Multi-objective Optimization and its Engineering Applications

Many practical optimization problems usually have several conflicting objectives. In those multi-objective optimization, no solution optimizing all objective functions simultaneously exists in general. Instead, Pareto optimal solutions, which are ``efficient" in terms of all objective functions, are introduced. In general we have many Pareto optimal solutions. Therefore, we need to decide a final solution among Pareto optimal solutions taking into account the balance among objective functions, which is called ``trade-off analysis". It is no exaggeration to say that the most important task in multi-objective optimization is trade-off analysis. Consequently, the methodology should be discussed in view of how it is easy and understandable for trade-off analysis. In cases with two or three objective functions, the set of Pareto optimal solutions in the objective function space (i.e., Pareto frontier) can be depicted relatively easily. Seeing Pareto frontiers, we can grasp the trade-off relation among objectives totally. Therefore, it would be the best way to depict Pareto frontiers in cases with two or three objectives. (It might be difficult to read the trade-off relation among objectives with three dimension, though). In cases with more than three objectives, however, it is impossible to depict Pareto forntier. Under this circumstance, interactive methods can help us to make local trade-off analysis showing a ``certain" Pareto optimal solution. A number of methods differing in which Pareto optimal solution is to be shown, have been developed. This paper discusses critical issues among those methods for multi-objective optimization, in particular applied to engineering design problems.