), to a as follows:

Definition 2.2. The vector function R(.). defined as above, is called a vectorranking function. Moreover, let a and b belong to i^(R), then: tt > 6 if and only if R(a) > R(6).

R a > 6 if and only if R(a) > R(6). n = & if and only if R(a) - R(6).

RI

a / 6 If and only if R() ^ R(6).R

Also we write a < 6 if and only if 6 > a; a < 6 if and only if 6 > a.R R R R

Example 2.1. Let d bo a fuzzy number. Some vecttor ranking functions may bedefined as follow:

(a) The Roubcns ranking function which is defined as:1 f '

R{a) = -x I (inf Or + supar)dr,2 Jo

560 M. Zangiabadi & H. R. Maleki

where ttr is an r-cuts of a, i.e., hr = {x eM. \a{x) > r}, 0 < r < 1. Roubensranking function is a vector ranking function with k ^ 1 when the decisionmaker choose Ri{.) as the only characteristic for ordering.

(b) It is possible that the decision maker considers expectation, E{a), and variance,Var(a), of the density function associated with a (Delgado et ai., 1998a) as thecharacteristics for ordering. In this case k = 2 and

(c) For fc = 3, based on the the decision maker preferences we may considerR(a) - {V{a), A{a), F(a))',

where V{a), A{a) and F{a) are value, ambiguity and fuzziness nof a, respectively,which are defined as:

V{a)= f r[La{r) + Ih{r)]dr,Jo

A{a)= f r[/2a(r)-La(r)]rfr,Jo

rl/2 .1F{a) = / r[Ih{r) ~ L^{r)] dr + / rlLa{r) - Ra{r)] dr,

Jo Jl/2where both La(.) and Ra{.) are from [0,1] to R and defined as:

'mi{x\x G Supp(a)} if r 0.ifrG{O,l],

| { l Supp(a)} if ; = 0,and Supp(a) ^ {x G R\a{x) > 0}.

Observe that since a is upper semi-continuous with bounded support, the r-cutsuf a are clustxl and bounded iiatervaLs. Observe, too, that the function La is increas-ing and the function Ra is decreasing. For more details, see Delgado et al. (1998a,1998b).

2.2. Multiobjective linear progmmming[n this section we briefly describe multiobjective Unear programming problem andthe concept of optimality for it.

The model:max z(x) Cx,

s.t. Ax 0.

wiiere C'isakxji matrix of coefficients of the linear objective functions, A G E"***"and b G K" and x G R", is called a muitiobjective Unear prograinining model(MOLP).

Solving Linear Programming Problems with Fuzzy Parameters 561

For the sake of simplicity, we set X - {x G K^'lAx < b, x > 0}. We review theconcept of optimaiity for MOLP as usual manner.

Definition 2.3. A point x* G X is called a complete optimal solution for MOLPif and only if z(x*) > z(x) for all x G X.Definition 2.4. A point x* G X is called a pareto optimal solution for MOLPif and only if there does not exist another x G X such that z(x) > z(x') andz(x)^z(x ' ) .Definition 2.5. A point x* G X is called a weak pareto optimal solution forMOLP if and only if there does not exist another x G X such that z(x) > z(x').

Let E''. EP and E'^P denote the set of all complete optimal solutions, paretooptimal solution and all weak pareto optimal solutions for MOLP, respectively, thenit is easy to show that E^ C EP C

3. Liner Programming Problem with Fuzzy ParametersIn this section, we consider a lijiear programming problem with fuzzy parametersand investigate some methods to solve it. Moreover, we suppose that R is a givenvector ranking function.

Definition 3.1. The modelmax z " cx,

R (3.1)s.t. X G X.

where c = (f?i,f2,... ,Cn) G (F(E))", Ls called a linear programming probiem withfuzzy parameters (FLP).Definition 3.2. A point x* G X is called an R-optimal solution for FLP (3.1) ifand only if cx* > cx for all x G X.

RDefinition 3.3. A point x* G X is called an R-efficient solution for FLP (3.1) ifand only if there does not exist another x G X such that cx > cx* and cx ^ ex.

R RDefinition 3.4. A point x* G X is called an R-weak efficient solution for FLP(3.1) if and tmlv if there does not exist another x G X such that cx ' >cx*.

In this paper X'^^, X'^ '^ and X^*^ denote R-optimal. R-efficieiit and R-weakefficient solution sets, respectively. In addition, from tlieir definitions, it can beeasily proved that X"" c X^ c X""".

