Evolutionary multi-objective algorithms have been widely shown to be successful when utilized for a variety of stochastic combinatorial optimization problems. Chance constrained optimization plays an important role in complex real-world scenarios, as it allows decision makers to take into account the uncertainty of the environment. We consider a version of the knapsack problem with stochastic profits to guarantee a certain level of confidence in the profit of the solutions. We introduce the multi-objective formulations of the profit chance constrained knapsack problem and design three bi-objective fitness evaluation methods that work independently of the specific confidence level required. We evaluate our approaches using well-known multi-objective evolutionary algorithms GSEMO and NSGA-II. In addition, we introduce a filtering method for GSEMO that improves the quality of the final population by periodically removing certain solutions from the interim populations based on their confidence level. We show the effectiveness of our approaches on several benchmarks for both settings where the knapsack items have fixed uniform uncertainties and uncertainties that are positively correlated with the expected profit of an item.

翻译：进化多目标算法已被广泛证明可在多种随机组合优化问题中成功应用。机会约束优化在复杂的现实场景中扮演重要角色，因为它允许决策者考虑环境的不确定性。我们研究了一种具有随机利润的背包问题变体，以保证解决方案利润达到一定置信水平。我们提出了利润机会约束背包问题的多目标公式，并设计了三种与特定置信水平无关的双目标适应度评估方法。我们采用著名的多目标进化算法GSEMO和NSGA-II对方法进行评估。此外，我们为GSEMO引入了一种过滤方法，通过定期根据置信水平移除临时种群中的特定解，从而提升最终种群的质量。我们在背包物品具有固定均匀不确定性以及不确定性与其预期利润正相关两种设定下，通过多个基准测试验证了方法的有效性。