Multi-objective optimization aims at simultaneously optimizing two or more objectives of a problem. Multi-objective evolutionary algorithms (MOEAs) are widely accepted and useful for solving real world multi-objective problems. When we have two or more conflicting objectives of a problem then we can apply MOEA. MOEA generates a set of non-dominated solutions at the end of run, which is called Pareto set. The Pareto front contains set of Pareto solutions. Any MOEA aims to improve (i) convergence of population towards true Pareto front and (ii) diversity of solutions belonging to Pareto set. Generally, an external archive is used by MOEAs to maintain a set of non-dominated Pareto set solutions. Sometimes, Pareto set contains more number of solutions than the size of archive. This paper presents survey of various methods used by different MOEAs for reducing the size of Pareto set while maintaining solutions diversity. It presents comparison of these methods along with their advantages and disadvantages. The paper concludes by giving limitation of crowding distance based method in various scenarios.
It is important for any MOEA (Multi Objective Evolutionary Algorithm) to improve convergence and diversity of solutions of Pareto front, which is obtained at the termination of MOEA. There are many MOEA available in the literature: NSGA-II, SPEA, SPEA2, PESAII and IBEA. This paper aims at improving solutions diversity of Pareto front of a well known multi-objective optimization algorithm, NSGA-II. The standard NSGA-II algorithm uses crowding distance based method for maintaining solutions diversity. The limitation of crowding distance based method is that it selects two nearer solutions from the Pareto front for the mating. The SPEA algorithm uses agglomerative hierarchical average linkage based clustering method for maintaining solutions diversity. The method sometimes may not preserve extreme solutions in Pareto front. In this paper, we propose a new diversity method based on agglomerative hierarchical clustering with extreme solutions preservation. The proposed method is tested on standard test problems of MOEA. It is observed that the proposed method gives good solution diversity on two objectives test problems compared to the existing diversity method of NSGA-II.
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