Amelia Bilbao

Linear programming with fuzzy parameters: An interactive method resolution

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.

Pareto-optimal solutions in fuzzy multi-objective linear programming

The problem of solving multi-objective linear-programming problems, by assuming that the decision maker has fuzzy goals for 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 goals is fully achieved, a fuzzy-efficient solution may not be Pareto-optimal and therefore we propose a general procedure to obtain a non-dominated solution, which is also fuzzy-efficient. Two numerical examples illustrate our procedure.

On constructing expert Betas for single-index model ☆

Abstract The aim of this paper is to design flexible decision making models for portfolio selection including expert’s knowledge and imprecise preferences provided by financial analysts and investors, respectively. Sharpe’s single-index model involves the estimation of Beta for each potential asset; these estimations are obtained based on past data and using statistical methods in order to obtain future Betas. The main contribution of our paper is the methodological proposal of an extension of Sharpe’s single-index model, called “Sharpe’s model with expert Betas”. This extension has been carried out through the construction of Betas obtained from both, statistical and imprecise expert estimations taking, also, into account several views of the market. Defined Betas, called “Expert Betas”, must have suitable features with respect to the quality and handling of information such that they can be incorporated in useful mathematical programming models. In this sense, the inclusion of these Expert Betas in a Goal Programming (GP) decision making model for portfolio selection is proposed.

Approximate resolution of an imprecise goal programming model with nonlinear membership functions

In standard goal programming (GP) it is assumed that the decision maker (DM) is able to determine goal values accurately. This is unrealistic; usually DM expresses his/her aspirations in an imprecise way. The imprecise nature of the DMs judgments has led to an important development of the fuzzy multiobjective approaches. In this paper, we will assume that the DMs goals may be expressed through fuzzy sets and therefore we deal with an imprecise goal programming (IGP). Determining the membership functions that represent the fuzzy goals of the DM use to be a difficult task and it is necessary to fit them in the sense of being robust; hence, some results on sensibility analysis are established. We show that small changes in membership functions produce small differences on the DMs global satisfaction degree and on the efficient frontier. On the other hand, when membership functions are nonlinear, IGP becomes a nonlinear program that may be difficult to solve. This difficulty may be overcome by approximating each nonlinear fuzzy membership function through functions belonging to a class of simpler/easier functions. In this paper, based on the sensibility analysis that we develop, we prove the goodness of subrogated functions. An illustrative example is also provided.

Fuzzy Extended Lexicographic Goal Programming

Goal Programming (GP) is perhaps the most widely used approach in the field of multicriteria decision making The major advantage of the GP model is its great flexibility which enables the decision maker to easily incorporate numerous variations of constraints and goals. Romero provides an unifying basis for GP and multiple objective programming approaches, Extended Lexicographic Goal Programming (ELGP) which is a rather general GP structure encompassing Archimedean and MINMAX (Tchebychev) GP variants as particular cases. In this work we propose the use of this general primary estructure (ELGP) for the resolution of fuzzy multiobjective programming problems.

Solving a Fuzzy Multiobjective Linear Programming Problem Through the Value and the Ambiguity of Fuzzy Numbers

In this paper we solve multiobjective programming problems with fuzzy parameters and flexible constraints. To work with fuzzy numbers we use two real indices, the value and the ambiguity. In order to rank two fuzzy numbers a lexicographic ranking procedure can be used: in the first step the values of fuzzy numbers are compared, if these values are ‘approximately equal’ then we compare their ambiguities. In this paper we apply this ranking procedure to a fuzzy programming problem with fuzzy coefficients. In the first step we solve a model in which the fuzzy coefficients have been defuzzified by its corresponding value. Now the question is to determine when two solutions of the first step are approximately equal. In order to answer this question we propose to reflect the decision makers (DMs) preferences through the natural language, establishing a semantic correspondence for the different satisfaction degrees. We consider as approximately equal all the solutions whose global satisfaction degrees belong to the same linguistic label. Then, in the second step, among all the solutions that belong to the same linguistic label as the solution obtained in the first step, we find those with minimum (or maximum) ambiguity, depending on DMs attitude when faced with the risk. We use one example to illustrate this procedure.

A Theory of Possibility Approach to The Solution of a Fuzzy Linear Programming

In conventional decision making problems, the estimation of the parameters of the model is often a problematic task. Normally they are either given by the decision maker (DM) who has imprecise information and/or expresses his/her considerations subjectively, or by statistical inference from past data and, consequently, their stability is doubtful. Therefore, it is reasonable to construct a model reflecting imprecise data or ambiguity in terms of fuzzy sets. This is the reason why a lot of fuzzy approaches to linear programming have been developed.