An Approach to Ordering Objectives and Pareto Efficient Solutions

TL;DR

This paper introduces a novel method using probability integral transform to map multi-objective optimization solutions into a common score space, enabling better ordering and trade-off analysis aligned with decision-maker preferences.

Contribution

It presents a new approach to order Pareto solutions by transforming objectives into scores, allowing for more accurate trade-off mapping and preference-based optimization.

Findings

Scores enable ordering of Pareto solutions.

Trade-offs closer to preferences when using score-based optimization.

Non-linear mapping further improves trade-off accuracy.

Abstract

Solutions to multi-objective optimization problems can generally not be compared or ordered, due to the lack of orderability of the single objectives. Furthermore, decision-makers are often made to believe that scaled objectives can be compared. This is a fallacy, as the space of solutions is in practice inhomogeneous without linear trade-offs. We present a method that uses the probability integral transform in order to map the objectives of a problem into scores that all share the same range. In the score space, we can learn which trade-offs are actually possible and develop methods for mapping the desired trade-off back into the preference space. Our results demonstrate that Pareto efficient solutions can be ordered using a low- or no-preference aggregation of the single objectives. When using scores instead of raw objectives during optimization, the process allows for obtaining trade-offs significantly closer to the expressed preference. Using a non-linear mapping for transforming a desired solution in the score space to the required preference for optimization improves this even more drastically.