Now, associated with the model (3.1), we consider the following MOLP problem:max z(x) = (i2i(cx)

s.t. X G X . (3.2)

562 M. Zangiabadi & H. R. Maleki

In a more compact format, MOLP (3.2) is written:max {z(x) = R(cx)|x G X}, (3.3)

where R(.) = (/?,(.), R2{.) Rk{.)y.The relationship between the optimal solutions ofthe MOLP (3.3) and the model

(3.2) can be characterized by the following theorems.Theorem 3.1. A point x* e X is an R-optimal solution for the model (3.1) ifand only ifx.* is a complete optimal solution for MOLP (3.3).

Proof. Assume that x ' G X is an R-optimal solution of the model (3.1), then forany X G X we have cx' >cx. Now, by definition of > we have R(cx') > R(cx),

a R V / ^ >, ^1for all X G X, and hence z(x*) > z(x) for all x G X.Therefore x* is a completeoptimal solution for MOLP (3.3).

Conversely, assume x* is a complete optimal solution of MOLP (3.3), then forall X G X we have z(x') > z(x), and hence R(cx*) > R(cx). Therefore ex* >cx,for all X G X. This implies x* Ls an R-optimal soiution for the model (3.1). n

Theorem 3.2. A point x* G X zs an R-effcient solution for model (3.1) // andonly ifx* is a pareto optirnat solution for MOLP (3.3).

Proof. Let X* G X be an R-efficient solution for the model (3.1). Oti the contrary,we suppose that x* is not a pareto optimal solution for MOLP (3.3), i.e., thereexists an X G X such that z(x) > z(x*) and z(x) ^ z(x'), hence R(cx) > R(cxV)and R(cx) ^ R(cx*). This impUe.s cx>cx* and cx?^cx', which contradicts that

R RX* G X is an R-efficient solution for the model (3.1).

On the other hand, let x" G X be a pareto optimal solution for MOLP (3.3).On the contrary, we suppose that x ' G X is not an R-efficient solution for themodel (3.1), i.e., there exists an x G X such that cx > cx* and cx ^ cx* this implies

R RR(cx) > R(cx ) and R(cx) ^ R(cx*). This means z(x) > z(x-) and z(x) ^ z(x*),which contradicts that x ' G X Ls a pareto optimal solution for MOLP (3.3). DTheorem 3.3. A point, x" G X is an R-weak effcient soiution for modei (3.1) ifand only if x* is a weak pareto optimal solution for MOLP (3.3).

Proof. Straight forward.

A classic method to generate a pareto (weak pareto) optimal solution of MOLP(3.3) is to use the weighted sums of objective functions, i.e., to consider the solutionsof tlic following weighted problem:

max {wR(cx)|x G X}, (3.4)where w G W - {w G !R*^ |w > 0, ^ ^ ^ j Wj = 1}. For finding R-efficient solutionor R-weak efficient solutions of the model (3.1J, it suffices to use the followingtheorems.

Solving Linear Programming Problems with Fuzzy Parameters 563

Theorem 3.4. Let a point x* e X be an optimal solution of weighted problem(3.4) for some w > 0, then x ' is an K-efficient solution for model (3.1).

Proof. On the contrary, let x* not be an R-efficient solution of model (3.1). FromTheorem 3.2, we conclude that x* is not a pareto optimal solution of MOLP (3.3).Therefore there exists an x G X such that z(x) > z(x') and z(x) ^ z(x'). HenceR(cx) > R(cx*) and R(cx) ^ R(cx*). This means /Ij(cx) > RJ{CJC') for allj e {l,2,...,k} and there exists at least a jo G {1.2,.. .,A;} such that i?j,,(cx) >i2j,,(cx'), now since w > 0 we have:

k fcwR(cx*) = ^WjRj{cK*) < "^WjRjic^) = wR(cx).

This contradicts that x ' is an optimal solution of weighted problem (3.4). D

Theorem 3.5. Let a point x* G X 6e an R-efficient solution for model (3.1), thenx ' is an optimal solution of weighted problem (3.4) for som.e w > 0.

Proof. Let x* be an R-efficieiit solution for model (3.1), by Theorem 3.2, it is apareto optimal solution for MOLP (3.3). By using Theorem 9.6 of Steuer (1986),x* Ls an optimal solution for weighted problem (3.4) for some w > 0. D

Theorem 3.6. A point x* is an optimal solution for weighted problem. (3.4) forsome w G W / and only if x" is an R-weak efficient solution for model (3.1).

Proof. Straight forward.

