Solving fuzzy optimization problems by evolutionary algorithms

Abstract

In this paper mathematical programming problems with fuzzy constraints are dealt with. Fuzzy solutions are obtained by means of a parametric approach in conjunction with evolutionary techniques. Some relevant characteristics of the evolutionary algorithm are for instance a real-coded representation of solutions and the preselection scheme as niche formation and elitist technique. Three test problems with fuzzy constraints and different structures are used in order to check and compare the proposed technique. The results obtained are very good in comparison with those from another methods.

Introduction

A constrained optimization problem can be mathematically formulated as

$$\text{Min}\text{f(x)}\text{s}\text{.}\text{t}\text{.}\text{g}{}_{\mathrm{j}}\text{(x)⩽0,}\text{j=1,…,m}\text{x∈X}$$

where f(x), the objective function, and g_j(x), j=1,…,m, the constraints, are defined on $\text{R}{}^{\mathrm{n}}$, X is a subset of $\text{R}{}^{\mathrm{n}}$, usually called solution space or feasible set, and x is a vector of n components x₁,…,x_n.

This problem must be solved for the values x₁,…,x_n belonging to the solution space that satisfy the constraints and minimize the function f. This is to be meant as one needs to find a feasible point $\text{x}{}^{\mathrm{\ast }}$, that is a point $\text{x}{}^{\mathrm{\ast }}\text{∈X}$ satisfying all the constraints, such that $\text{f(x}{}^{\mathrm{\ast }}\text{)⩽f(x)}$ for each feasible point x. Then $\text{x}{}^{\mathrm{\ast }}$ is an optimal solution. Various optimization problems can be categorized based on the characteristics of X, f(x) and g_j(x), j=1,…,m. Thus if f(x) and g_j(x), j=1,…,m are linear functions the model above describes a linear optimization problem. The very best known example of a linear optimization problem is the linear programming problem which, by means of the simplex algorithm, can be easily solved in a finite number of steps. Otherwise the model becomes a nonlinear optimization problem (fractional, quadratic, etc.) which, in contrast to the linear case, in general is difficult to solve, and usually deterministic algorithms are not applicable.

Although the efforts made to solve efficiently nonlinear problems have produced important progress along the last years, unfortunately there is not a universal method, like in the linear case with the simplex algorithm happens, to solve a general nonlinear programming problem. Consequently many relevant scientific and engineering optimization problems, for which real solutions are needed, cannot be adequately solved. This need of a solution leads to solve this kind of nonlinear programming problems by proposing nonoptimal solutions which may be obtained by different methods, typical and generally called heuristics (neural networks, simulated annealing, tabu search,…). Is just at this point where evolutionary algorithms (EA) [1], [3], [6] appear just as another heuristic tool to try to solve the problems in which here we are interested, i.e., general nonlinear programming problems.

Along the last years EA have shown a good performance in solving optimization problems with complex solution spaces. As it is well known EA imitate, on an abstract level, biological principles of natural selection and genetics such as a population based approach, the inheritance of information, the variation of information via crossover/mutation, and the selection of individuals based on fitness. EA are stochastic search algorithms which often can find better solutions than classical optimization techniques, mainly when an optimal solution for a concrete nonlinear programming problem is difficult to obtain.

Besides the actual difficulty of the problems to be here considered, mainly due to its nonlinear nature, in many real optimization problems encountered in engineering and other areas like artificial intelligence, operations research, etc., coefficients and data taking part into the problem may be affected of an inherent vagueness on their exact values, which often is bridged by providing approximations of the true values that permit to approach solution methods. However the problem obtained as a result of these numerical approximations may be rather than far from the former problem. Among the different types of uncertainty that can appear in the parameters defining the problems, we are here interested in the case in which this vagueness has a fuzzy nature, i.e., when this vagueness is due to a lack of precision, and hence it can be modeled by means of fuzzy sets concepts. The corresponding model is usually referred as a fuzzy constrained optimization problem.

Fuzzy constrained optimization problems have been extensively studied since the seventies. In the linear case, the first approaches to solve the so-called fuzzy linear programming problem were made in [9], [11]. Since then, important contributions solving different linear models have been done and these models have been recipients of a great dealt of work. In the nonlinear case the situation is quite different, as there is a wide variety of specific and both practical and theoretically relevant nonlinear problems, each having a different solution method. Consequently in the following we will focus on fuzzy nonlinear programming problems in which, as said, coefficients and/or constraints defining the problem are given as fuzzy ones.

To concrete, consider a nonlinear programming problem with fuzzy constraints. From a mathematical point of view the problem can be addressed as:

$$\text{Min}\text{f(x)}\text{s}\text{.}\text{t}\text{.}\text{g}{}_{\mathrm{j}}\text{(x)≲b}{}_{\mathrm{j}}\text{,}\text{j=1,…,m}\text{x}{}_{\mathrm{i}}\text{∈[l}{}_{\mathrm{i}}\text{,u}{}_{\mathrm{i}}\text{],}\text{i=1,…,n,}\text{l}{}_{\mathrm{i}}\text{⩾0}$$

where $\text{x=(x}{}_1\text{,…,x}{}_{\mathrm{n}}\text{)∈}\text{R}{}^{\mathrm{n}}$ is a n dimensional real-valued parameter vector, $\text{[l}{}_{\mathrm{i}}\text{,u}{}_{\mathrm{i}}\text{]⊂}\text{R}$ (i=1,…,n), $\text{b}{}_{\mathrm{j}}\text{∈}\text{R}$, f(x), g_j(x) are continuous arbitrary functions, and the symbol ≲ indicates that the corresponding constraint is a fuzzy constraint [11].

It is patent that EA could be used to solve fuzzy nonlinear programming problems like the above one because of EA are solution methods potentially able of solving general nonlinear programming problems or, at least, of approaching theoretic solution ways that, each case, are to be specified according to the concrete problem to be solved.

Therefore, from this background, the main aim of this paper is to propose and present an EA-based solution method for fuzzy nonlinear programming which eventually can be easily adapted to solve general nonlinear programming problems. Consequently the paper is organized as follows. Next section describes an evolutionary-parametric solution approach to solve fuzzy programming problems as in (2), Section 3 presents simulation results for several test problems, and finally, Section 4 points out the main conclusions.