Fuzzy programming with nonlinear membership functions: Piecewise linear approximation

Abstract

One promising approach that has been proposed for dealing with multicriteria (i.e., multiple objective) programming models has been fuzzy linear programming (FLP). In essence, the FLP approach involves the replacement of the multiple objectives with goals by means of assigning an aspiration level to each objective. Fuzzy membership functions are then introduced to represent the measure of the achievement of the actual solution for each goal — relative to its aspired level. It has been shown that, if the FLP problem is linear, and if linear membership functions are employed to reflect goal achievement, then the FLP problem may be transformed into a conventional (i.e., single objective) linear programming model. As such, one may use any conventional linear programming algorithm to solve the transformed model. However, in the more general case of nonlinear membership functions, the transformation process becomes considerably more involved — and has, in the past, typically led to less desirable formulations. In this paper, we present what we believe to be a straightforward and computationally efficient procedure for dealing with the FLP problem with any general class of nonlinear membership functions. Conversion of such a FLP will result in either a regular linear programming model or a linear integer programming model, depending upon the specific characteristics of the membership function. In either case, existing commercial software is readily available to solve such models.