Fuzzy Sets and Systems 160 (2009) 2714–2721www.elsevier.com/locate/fss

Pareto-optimal solutions in fuzzy multi-objective linearprogramming

Mariano Jiméneza,∗, Amelia Bilbaob

aDpto. de Economía Aplicada I, Universidad del País Vasco, Plaza Oñati 1. 20018 San Sebastián, SpainbDpto. de Economía Cuantitativa, Universidad de Oviedo, Avda. del Cristo s/n. 33006 Oviedo, Spain

Received 26 November 2007; received in revised form 19 December 2008; accepted 19 December 2008Available online 24 December 2008

Abstract

The problem of solving multi-objective linear-programming problems, by assuming that the decision maker has fuzzy goalsfor each of the objective functions, is addressed. Several methods have been proposed in the literature in order to obtain fuzzy-efficient solutions to fuzzy multi-objective programming problems. In this paper we show that, in the case that one of our goalsis fully achieved, a fuzzy-efficient solution may not be Pareto-optimal and therefore we propose a general procedure to obtain anon-dominated solution, which is also fuzzy-efficient. Two numerical examples illustrate our procedure.© 2008 Elsevier B.V. All rights reserved.

Keywords: Multi-objective programming; Goal programming; Fuzzy mathematical programming; Fuzzy-efficient solution; Pareto-optimalsolution; Two-phase method

1. Introduction

Let a multi-objective linear programming (MOLP) problem with k objective functions zi (x) = ci x , i = 1, . . . , k be

Min z(x) = (z1(x), z2(x), . . . , zk(x))

S.t. x ∈ X, (1)

where X = {x ∈ Rn|Ax�b, x�0}, ci = (ci1, . . . , cin) ∈ Rn , i = 1, . . . , k; b = (b1, . . . , bm) ∈ Rm and A is a Rm×n

matrix.For the sake of simplicity we suppose that we want to minimize all the objective functions. However, the procedure

demonstrated is easily extendable were the case to include some maximizing objectives.In problem (1) it is unlikely that all objective functions will simultaneously achieve their optimal value. Therefore

in practice the Decision maker (DM) chooses a satisficing solution, according to the aspiration level fixed for eachobjective.

∗Corresponding author. Tel.: +34943015818; fax: +34943018360.E-mail addresses: mariano.jimenez@ehu.es (M. Jiménez), ameliab@uniovi.es (A. Bilbao).

0165-0114/$ - see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.fss.2008.12.005

M. Jiménez, A. Bilbao / Fuzzy Sets and Systems 160 (2009) 2714–2721 2715

Assuming that theDMproposes imprecise aspiration levels such as, “the objective function zi (x) should be essentiallyless than or equal to some value gi”, model (1) can be written as

Find x

Such that zi (x)�gi i = 1, 2, . . . , k

x ∈ X. (2)

Each expression ci x�gi is represented by a fuzzy set called fuzzy goal, whose membership function, �i (zi ),�i : R → [0, 1], provides the satisfaction degree �i to which the i th fuzzy inequality is satisfied. In order to de-fine the membership function �i (zi ) the DM has to provide the tolerance margins gi + ti that he is willing to accept. So�i (zi ) should be equal to 1 if zi �gi , strictly monotone decreasing from 1 to 0 over the interval (gi , gi + ti ) and equalto 0 if zi �gi + ti .

Models like (2) are named fuzzy multi-objective linear programming (FMOLP) problems. As it has been widelyseen, a FMOLP problem, using the fuzzy decision (max–min) of Bellman and Zadeh [1] and introducing the auxiliaryvariable � adopts the following formulation [13]:

Max �

S.t. 1��i (zi (x))���0, i = 1, . . . , k,

x ∈ X. (3)

If there exists an unique optimal solution of (3), then it is a fuzzy-efficient solution to the FMOLP problem (2) (seeDefinition 1). However if the uniqueness of a solution is not satisfied, the fuzzy-efficiency is not guaranteed for allsolutions of model (3), but at least one of the multiple optimal solutions is fuzzy-efficient [11]. In order to produce afuzzy-efficient solution several approaches have been proposed in the literature. Guu and Wu [3,4], proposed a secondstep in which they use the additive criterion to aggregate the fuzzy goals. Sakawa et al. [9,10] proposed an interactiveprocedure. However, during some stages of this process they use the max–min approach whereas during other stagesthey use the additive criterion. As Dubois and Fortemps [2] say these kinds of approaches are partially inconsistentbecause they change from a non compensatory criterion (min) to a compensatory criterion (sum) during different stages.In order to overcome this drawback Dubois and Fortemps [2] propose a multi-step procedure which involves solvingsequentially, several max–min optimization problems.In order to illustrate our procedure, for the sake of simplicity, in this paper we only put emphasis on the procedures

