The multiple knapsack problem (MKP) generalizes the classical knapsack problem by assigning items to multiple knapsacks subject to capacity constraints. It is used to model many real-world resource allocation and scheduling problems. In practice, these optimization problems often involve stochastic and dynamic components. Evolutionary algorithms provide a flexible framework for addressing such problems under uncertainty and dynamic changes. In this paper, we investigate a stochastic and dynamic variant of MKP with chance constraints, where the item weights are modeled as independent normally distributed random variables and knapsack capacities change during the optimization process. We formulate the problem as a bi-objective optimization formulation that balances profit maximization and probabilistic capacity satisfaction at a given confidence level. We conduct an empirical comparison of four widely used multi-objective evolutionary algorithms (MOEAs), representing both decomposition- and dominance-based search paradigms. The algorithms are evaluated under varying uncertainty levels, confidence thresholds, and dynamic change settings. The results provide comparative insights into the behavior of decomposition-based and dominance-based MOEAs for stochastic MKP under dynamic constraints.

翻译：多背包问题（MKP）是经典背包问题的一般化形式，通过将物品分配至多个受容量约束的背包来建模。该问题被广泛应用于建模实际资源分配与调度问题。在实践中，这些优化问题通常包含随机性和动态性成分。进化算法为应对不确定性与动态变化提供了灵活框架。本文研究带机会约束的随机动态MKP变体，其中物品重量被建模为独立正态随机变量，且背包容量在优化过程中动态变化。我们将该问题构建为双目标优化形式，在给定置信水平下平衡利润最大化与概率容量可行性。通过对比四种主流多目标进化算法（MOEAs），分别代表基于分解与基于支配的搜索范式，我们在不同不确定性水平、置信阈值及动态变化场景下评估算法性能。实验结果揭示了基于分解与基于支配的MOEAs在处理动态约束随机MKP时的行为差异。