A Novel Graph-Theoretical Approach of Selecting Representative Pareto Optimal Solutions for Multi-objective Optimization Problems

Abstract

AbstractThe real-world problems often pose as multi-objective with competing objectives. Unlike single-objective optimization problems, multi-objective problems result in a large set of solutions called Pareto optimal solutions (Pareto set). All the solutions in this set are considered equally good with some trade-offs. Therefore, the decision-makers face the challenge of choosing a solution especially in the absence of subjective or judgmental information. On the other hand, analyzing all the solutions is not practical due to the time complexity. This means that a pruning method is needed to tackle this problem. Several methods have been proposed in the literature. These methods include clustering (e.g., K-means) and ranking (e.g., hierarchy process-based) of Pareto optimal solutions to reduce the number of solutions to a promising set with smaller cardinality. In clustering methods, a representative solution is extracted from each cluster to form the reduced set (e.g., the solution at the cluster center or one closest to the ideal solution of the cluster). However, the point closest to the ideal solution may not be a good representation for the entire cluster. Moreover, the reduced set may not contain the extreme solutions and, hence, does not capture the diversity of the entire Pareto set. Therefore, to alleviate the shortcomings of the existing approaches, we propose a novel graph-theoretical approach, which is based on the connectivity (e.g., degree) in the objective space, to obtain the representative solutions from each cluster. We test the applicability of the proposed method on the Pareto optimal solutions obtained from a multi-objective optimization model for a realistic case study. We show both qualitatively and quantitatively that the reduced set obtained from the proposed method better represents the entire Pareto set.KeywordsMulti-objective optimization problemsPareto optimal solutionsClusteringGraph theory