suggested by Guu and Wu and Dubois and Fortemps, but any other procedure that supplies a fuzzy-efficient solutioncould be used.Indeed, in this paper we propose an extension to the Guu andWu and Dubois and Fortemps approaches. Our method

finds an efficient solution to a more general case than that of the aforementioned authors. In order to achieve this,we use a modified technique for generating a Pareto-optimal solution to the MOLP problem (1) which preserves theproperty of being a fuzzy-efficient solution to the FMOLP problem (2).The paper is organized as follows. In Section 2, we show that a fuzzy-efficient solution for the FMOLP model (2),

in which one of the goals is fully achieved, may or may not be a Pareto-optimal solution for the MOLP model (1). Wethen propose a method that improves the aforementioned fuzzy-efficient solution generating a Pareto-optimal solutionthat preserves the property of being fuzzy efficient. Finally, in Section 3, we show how our method can be used as anextension to several published procedures and, to further explain our approach, we solve two numerical examples, thesecond example being the same as that suggested by Guu and Wu [4] in illustrating their method.

2. A model to obtain Pareto-optimal solutions in case some goals are fully achieved

Within the scope of themulti-criteria decisionmaking (MCDM) theory, the Pareto-optimality is a necessary conditionin order to guarantee the rationality of a decision. Therefore a “reasonable” solution to a MOLP problem should bePareto-optimal and, as FMOLP is an approach which is used in order to be able to solve MOLP problems, any optimalsolution of a FMOLP problem should be also Pareto-optimal.

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Definition 1. A decision plan xo ∈ X is said to be a Pareto-optimal solution to the MOLP problem (1) if and only ifthere does not exist another y ∈ X such that zi (y)� zi (xo) for all i and z j (y) < z j (xo) for at least one j.

It should be stated that, as Pareto order is partial, it makes more sense to speak of Pareto-maximal as opposedto Pareto-optimal. However, in the reference significant literature Pareto-optimal is the term most extensively used.Therefore, for the sake of simplicity and uniformity, we use it here.

Definition 2 ([11,12]). A decision plan xo ∈ X is said to be a fuzzy-efficient solution to the FMOLP problem (2) ifand only if there does not exist another y ∈ X such that �i (zi (y))��i (zi (x

o)) for all i and � j (z j (y)) > � j (z j (xo)) for

at least one j.

It is obvious that any fuzzy-efficient solution x0 to the FMLP problem (2) such that zi (xo) ∈ (gi , gi + ti ) for all i, isa Pareto-optimal solution to the MOLP problem (1). However, in the case of the saturation (membership degree = 1)of a goal, fuzzy-efficiency does not guarantee Pareto-optimality, as we emphasize in the follow observation:

Observation 1. Let xo be a fuzzy efficient solution to the FMOLP problem (2) such that � j (z j (xo)) = 1 for some j,

that is to say z j (xo)�g j , then it could be the case that xo is not a Pareto-optimal solution. This is due to the fact thaton the left of g j the membership function � j is constantly equal to 1. Suppose e.g. that �1(z1(x

o)) = 1, then it couldbe some y ∈ X such that �i (zi (y)) = �i (zi (x

o)) ∀i , where z1(y) < z1(xo).

2.1. Obtaining a fuzzy-efficient solution

Several authors have proposed procedures in order to achieve fuzzy-efficient solutions. We refer only the Guu andWu’s and the Dubois and Fortemps’ approaches.Guu and Wu [3,4] proposed a two-phase method. Phase 1 uses max–min operator, that is to say, it solves model (3).

Let x∗ be the optimal solution obtained in phase I. If x∗ is a unique optimal solution, then it is fuzzy-efficient and theprocedure stops. If x∗ is not a unique optimal solution, they go on to the second phase where they solve the followingmodel

Maxk∑

i=1

�i

S.t. 1��i (zi (x))��i ��i (zi (x∗)), i = 1, . . . , k,

x ∈ X. (4)

Guu and Wu [4] showed that any optimal solution x∗∗ to problem (4) is a fuzzy-efficient solution to the FMOLPproblem (2).Dubois and Fortemps [2], with the intention of comparing two feasible solutions, introduce the discrimin-partial

order.With the discrimin-partial order they aim to identify the lowest amongst the satisfaction degrees of the constraintsnot equally satisfied by the two compared solutions. Based on the discrimin-partial order, they propose an algorithmconsisting on defuzzifying the constraints that are equally satisfied by all feasible solutions, and solving the max–minoptimization problem involving the defuzzified constraints and the other fuzzy constraints. This procedure is appliedsequentially until all constraints are defuzzified. The obtained solution is called a discrimin-optimal solution and theyshow that it is fuzzy-efficient.

