Decision making for multi-objective problems: Mean and median metrics

When dealing with problems with more than two objectives, sophisticated multi-objective optimization algorithms might be needed. Pareto optimization, which is based on the concept of dominated and non-dominated solutions, is the most widely utilized method when comparing solutions within a multi-objective setting. However, in the context of optimization, where three or more objectives are involved, the effectiveness of Pareto dominance approaches to drive the solutions to convergence is significantly compromised as more and more solutions tend to be non-dominated by each other. This in turn reduces the selection pressure, especially for algorithms that rely on evolving a population of solutions such as evolutionary algorithms, particle swarm optimization, differential evolution, etc. The size of the non-dominated set of trade-off solutions can be quite large, rendering the decision-making process difficult if not impossible. The size of the non-dominated solution set increases exponentially with an increase in the number of objectives. This paper aims to expand a framework for coping with many/multi-objective and multidisciplinary optimization problems through the introduction of a min-max metric that behaves like a median measure that can locate the center of a data set. We compare this metric to the Chebyshev norm L_infinity metric that behaves like a mean measure in locating the center of a data set. The median metric is introduced in this paper for the first time, and unlike the mean metric is independent of the data normalization method. These metrics advocate balanced, natural, and minimum compromise solutions about all objectives. We also demonstrate and compare the behavior of the two metrics for a Tradespace case study involving more than 1200 CubeSat design alternatives identifying a manageable set of potential solutions for decision-makers.