By Theorem 3.4, among the other results, each x G X which is an optimalsolution of the weighted problem (3.4) for any w > 0 is an R-efficient optimalsolution for model (3.1). On the other hanrl. under the conditions of Theorem 3.5,at least one w > 0 is associated with every R-efRcient optimal sohition, that causesthe weighted problem (3.4), to have it as an optimal solution. Note that differentmethods exist to help the decision maker for assigning weights to the objectivefunc-tions (Diakoulaki et aL, 1995; Nutt, 1980; Pekelman and Sen, 1974; Solyimosimid Domhi, 198; Steuer.

Example 3.1. To illustrate the efficiency of the propostjd method we consider thefollowing problem which is used by Zhang et al. (2003).

s.t. xi + 4x2 ^ 14,

4x1 + IO12 < 38, (3.5)2811 - 5x2 < 56,

Xl > 0, X2> 0,

564 M. Zangiabadi & H. R. Maleki

where the membership functions of c^ and C2 are

x-5 5 I), then x" is an R-efficiout solution for model (3.5). The solution of theproblem depends on the choice of the weights in problem (3.7). For example, if the

Solving Linear Programming Problems with Fuzzy Parameters 565

decision maker sets wi = 0.5, ziJ2 = 0.25, wz = 0.25, then the optimal solution is(xl,x5) = (0,3.5).

We denote that our method includes Zhang et al. method (2003) as a specialcase. To this end we consider the following examples.

Example 3.2. Consider FLP problem (3.5). If we set A; ^ 4 and

where i2i(a), R2{a), Raia) and R4{a) are defined as:Ri{a) - i n f { x e R | a ( x ) > 0 } ,i?2(a) - i n f { x G E | a ( x ) > l } ,/?3(a) = sup{x G M[a(x) > 1},R4{d) = sup{x G R|a(x) > 0},

then

The following MOLP is associated with problem (3.5).max z(x) = (5X1 + 16x2. 6x1 + 17x2, 7xi + 18x2, 20xi + 40x2)',s.t. xi + 4X2 < 14,

4x1 + 10X2 < 38, (3.8)28xi - 5x2 < 56,Xi > 0, X2 > 0.

The MOLP (3.8) is exactly the same as an MOLP problem obtained by the methodof Zhang et al. (2003, example 2).Example 3.3. Consider FLP problem (3.5). If we set fc = 5 and

where Ri{a),R2{a},Rz{a),R4{a) and R.^{d) are defined as:/2i(a) = inf{x G Ria(x) > 0},R2{d) = inf{x G Rla(x) > 1},

z{h) - sup{x G M(a(x) > 1},ia) = sup{x G Rjd(x) > 0},ia) = A{d),

566 M. Zangiabadi & H. R. Maleki

where A{d) is defined in the previous section. Then

=7,

i25(e2) = 13/3.The following MOLP is associated with problem (3.5).

max z(x) = (5xi + 16x2,6xi + 17x2,7xi + 18x2,20xi4-40x2,17/6x] + 13/3x2)',

s.t. xi + 4x2 < 14,4x, -h 10x2 < 38, ^^'^^28xi - 5x2 < 56,3:1 > 0, X2 > 0.

To solve the above problem, the following weighted problem is considered:max w;i(5xi -H 16x2) -l-u;2(6xi -H 17x2) + ti;3(7xi + 18x2)

+ W4(20xi + 40x2) + W5il7/6xi -I- 13/3x2),s.t. xi + 4x2 < 14,

4X1 + 10x2 < 38, ^^'^^^28xi - 5x2 < 56,Xl > 0, X2 > 0.

From Theorem 3.4, if x* is an optimal solution to the weighted problem (3.10) forsome w > 0, then x* is an R-efficient solution for model (3.5). The solution of theproblem depends on the choice of the weights in problem (3.10). For example, if thedecision maker sets Wi - w?2 = "^ 3 - W4 ^ O.l,w^ ^ 0.6, then the optimal solutionis(:rt,.-c^)-(2.5,2.8).

4. Delgado and Verdegays' Method

In this section, we briefly described Delgado and Verdegays' method (1987). Weconsidered the following FLP problem

max{cixi +C2X2-!--+CnXn|xe X} (4.1)Where c = (o], co... . , Cn), are vaguely stated estimates of the coefficients which aretiefined by means of membership fun('t ion like

'0 li Rj < x < r j ,hi{x) if r,

Solving Linear Programming Problems with Fuzzy Parameters 567

Delgado and Verdegay (1987) showed that, the fuzzy solution to FLP (4.1) can beobtained from the solution of the MPLP problem.

max (c'x, c^x,...,s.t. X e X,

where (1 -a) is the subset constituted by vectors whose jtk component is equalto either the upper or the lower bound of C j , ^ j ( l - a ) =g~^{l-a) or ipj{l-a) =/i~'(l - a ) , j = 1,2,...,n, that is,

c= (CI ,C2, . . . ,CH) G :(1 -a)-^Cj = t^j(l - a ) or ipj{l - a ) , j" = l ,2 , . . . , n .Notice that several approaches can be considered to solve muitiobjective linear

programming model (4.2). For (nore detail, see Delgado and Verdegay (1987).Although this method is applicable to solve FLP problem in which the co.sts are

fuzzy ones with trapezoid membership functions, but one of the most advantage ofthis method is giving a fuzzy solution to the FLP problem.

