The chance-constrained knapsack problem is a variant of the classical knapsack problem where each item has a weight distribution instead of a deterministic weight. The objective is to maximize the total profit of the selected items under the condition that the weight of the selected items only exceeds the given weight bound with a small probability of $\alpha$. In this paper, consider problem-specific single-objective and multi-objective approaches for the problem. We examine the use of heavy-tail mutations and introduce a problem-specific crossover operator to deal with the chance-constrained knapsack problem. Empirical results for single-objective evolutionary algorithms show the effectiveness of our operators compared to the use of classical operators. Moreover, we introduce a new effective multi-objective model for the chance-constrained knapsack problem. We use this model in combination with the problem-specific crossover operator in multi-objective evolutionary algorithms to solve the problem. Our experimental results show that this leads to significant performance improvements when using the approach in evolutionary multi-objective algorithms such as GSEMO and NSGA-II.

中文翻译：

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概率约束背包问题的特定单目标和多目标进化算法

机会约束背包问题是经典背包问题的一个变体，其中每个项目都有一个权重分布，而不是一个确定性的权重。目标是在所选物品的权重仅以小概率$\alpha$超过给定权重界限的情况下，使所选物品的总利润最大化。在本文中，考虑针对问题的特定问题的单目标和多目标方法。我们研究了重尾突变的使用，并引入了一个特定于问题的交叉算子来处理机会受限的背包问题。单目标进化算法的经验结果表明，与使用经典算子相比，我们的算子是有效的。而且，我们为机会约束背包问题引入了一种新的有效多目标模型。我们将此模型与多目标进化算法中特定问题的交叉算子结合使用来解决问题。我们的实验结果表明，在 GSEMO 和 NSGA-II 等进化多目标算法中使用该方法时，这会带来显着的性能改进。