A novel graph-theoretical clustering approach to find a reduced set with extreme solutions of Pareto optimal solutions for multi-objective optimization problems

Abstract

Multi-objective optimization problems and their solution algorithms are of great importance as single-objective optimization problems are not usually a true representation of many real-world problems. In general, multi-objective optimization problems result in a large set of Pareto optimal solutions. Each solution in this set is optimal with some trade-offs. Therefore, it is difficult for the decision-maker to select a solution, especially in the absence of subjective or judgmental information. Moreover, an analysis of all the solutions is computationally expensive and, hence, not practical. Thus, researchers have proposed several techniques such as clustering and ranking of Pareto optimal solutions to reduce the number of solutions. The ranking methods are often used to obtain a single solution, which is not a good representation of the entire Pareto set. This paper deviates from the common approach and proposes a novel graph-theoretical clustering method. The quality of the clustering based on the Silhouette score is used to determine the number of clusters. The connectivity in the objective space is used to find representative solutions for clusters. One step forward, we identify ‘extreme solutions’. Hence, the reduced set contains both extreme solutions and representative solutions. We demonstrate the performance of the proposed method by using different 3D and 8D benchmark Pareto fronts as well as Pareto fronts from a case study in Royal Australian Navy. Results revealed that the reduced set obtained from the proposed method outperforms that from the K-means clustering, which is the most popular traditional clustering approach in Pareto pruning.