Fitness evaluation using generalized data envelopment analysis in MOGA

Abstract

Most practical problems are formulated as multiobjective optimization problems (MOP) so as to meet the diversified demands of a decision maker. Usually, there is a trade-off relation among objective functions, and thus there does not necessarily exist the solution that optimizes all objective functions simultaneously in MOP. Therefore, Pareto optimal solution is used as a definition of solution to MOP. Recently, evolutionary algorithms have been developed remarkably in order to obtain approximate solutions to optimization problems. Particularly, multiobjective genetic algorithms (MOGA) have been developed for generating Pareto optimal solutions. However, there are two problems in MOGA: how to assign the fitness to individuals, and how to keep the diversification of individuals. Many existing MOGAs have made an effort in order to overcome these problems, and so does this paper. First, this paper suggests a fitness function in MOGA using generalized data envelopment analysis (GDEA) which was suggested for evaluating the relative efficiency of individuals under several items of assessment management science. It is shown that the GDEA method can approximate Pareto optimal solutions more effectively and faster than the ranking method which is mostly used in MOGA, and generate well-distributed Pareto optimal solutions. Furthermore, this paper suggests the aspiration-level based GDEA method to generate the most interesting part (not the whole Pareto optimal solutions) to an aspiration level of decision maker for choosing a final solution from many Pareto optimal solutions. Finally, this paper illustrates the effectiveness of the methods using GDEA through several numerical examples.