Metaheuristics, Generalized DEA and Aspiration-based Method for Multi-objective Optimization

Many decision making problems can be formulated as multi-objective optimization problems (MOP). There hardly exists the solution that optimizes all objective functions in MOP, and then the concept of Pareto optimal solution (or efficient solution) is introduced. Usually, there exist a lot of Pareto optimal solutions, which are considered as candidates of final decision making solution. It is an issue how decision makers decide the final solution from the set of Pareto optimal solutions. Interactive multi-objective optimization methods have been developed to this end [9]. These methods search a decision making solution by processing the following two stages repeatedly: 1) solving auxiliary optimization problem to obtain a Pareto optimal solution closest to the given aspiration level, and 2) revising their aspiration levels by making trade-off analysis. Problems such as engineering design problems, however, criteria functions cannot be given explicitly in terms of design variables. Under this circumstance, values of criteria functions are usually obtained by some analyses such as structural analysis, thermodynamic analysis, fluid mechanical analysis, and so on. These analyses require considerably expensive computational time, that is, it takes too much time to obtain Pareto optimal solutions. Therefore, this makes the interaction between decision makers and computer almost impossible practically. In order to overcome this difficulty, genetic algorithms (GAs) for multiobjective optimization problems have been researched by many authors [1, 2, 4, 5, 7, 8]. GAs are useful for generating the set of Pareto optimal solutions to MOP. Ranking methods [4, 5], which can be applied to several optimization problems without any restriction, provide the index of the individual how many individuals it is dominated by. Due to it, ranking methods need a large generation number to provide good approximate efficient frontiers. In order to settle the problem of ranking methods, the DEA method [1] using data envelopment analysis (DEA [3]) was suggested. The DEA method provides the index how far (or close) an individual is from the efficient frontier. Additionally, it is seen through many illustrations that the DEA method makes Pareto optimal solutions scattered. However, the DEA method has the shortcoming not being able to generate the non-convex part of efficient frontiers. In the cir-