The Best Compromise Solution for Multi-objective Programming Problems

Abstract

This paper focuses on multi-objective optimization problems that are an important part of operations research. This part is concerned with mathematical optimization problems involving two or more interdependent objectives to be optimized simultaneously. Thus, there is not a single optimal solution for multi-objective problems, but a set of solutions that represents the compromise (Pareto-optimal, efficient, non-dominated, trade-off, or non-inferior) solutions and can be visualized as Pareto front in the objective space. The best solution of this set has the shortest distance to the ideal (utopian) solution, whereas the ideal solution optimizes all objective functions, which often cannot be found. The main contribution of this paper is to introduce some methods to find the best compromise solution. These methods depend on new calculations for the normal of objectives. They can help to reduce the overall computational distance of the searching process. Therefore, they are flexible and stable. Besides, some numerical examples are presented to demonstrate the effectiveness of proposed methods with discussing their similarities and differences. The experimental results show that the proposed methods are effective and efficient for many different multi-objective (convex and non-convex) problems.