2.2. The model

When we have several objective functions it can be a difficult task for the decision maker to determine coherentaspiration levels, so it is possible that some goals are easily achieved, that is to say � j (z j (x

∗∗)) = 1 for some j , wherex∗∗ could be either an optimal solution of model (4) or a discrimin-optimal solution of Dubois and Fortemps. In thesecircumstances, as we have seen in Observation 1, fuzzy-efficiency does not guarantee the Pareto-optimality. Observethat at this point we are in a zone of indifference (satisfaction degree equal to 1) in which the fuzzy constraints areuseless. Then we can revert back to the original, and conventional, MOLP (1). In accordance with which, we now

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consider the problem like a conventional goal programming (GP) problem. The most common method, found in theliterature of conventional GP, to avoid the problem of obtaining non-Pareto-optimal solutions, proposes maximisingthe sum of the overachievement of the goals ([7], [8, p. 16]). In line with this approach, we propose looking for aPareto-optimal solution by solving the following model

Maxp∑

s=1

ns

S.t. zs(x) + ns = zs(x∗∗), s = 1, . . . , p

�r (zr (x)) = �r (zr (x∗∗)), r = p + 1, . . . , k,

x ∈ X, ns �0, (5)

where subscripts s refer to objective functions such that �s(zs(x∗∗)) = 1 and subscripts r refer to objective functions

such that �r (zr (x∗∗)) < 1. The variables ns have the same significance as the negative deviation in conventional GP.

We intend to demonstrate that the optimal solutions yielded by model (5) are Pareto-optimal.

Lemma 1. Let xo be an optimal solution of problem (5), then xo is a fuzzy efficient solution to the FMOLP problem(2) and a Pareto optimal solution to the MOLP problem (1).

Proof. The fuzzy efficiency of xo is evident from the constraints of model (5) and the fuzzy efficiency of x∗∗. Suppose,on the contrary, that xo is not a Pareto optimal solution. Then there exists a feasible solution to the MOLP problemy such that zi (y)� zi (xo), ∀i = 1, . . . , k, and z j (y) < z j (xo) for some j. Then, as �i (·) are decreasing functions,�i (zi (y))��i (zi (x

o)) ∀i ∈ {1, . . . , k}, which implies that y is feasible for the FMOLP model. Besides it is obviousthat ns(y)�ns(xo) ∀s. Suppose now that j belongs to subscripts r , then, as � j (·) is strictly monotone decreasingover the interval (g j , g j + t j ), we have � j (z j (y)) > � j (z j (x

o)), which contradict the fact that xo is fuzzy-efficient.Therefore j only can belong to subscripts s and consequently n j (y) > n j (xo), thus the following inequality holds∑p

s=1 ns(y) >∑p

s=1 ns(xo). This implies that xo is not an optimal solution to problem (5), a contradiction. �

3. A general framework for solving multi-objective programming problems with fuzzy goals

The above ideas can be integrated into a general procedure or algorithm to obtain a Pareto-optimal solution to theMOLP problem which has the additional property of being fuzzy-efficient.The steps of the proposed algorithm can be synthesized as follows (we notice that the two first steps coincide with the

two-phase approach of Guu andWu [4]. However, as we have said before, any other procedure to obtain a fuzzy-efficientsolution could be used here).Step 1: Solve the max–min model (3):

(1) In the case that the optimal solution is unique:

(a) If all the satisfaction degrees are strictly less than 1 the solution is fuzzy-efficient and Pareto-optimal too, soit can be chosen. The algorithm is finished.

(b) If some satisfaction degrees are equal to 1, that is to say if one or more targets is or are fully achieved, thesolution is fuzzy-efficient but it may not be Pareto-optimal. Then go to step 3.

(2) In the case that there exist multiple optimal solutions: go to step 2.

Step 2: Solve the second phase model, that is to say maximize the sum of satisfaction degrees without making theachievement degrees obtained in the previous step worse, see model (4):

(1) If all the satisfaction degrees are strictly less than 1 the solution is fuzzy-efficient and Pareto-optimal too, so it canbe chosen. The algorithm is finished.

(2) If some satisfaction degrees are equal to 1, that is to say if one or more targets is or are fully achieved, the solutionis fuzzy-efficient but it may not be Pareto-optimal. Then go to step 3.

