Mariano Jiménez

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.

A fuzzy linear programming based approach for tactical supply chain planning in an uncertainty environment

This paper models supply chain (SC) uncertainties by fuzzy sets and develops a fuzzy linear programming model for tactical supply chain planning in a multi-echelon, multi-product, multi-level, multi-period supply chain network. In this approach, the demand, process and supply uncertainties are jointly considered. The aim is to centralize multi-node decisions simultaneously to achieve the best use of the available resources along the time horizon so that customer demands are met at a minimum cost. This proposal is tested by using data from a real automobile SC. The fuzzy model provides the decision maker (DM) with alternative decision plans with different degrees of satisfaction.

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.

PROMETHEE: an extension through fuzzy mathematical programming

PROMETHEE multi-criteria methods are based on fuzzy evaluations of the differences between pairs of alternatives for each criterion. PROMETHEE II associates a crisp number to each action. PROMETHEE III associates an interval to each action and two actions are considered indifferent when they are very close to each other. PROMETHEE V applies Integer Linear Programming in order to select the subset of alternatives that maximizes the sum of PROMETHEE II scorings, subject to a set of constraints. In order to make the model more realistic, this paper proposes that some constraints are soft and some coefficients are estimated by fuzzy numbers. Fuzzy Integer Linear Programming is applied, using the PROMETHEE III scorings as objective function coefficients, in order to find the subsets of non-outranked alternatives that best satisfy the soft constraints. The new model is more realistic and fits better the fuzzy philosophy of PROMETHEE. The method is illustrated with an example.

An extension of Sharpe's single-index model: portfolio selection with expert betas

This paper presents an approach to the portfolio selection problem based on Sharpes single-index model and on Fuzzy Sets Theory. In this sense, expert estimations about future Betas of each financial asset have been included in the portfolio selection model denoted as ‘Expert Betas’ and modelled as trapezoidal fuzzy numbers. Value, ambiguity and fuzziness are three basic concepts involved in the model which provide enough information about fuzzy numbers representing ‘Expert Betas’ and that are simple to handle. In order to select an optimal portfolio, a Goal Programming model has been proposed including imprecise investors aspirations concerning assets proportions of both, high-and low-risk assets. Semantics of these goals are based on the fuzzy membership of a goal satisfaction set. To illustrate the proposed model a real portfolio selection problem is presented.

Incorporating additional meta-objectives into the extended lexicographic goal programming framework

This paper introduces two new meta-objectives into the extended goal programming framework. The first meta-objective is the number of unmet goals and the second is a measure of closeness to the pairwise comparisons given by the decision maker. These complement the original two meta-objectives of the weighted sum of deviations and the maximal weighted deviation to provide a flexible four meta-objective framework. Lexicographic and non-lexicographic representations of the framework are developed. An example relating to transportation is solved in both the lexicographic and non-lexicographic forms. Weight sensitivity analysis is applied to the meta-weight parameters for the non-lexicographic case in order to find a range of available distinct solutions.

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 number approximation

As the number of parameters involved in an economic model is often uncertain, we propose that it be estimated using fuzzy numbers. Since we move in an environment of uncertainty, it is logical to leave room for deviation in estimating membership functions. We should recall that when soft max-min operators are used, the resulting deviation is never greater than the variation introduced in estimating the initial data. Often, the result of our calculations is not a triangular fuzzy number. In this paper we study the value of approximating the resulting non-linear fuzzy number using a triangular fuzzy number having the same support and kernel. Finally, we present a simple method for weighing this approximation.

A sequential goal programming model with fuzzy hierarchies to sustainable and responsible portfolio selection problem

Sustainable and responsible (SR) investors have to address two criteria types: both financial ones and those pertaining to sustainability and social responsibility. We present a comfortable tool for SR investors that allow them to express their preferences at two levels: first, by comparing criteria of the same nature, and second, via the comparison between the two superior level criteria (the financial and the SR objectives). Owing to the difficulty involved in determining a precise preference between the conflicting objectives, we address this by goal programming with fuzzy hierarchies (GPFH) modelling. This methodology is a modification of the lexicographic GP approach whereby the relative importance relations among the criteria are modelled by fuzzy relations. The proposed sequential handling for the SR portfolios selection provides information to the investors on the best result they can achieve in regard to their goals. An application to a set of UK-SR mutual funds is presented.

A rolling horizon approach for material requirement planning under fuzzy lead times

This paper proposes a fuzzy multi-objective integer linear programming (FMOILP) approach to model a material requirement planning (MRP) problem with fuzzy lead times. The objective functions minimise the total costs, back-order quantities and idle times of productive resources. Capacity constraints are included by considering overtime resources. Into the crisp MRP multi-objective model, we incorporate the possibility of occurrence of each uncertain lead time using fuzzy numbers. Then FMOILP is transformed into an auxiliary crisp mixed-integer linear programming model by a fuzzy goal programming approach for each fuzzy lead time combination. In order to defuzzify the set of solutions associated with each fuzzy lead time combination, a solution method based on the centre of gravity concept is addressed. Model validation with a numerical example is carried out by a novel rolling horizon procedure where uncertain lead times are updated during each planning period according to the centre of gravity obtained. For illustration purposes, the proposed solution approach is satisfactorily compared to a rolling horizon approach in which lead times are allocated when the possibility of occurrence is established at one.