Linear programming with fuzzy parameters: An interactive method resolution

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Abstract

This paper proposes a method for solving linear programming problems where all the coefficients are, in general, fuzzy numbers. We use a fuzzy ranking method to rank the fuzzy objective values and to deal with the inequality relation on constraints. It allows us to work with the concept of feasibility degree. The bigger the feasibility degree is, the worst the objective value will be. We offer the decision-maker (DM) the optimal solution for several different degrees of feasibility. With this information the DM is able to establish a fuzzy goal. We build a fuzzy subset in the decision space whose membership function represents the balance between feasibility degree of constraints and satisfaction degree of the goal. A reasonable solution is the one that has the biggest membership degree to this fuzzy subset. Finally, to illustrate our method, we solve a numerical example.

Introduction

Linear programming (LP) is the optimisation technique most frequently applied in real-world problems and therefore it is very important to introduce new tools in the approach that allow the model to fit into the real world as much as possible.

Any linear programming model representing real-world situations involves a lot of parameters whose values are assigned by experts, and in the conventional approach, they are required to fix an exact value to the aforementioned parameters. However, both experts and the DM frequently do not precisely know the value of those parameters. If exact values are suggested these are only statistical inference from past data and their stability is doubtful, so the parameters of the problem are usually defined by the DM in a uncertain way or by means of language statement parameters. Therefore, it is useful to consider the knowledge of experts about the parameters as fuzzy data.

This paper considers LP problems whose parameters are fuzzy numbers but whose decision variables are crisp. The aim of this paper is to introduce a resolution method for this type of problems that permits the interactive participation of DM in all steps of decision process, expressing his/her opinions in linguistic terms.

Two key questions may be found in these kinds of problems: how to handle the relationship between the fuzzy left and the fuzzy right hand side of the constraints, and how to find the optimal value for the fuzzy objective function. The answer is related to the problem of ranking fuzzy numbers.

A variety of methods for comparing or ranking fuzzy numbers have been reported in the literature (Wang and Kerre, 1996) and ranking methods do not always agree with each other. Different properties have been applied to justify ranking methods, such as: distinguishability (Bortolan and Degani, 1985), rationality (Nakamura, 1986), fuzzy or linguistic presentation (Delgado et al., 1988, Tong and Bonissone, 1980) and robustness (Yuan, 1991). In this paper we use a method (Jiménez, 1996) that verifies all the above properties and that, besides, is computationally efficient to solve an LP problem, because it preserves its linearity.

Looking at the property of representing the preference relationship in linguistic or fuzzy terms, ranking methods can be classified into two approaches. One of them associates, by means of different functions, each fuzzy number to a single point of the real line and then a total crisp order relationship between fuzzy numbers is established. The other approach ranks fuzzy numbers by means of a fuzzy relationship. It allows DM to present his/her preferences in a gradual way, which in an LP problem allows it to be handled with different degrees of satisfaction of constraints and, with regard to objective value, it allows us to look for a non-dominated satisfying solution. In Section 3 we show how we use our method to rank fuzzy numbers in order to define the feasibility degree of the decision vector and to define the acceptable optimal solution concept.

Obviously if the DM establishes a high degree of satisfaction of constraints for a solution, the feasible solution set becomes smaller and, consequently, the objective optimal value is worse. So, the DM has to find a balanced solution between two objectives in conflict: to improve the objective function value and to improve the degree of satisfaction of constraints. In Section 4 we show how we can operate in an interactive way in order to evaluate the two aforementioned conflicting factors. Finally in Section 5 we solve a numerical example.

Section snippets

Notation and basic definitions

A fuzzy set A of a universe Ω is characterized by its membership function μA˜:Ω[0,1]. Where r=μA˜(x); x  Ω, is the membership degree of x to A (Zadeh, 1965).

When A is an uncertain value parameter, the membership degree μA˜(x) can be viewed as the plausibility degree of A taking value x. Zadeh (1978) defines a possibility distribution associated with A as numerically equal to μA˜.

A fuzzy number is a fuzzy set a˜ on the real line R whose membership function μa˜ is upper semi-continuous (we

Presentation of the problem

Let us consider the following linear programming problem with fuzzy parameters:minimizez=c˜txsubjecttox(A,b˜)={xRn|a˜ixb˜i,i=1,,m,x0},where c˜=(c˜1,c˜2,,c˜n),A˜=[a˜ij]m×n,b˜=(b˜1,b˜2,,b˜m)t represent, respectively, fuzzy parameters involved in the objective function and constraints. The possibility distribution of fuzzy parameters is assumed to be characterized by fuzzy numbers. x = (x1, x2, …, xn) is the crisp decision vector.

The uncertain and/or imprecise nature of the parameters of the

Interactive resolution method

From (14) the obtaining of a better value to the optimal objective function implies a lesser degree of feasibility of the constraints. Then the DM runs into two conflicting objectives: to improve the objective function value and to improve the degree of satisfaction of constraints. In Jiménez et al. (2000) we proposed to solve this problem through compromise programming, now we show how it can be solved in an interactive way as well.

The best way to reflect DM preferences is to express them

Numerical example

To illustrate our method, we will solve the following linear programming problem with fuzzy parameters, which is the same as that proposed in Jiménez et al. (2000):min(19,20,21)x1+(29,30,31)x2s.t.(4.5,5,5.5)x1+(2.5,3,4)x2(194,200,206),(3,4,5)x1+(6.5,7,7.5)x2(230,240,250),x10,x20.For simplicity we have supposed that all imprecise parameters are triangular fuzzy numbers, but any other fuzzy number could be used.

Bearing in mind the expression (13) and the Definition 4, we will calculate the α-

Conclusions

In this paper we have proposed a resolution method, for a linear programming problem with fuzzy parameters, which allows us to take a decision interactively with the DM. Through the idea of feasible optimal solution in degree α, the DM has enough information to fix an aspiration level. The DM can also choose the degrees of feasibility that he/she is willing to admit depending on the context. It is important to highlight that the acceptable optimal solutions in degree α are not fuzzy quantities,

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    This work was supported by the Spanish Ministry of Education and Science (project MEC-04-MTM-07478) and by the University of the Basque Country. This support is gratefully acknowledged.

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