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Step 3: Maximize the sum of negative deviations, for targets that are fully achieved, without making the valuesobtained in the previous step worse, see model (5). The solution is fuzzy-efficient and Pareto-optimal. The algorithmis finished.

3.1. Numerical examples

To check the usefulness of our approach we consider two numerical examples. The second one is the same assuggested by Guu and Wu [4].

3.1.1. Example 1Let the following be a MOLP problem:

Min z1 = 3x1 + 3x2 + 3x3

Min z2 = 2x1 + x2 + 2x3

Min z3 = 4x1 + 4x2 + 2x3

S.t. 4x1 + 2x2 + 4x3�18

x1�1,

x2�0,

0�x3�3. (6)

Assume the DM proposes the following imprecise aspiration levels: “z1 should be essentially less than or equalto 21”, “z2 should be essentially less than or equal to 8” and “z3 should be essentially less than or equal to 13”,the respective tolerance threshold being 24, 10 and 15. For the sake of simplicity we work with linear membershipfunctions. Although we present the approach for solving problems with linear membership functions, our method canbe easily extended to a more general shape of membership functions (Jiménez et al. [6] showed how any nonlinearmembership function can be approximated by a piecewise linear function and Inuiguchi et al. [5] developed a generalprocedure to obtain a max–min solution of a FMOLP problem with piecewise linear membership functions).Step 1: We solve the max–min problem (see model (3))

Max �

S.t. �� 13 (24 − (3x1 + 3x2 + 3x3))

�� 12 (10 − (2x1 + x2 + 2x3)),

�� 12 (15 − (4x1 + 4x2 + 2x3)),

4x1 + 2x2 + 4x3�18,

x1�1, x2�0, 0�x3�3,

0���1.

We obtain the following optimal solution x∗1 = 1, x∗

2 = 1, x∗3 = 3. The optimal value is �∗ = 0.5 and z1(x∗) = 15,

z2(x∗) = 9, z3(x∗) = 14.We note that �1(x

∗) = 1, �2(x∗) = �3(x

∗) = �∗ = 0.5.Step 2: We look for a fuzzy-efficient solution by solving the following problem (see model (4))

Max �1 + �2 + �3

S.t. �1� 13 (24 − (3x1 + 3x2 + 3x3))

�2� 12 (10 − (2x1 + x2 + 2x3)),

M. Jiménez, A. Bilbao / Fuzzy Sets and Systems 160 (2009) 2714–2721 2719

�3� 12 (15 − (4x1 + 4x2 + 2x3)),

4x1 + 2x2 + 4x3�18,

x1�1, x2�0, 0�x3�3,

�1 = 1, 0.5��2�1, 0.5��3�1.

We get x∗∗1 = 1.25, x∗∗

2 = 0.5, x∗∗3 = 3. Where �∗∗

1 = �∗∗3 = 1, �∗∗

2 = 0.5 and z1(x∗∗) = 14.25, z2(x∗∗) = 9,z3(x∗∗) = 13. We observe that the decision x∗∗ improves the solution x∗ obtained in the previous step becausez1(x∗∗) < z1(x∗), z2(x∗∗) = z2(x∗), z3(x∗∗) < z3(x∗). But there is at least one target that is fully achieved (�1(x∗∗) =�3(x

∗∗) = 1) then, according to Observation 1, the solution x∗∗ may not be Pareto-optimal. So we proceed to theStep 3.If, in order to obtain a fuzzy-efficient solution we apply (for step 2) the Dubois and Fortemps method (instead of Guu

and Wu’s) then we proceed as follows: From Step 1 we conclude that the consistency degree (see [2]) of our problemis equal to 0.5 and that {2} (i.e. the associate to the second goal) is the critical subset of constraints (see [2] p. 109 andp. 120). We defuzzify the constraint {2}, transforming it into the classical constraint 2x1 + x2 + 2x3 = 0.5, and solvethe resultant max–min optimization problem. The consistency degree of the new problem is equal to 1 and we find thediscrimin-optimal solution x∗∗ coincides, in this case, with that obtained by employing Guu and Wu’s approach.

Step 3: We look for a Pareto-optimal solution by solving the following problem (see model (5))

Max n1 + n3

S.t. 3x1 + 3x2 + 3x3 + n1 = 14.25

4x1 + 4x2 + 2x3 + n3 = 13,

12 (10 − (2x1 + x2 + 2x3)) = 0.5,

4x1 + 2x2 + 4x3�18,

x1�1, x2�0, 0�x3�3,

nx�0, n3�0.

