# An Approach to Ordering Objectives and Pareto Efficient Solutions

arXiv: https://arxiv.org/abs/2205.15291 · HTML: https://mlsanity.com/paper/2205.15291/

## Key facts

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- First posted on arXiv 30 May 2022.

## Abstract (arXiv)

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.