Particle Swarm Optimization with External Archives for Interactive Fuzzy Multiobjective Nonlinear Programming

In general nonlinear programming problems to find a solution which minimizes an objective function under given constraints, one whose objective function and constraint region are convex is called a convex programming problem. For such convex programming problems, there have been proposed many efficient solution method as the successive quadratic programming method and the general gradient method. Unfortunately, there have not been proposed any decisive solution method for nonconvex programming problems. As practical solution methods, meta-heuristic optimization methods as the simulated annealing method and the genetic algorithm have been proposed. In recent years, however, more speedy and more accurate optimization methods have been desired because the size of actual problems has been increasing. As a new optimization method, particle swarm optimization (PSO) was proposed (Kennedy & Eberhart, 1995). PSO is a search method simulating the social behavior that each individual in the population acts by using both the knowledge owned by it and that owned by the population, and they search better points by constituting the population. The authors proposed a revised PSO (rPSO) by incorporating the homomorphous mapping and the multiple stretching technique in order to deal with shortcomings of the original PSO as the concentration to local solution and the inapplicability of constrained problems (Matsui et al., 2008). In recent years, with the diversification of social requirements, the demand for the programs with multiple objective functions, which may be conflicting with each other, rather than a single-objective function, has been increasing (e.g. maximizing the total profit and minimizing the amount of pollution in a production planning). Since there does not always exist a complete optimal solution which optimizes all objectives simultaneously for multiobjective programming problems, the Pareto optimal solution or non-inferior solution, is defined, where a solution is Pareto optimal if any improvement of one objective function can be achieved only at the expense of at least one of the other objective functions. For such multiobjective optimization problems, fuzzy programming approaches (e.g. (Zimmermann, 1983), (Rommelfanger, 1996), considering the imprecise nature of the DM's judgments in multiobjective optimization problems, seem to be very applicable and promising. In the application of the fuzzy set theory into multiobjective linear programming problems started