Pareto-Set Analysis: Biobjective Clustering in Decision and Objective Spaces

Multiobjective problems usually contain conflicting objectives. Therefore, there is no single best solution but a set of solutions that represent different tradeoffs between these objectives. Knowledge of this front can help in understanding the optimization problem better, as promising designs can be identified, and it can be seen what the achievable tradeoffs between the objective values are. Although for real-world problems, this interpretation of the front is usually not straightforward. This paper proposes a method to help the decision maker by clustering a given set of tradeoff solutions. It does so by extending the standard approach of clustering the solutions in objective space, such that it finds clusters that are compact and well separated both in decision space and in objective space. It is not the goal of the method to provide the decision maker with a single preferred solution. Instead, it helps the decision maker by structuring the tradeoff solutions such that he or she can learn about the problem. More precisely, a good clustering of the tradeoff solutions both in decision space and in objective space elicits information from the front about what design types lead to what regions in objective space. The novelty of the presented approach over existing work is its general nature, as it does not require the identification of distinct design variables or feature vectors. Instead, the proposed method only requires that a distance measure between a given pair of solutions can be calculated both in decision space and in objective space. As good clusters in decision space do not necessarily correspond to good clusters in objective space, we formulate this clustering problem as a biobjective optimization problem and propose PAN, a multiobjective evolutionary algorithm, to generate promising partitionings. Tests on artificial datasets are used to identify a suitable representation and a suitable partitioning goodness measure for PAN. Results from applying PAN to a knapsack problem and a bridge construction problem show that PAN is able to find multiple tradeoffs between good clustering in decision space and in objective space. Copyright © 2012 John Wiley & Sons, Ltd.