Multiobjective Fuzzy Satisficing Methods for 0–1 Knapsack Problems through Genetic Algorithms

Abstract

In this paper, multiobjective fuzzy satisficing methods for multidimensional 0–1 knapsack problems are presented by incorporating the desirable features of both fuzzy programming methods and genetic algorithms. Considering the vague or fuzzy nature of human judgements, fuzzy goals of the decision maker (DM) for objective functions are quantified by eliciting the corresponding linear membership functions. By adopting the fuzzy decision, a compromise solution for the DM can be derived efficiently through a genetic algorithm with double strings which generates only feasible solutions without using penalty functions for treating the constraints. There remains, however, such a problem that no interaction with the DM is considered once the membership functions have been determined. Realizing such drawbacks, an interactive fuzzy satisficing method for multiobjective multidimensional 0–1 knapsack problems is proposed by incorporating the desirable features of genetic algorithms with double strings and interactive fuzzy satisficing methods both proposed by the authors. The basic idea behind an interactive fuzzy satisficing method is to derive a satisficing solution for the DM from a set of Pareto optimal solutions efficiently by interactively updating reference membership functions. For obtaining an optimal solution not dominated by the solutions before interaction, the genetic algorithm is revised by introducing some new mechanisms for forming an initial population. Illustrative numerical examples demonstrate both feasibility and effectiveness of the proposed methods.