Example 4.1. We consider the following problem which is used by Detgado andVerdegay (1987).

max z{x) = ciXi 4-2x2,s.t. 2x1 - i 2 < 6 ,

(4.3)Xi -I- 5x2 < 10,Xi > 0, X2 > 0.

where the membership functions of ci is defined as:

{x-2 2

568 Af. Zangiabadi & H. R. Maleki

5. Using Lexicographic Ranking Function for Solving FuzzyNumber Linear Programming Problem

Suppose that the decision maker orders the components ofthe vector ranking func-tion based on some preferences, by starting with the most important component.In this case we associate a lexicographic ranking function to a fuzzy number.

Definition 5.1. A vector x is called lexicographically positive, denoted by x ;^ 0,if the following two conditions hold:

1. X is not identically zero.2. The first nonzero component of x is positive.

A lexicographically nonnegative vector, denoted by >rO, is either the zero vector ora lexicographically positive vector.

Definition 5.2. A vector x is lexicographically greater than y, denoted by x ;^ y,if and only if x y ;^ 0.Also X is said to 6e lexicographically greater than or equal to y, denoted by x >:: y,if and only if x y >: 0.

Definition 5.3. The vector ranking function 3? whose components are orderedon the basis of the decision maker's preferences, starting with the most imjx)rtant,is called a lexicographic vector ranking function. Moreover, let a and 6 belong toF{ R ), then we define

d > h if and only if!)?() a > I) if and only if 3?(a) >- ! a h if and only if 9?(a) = a^b'ii and only if ^{a) /

3?

Also we write o < 6 if and only if 6 > a; o < & if and only if 6 > a.

Now we return to model (3.1) and apply lexicographic vector ranking fim

Solving Linear pTogramming Problems with Fuzzy Parameters 569

following MOLP problem:

max (Si(ex), 3?2(cx),..., Mk{cx))\ ,^ 2)s.t. X e X.

Which in a more compact format can be written as follow:

max{3?(cx)|xGX}. (5.3)For solving model (5.2), by using lexicographic maximization process (Steuer, 1986),we recursively construct the following reduced feasible regions:

50 = X,51 = {y So : Ki(cy) = max{!R,(cx) : x G So}},Sa - {y G S, ; 5ft2(cy) - max{2(cx) : x G Si}},

Sfc - (y G Sfc_i : JJfc(cy) = max{3efc(cx) : x e Sfc_i}}.The process begins by maximizing objective function 5Ri(cx) which is boundedover So = X. By constraining So to the maximal value of the first criterion. Si isobtained. Then, the second objective function 3?2(cx) which is bounded over Si ismaximiztxl. By constraining Si to the maximal value of the second criterion, S2is obtained. The process continues in this fashion until we obtain either Sit 7^ or for some l{l < I < k), Si,Si+i,... ,Sk = c x " and cx?^cx*. It follows that 3?(cx) >: K(cx*) andi?(cx) ^ !R(cx'). Hence 3?(cx) - 3?(cx') t 0, and 3?(cx) - 3(cx*) / 0. Now, weconsider the following two cases:

Case 1: Let !'f?](cx) > 3fi(cx*). Hence x* does not belong to Si . From Sjt CSfc-i C - c S(] = X, it follows that x* does not belong to Sfc, which contradictsthat x ' G Sfc.

Case 2: There exists a ( e {2 k] such that %(cx.) - ?R((cx*) > 0 and%{c5(.) - %{cx') - 0 for aU i G {1 ,2 , . . . ,f - 1}. Since ?R,(cx) - >R,(cx') = 0 for alli G i l . 2 , . . . , i - 1} and X* e Sfc C S^-i C C So - X. we conclude that xG S^,for all i G {1 ,2 , . . . , t - 1}. On the other hand, 3?,(cx) > U,.{cx*} and x e Si,i E{1.2, . . . , i - l } . imply that x* does not belong to St. This contradicts that x* G Sk-

D

570 M. Zangiabadi & If. R. Maleki

From Theorem 5.2, if x* e Sfc, then x ' is an S-efficient solution for the model(5.1). A situation may happen that the set Sfc is empty. It occurs wfien for anl{l < I < k)^i{cx) is unbounded over Si_i. In this case, we have the option ofgaining the flexibility of the decision maker to either change the order of componentsof vet^ tor ranking function corresponding with the remaining objective functions orapply the weighted method in the previous section for solving FLP.

