A Normalized Circle Intersection Method for Bi-Objective Optimization Programming

Abstract

Multi-objective optimization (MOO) problems are encountered in many applications and a number of approaches have been proposed to deal with this kind of problems. Despite the computational efforts, the quality of the Pareto front is also a considerable issue. An evenly distributed Pareto front is desirable for developing analytical expressions. In this paper, a brand new approach called Normalized Circle Intersection (NCI) is proposed, which is able to efficiently generate a Pareto front with evenly-distributed Pareto points for bi-objective problems, no matter the feasible boundary is convex or not. Firstly, the anchor points are computed using an existing sequential MOO (SMOO) approach. Then in the normalized objective space, a circle with a radius of r centering at one of the anchor points or the latest obtained Pareto point is drawn. The intersection of the circle and the feasible boundary, which exists for sure, can be determined whether it is a Pareto point or not. For a convex or concave feasible boundary, the intersection is exactly the Pareto point to be found, while for a non-convex boundary the intersection can provide useful information for searching the true Pareto point even if it self is not a Pareto point. A novel MOO formulation is proposed for NCI correspondingly. Four examples, including two numerical and two engineering examples, are provided to demonstrate the applicability of the proposed method. Comparison of the computational results with WS, NNC and SMOO shows the effectiveness of the proposed method.