We get xo1 = 1.5, xo2 = 0, xo3 = 3 and z1(xo) = 13.5, z2(xo) = 9, z3(xo) = 12. The decision plan xo improves x∗∗,because z1(xo) < z1(x∗∗), z2(xo) = z2(x∗∗), z3(xo) < z3(x∗∗). Besides, according to Lemma 1, xo = (1.5, 0, 3) is aPareto-optimal, and fuzzy-efficient, solution to the proposed problem (6) and so it can be selected.

3.1.2. Example 2Now we shall consider the example used by Guu and Wu [4] to illustrate their two-phase approach. They propose

the following fuzzy linear programming problem

Max z(x) = 4x1 + 5x2 + 9x3 + 11x4

S.t. g1(x) = x1 + x2 + x3 + 1x4�15

g2(x) = 7x1 + 5x2 + 3x3 + 2x4�80,

g3(x) = 3x1 + 4.4x2 + 10x3 + 15x4�100,

x1, x2,x3,x4�0.

The membership functions, �i (x) i = 1, 2, 3, for the three fuzzy constraints are linear and the tolerance thresholdsare 20 for g1(x), 120 for g2(x) and 130 for g3(x). The aspiration level for the objective function is 130 and its tolerancethreshold is 99.28571, the membership function, �0(x), of the fuzzy goal is linear too.

In order to solve this problem Guu and Wu apply their two phase method, which coincide with the two firststeps of our procedure. That is to say, with Step 1 they solve the corresponding max–min model, obtaining the

2720 M. Jiménez, A. Bilbao / Fuzzy Sets and Systems 160 (2009) 2714–2721

following solution

x∗ = (8.571428, 0, 8.928572, 0); �0(x∗) = �1(x

∗) = �3(x∗) = �∗ = 0.5, �2(x

∗) = 0.8303572;z(x∗) = 114.64286, g1(x

∗) = 17.5, g2(x∗) = 86.784, g3(x

∗) = 115.

With Step 2 they look for a fuzzy efficient solution by solving the corresponding model (4). Guu and Wu obtainthe following optimal solution: x∗∗ = (4.048, 5.655, 7.798, 0); �0(x

∗∗) = �1(x∗∗) = �3(x

∗∗) = 0.5, �2(x∗∗) = 1;

z(x∗) = 114.64286, g1(x∗∗) = 17.5, g2(x∗∗) = 80, g3(x∗∗) = 115.If we use the method of Dubois and Fortemps, instead of Guu and Wu’s, we proceed as follows: The consistency

degree is equal to 0.5 and {0, 1, 3} is the critical subset of constraints. We defuzzify the constraints {0, 1, 3} and solvethe resultant max–min optimization problem. The consistency degree of the new problem is equal to 1. We get the samesolution as that obtained by employing Guu and Wu’s approach.Obviously x∗∗ is better than x∗ because it achieves the same objective value using less units of the second resource.

Guu and Wu propose x∗∗ as best decision.But as �2(x

∗∗) = 1, according to our Observation 1, the decision x∗∗ may not be Pareto-optimal. Then, in order tolook for a Pareto optimal solution, we solve the following problem (see model (5))

Max n2

S.t. 7x1 + 5x2 + 3x3 + 2x4 + n2 = 80

130.71429 (4x1 + 5x2 + 9x3 + 11x4 − 99.28571) = 0.5,

15 (20 − x1 + x2 + x3 + x4) = 0.5,

130 (130 − 3x1 + 4.4x2 + 10x3 + 15x4) = 0.5,

x1, x2,x3,x4�0, n2�0.

We obtain xo = (0, 10.714287, 6.785714, 0); z(xo) = 114.64286, g1(xo) = 17.5, g2(xo) = 73.928574, g3(xo) =115. This solution xo improves the decision x∗∗ proposed by Guu and Wu, because it requires less units of the secondresource with the same objective value.

4. Conclusions

We have shown that, in a multi-objective programming problem with fuzzy goals, a fuzzy-efficient solution in whichone of the goals is fully achieved may not be Pareto-optimal. From this we conclude that, under these circumstances,the solution yielded by the several methods published in the literature on the subject thus far can be improved upon.Accordingly we have proposed a general procedure to obtain a Pareto-optimal solution that has the additional propertyof being fuzzy-efficient.

Acknowledgment

This work was supported in part by the Universidad del País Vasco under grant no. UPV05/45, and by the SpanishMinisterio de Educación y Ciencia under Grant no. MTM2007-67634.The authors are very grateful to the anonymous referees for their comments and suggestions.

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