Evolutionary multi-objective algorithms have successfully been used in the context of Pareto optimization where a given constraint is relaxed into an additional objective. In this paper, we explore the use of 3-objective formulations for problems with chance constraints. Our formulation trades off the expected cost and variance of the stochastic component as well as the given deterministic constraint. We point out benefits that this 3-objective formulation has compared to a bi-objective one recently investigated for chance constraints with Normally distributed stochastic components. Our analysis shows that the 3-objective formulation allows to compute all required trade-offs using 1-bit flips only, when dealing with a deterministic cardinality constraint. Furthermore, we carry out experimental investigations for the chance constrained dominating set problem and show the benefit for this classical NP-hard problem.

翻译：进化多目标算法在将给定的约束条件放宽为一个额外目标的 Pareto 优化中已经得到成功应用。本文中，我们探讨了在具有随机约束条件的问题中使用 3 目标公式的应用。我们的公式交换了随机组成部分的预期成本和方差，以及给定的确定性约束条件。我们指出了这个 3 目标公式相对于最近研究的具有正态分布随机因素的双目标公式的好处。我们的分析表明，当处理确定的基数约束时，这个 3 目标公式只需要使用 1 位翻转就可以计算所有必需的权衡。此外，我们对具有随机约束支配集问题进行了实验调查，并展示了这个经典的 NP-难问题的好处。