The Skewboid Method: A Simple and Effective Approach to Pareto Relaxation and Filtering

The concept of Pareto optimality is the default method for pruning a large set of candidate solutions in a multi-objective problem to a manageable, balanced, and rational set of solutions. While the Pareto optimality approach is simple and sound, it may select too many or too few solutions for the decision-maker’s needs or the needs of optimization process (e.g. the number of survivors selected in a population-based optimization). This inability to achieve a target number of solutions to keep has caused a number of researchers to devise methods to either remove some of the non-dominated solutions via Pareto filtering or to retain some dominated solutions via Pareto relaxation. Both filtering and relaxation methods tend to introduce many new adjustment parameters that a decision-maker (DM) must specify. In the presented Skewboid method, only a single parameter is defined for both relaxing the Pareto optimality condition (values between −1 and 0) and filtering more solutions from the Pareto optimal set (values between 0 and 1). This parameter can be correlated with a desired number of solutions so that this number of solutions is input instead of an unintuitive adjustment parameter. A mathematically sound derivation of the Skewboid method is presented followed by illustrative examples of its use. The paper concludes with a discussion of the method in comparison to similar methods in the literature.