Example 5.1. Consider the model (3.5) given in Example 3.1. Let

whose components are ordered based on the decision maker's preferences. V{d), A{d)and F{d) are defined in Sec. 2.Note that:

V{ci) - 8.5, V{C2} = 21.

To find S], the following LP problem is solved.

max 2(x) = 8.5xi -I- 21x2,s.t. Xl -!-4x2 < 14,

4xi + 10x2 < 38, (5.4)28X1 - 5x2 < 56,3^1 > 0, X2 > 0.

We obtain Si = {{x\,xl) = (0,3.5)}. Since S] Is a singleton, then Si = S2 = S.-^ .Hence, from Theorem 5.2. (x;,x^) = (0.3.5) is an 9?-efficient solution formodel (3.5).Example 5.2. To illustrate the efficiency of the proposed method in comparisonwith the method proposed by Delgado and Verdegay (1987), Consider the model(4.3) given in Example 4.1. Let. based on the decision maker's preferences, weconsider k = 'S and

whose components are ordered based on the decision maker's preferences. V(a), A{a.)and F(o) are defined in Sec. 2.Note that

Solving Linear Programming Problems with Fuzzy Parameters 571

To find S|. the following LP problem is solved.max z(x) ^ 19/6x1 + 2x2,s.t. 2xi - X2 < 6,

(5.5)Xl + 5x2 < 10,X\ > 0, X2 > 0.

We obtain Si = {(Xi,x5) = (3.64,1.27)}. Since Si is a singleton, then Si = S2 =S.v Hence, from the Theorem 5.2, (xj,x5) = (3.64,1.27), 2* = 14.05 is an 9?-efficientsolution for model (4.3). Although the optimal solution obtained by our method andthe Delgado and Verdegays' method for model (4.3) are the same, but the optimalvalues of the objective function are different, see Example 4.1.

In comparison with our method, Delgado and Verdegay developed an auxil-iary 2" objective parametric linear programming (MPLP) problem related to FLPproblem. Clearly, when n is quit large, this auxiliary MPLP problem has so manyobjective functions. But in our method, the number of objective functions in MOLPcorrespond to FLP problem is based on the decision maker's preferences.

Delgado and Verdegays' method gives a fuzzy solution to the FLP problem, butsolution which can be found in our method unfortunately is crisp.

However Delgado and Verciegays' metiiod (1987) is only ai)plicable to solve FLPproblem in which the costs are fuzzy ones with trapezoid membership functions. Ourmethod is applicable to solve FLP problem in which the costs are fuzzy numberswith any form of membership functions.

Remark 5.1. Notice that .several approaches can be considered to solve FLPproblems. However some methods may be more suitable than others for particulai"FLP problems.

6. ConclusionIII this paper we consider a linear programming problem with fuzzy parametersin the objective function. There are several approaches for .solving this problemwhicli use different ranking functions. To improve the draw back of using a singlecharacteristic, we associated a ^-dimensional vector ranking function to a fuzzynumber. However when the decision maker has jireforonce in the components ofvector ranking function that it can be interpreted by an ordering, an alternativemethod that gains the privilege of the lexicographic vector ranking function wasdeveloped.

Acknow ledgmentsThis paper is supported in part by Fuzzy systems and Ai>i)licatioiis Center of Excel-lence, Shahid Bahonar University of Kerman, Kerman. I.R. of Iran.

572 M. Zangiabadi & H. R. Maleki

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Hamid Reza Maleki received his BSc from Sliiraz University in Iran, MSc andPhD from Kerman University in Iran. He is an Assistant Professor of Mathematics inShiraz University of Technology. His research interests are fuzzy operation researchand graph theory. He has published about 17 papers in international journals.Maryam Zangiabadi ret^ eived her BSc and MSc both from Kerman University inIran. She is pursuing a doctoral degret; from the University of Kerman, Iran. She isa Lecturer at the Department of Mathematical Science, Shahrekord University. Herresearch interests are fuzzy operation rese.arch and graph theory. She has published2 papers In the proceedings of international conferences. She has published 1 paperin an international journal.