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.
翻译:由于缺乏单一目标的可调适性,通常无法比较或下令解决多目标优化问题。此外,决策者往往认为可以比较规模化目标。这是一个谬误,因为解决方案的空间实际上不尽相同,没有线性权衡。我们提出了一个方法,利用概率整体转换方法,将问题的目标映射成所有都具有相同范围的得分。在得分空间中,我们可以了解哪些交易是实际可能的,并制定方法将所期望的取舍重新映射到优惠空间中。我们的结果表明,Pareto高效解决方案可以使用一个低或无基准的单一目标组合来排序。在优化过程中,当使用分数而不是原始目标时,这一过程允许获得大大接近所表示的偏好。使用非线性映射方法将分数空间的预期解决方案转换为最优化所需的偏好,可以更大幅度地改进这一选择。