Quadratic multiple knapsack problem (QMKP) is a combinatorial optimisation problem characterised by multiple weight capacity constraints and a profit function that combines linear and quadratic profits. We study a stochastic variant of this problem where profits are considered as random variables. This problem reflects complex resource allocation problems in real-world scenarios where randomness is inherent. We model this problem using chance constraints to capture the stochastic profits. We propose a hybrid approach for this problem, which combines an evolutionary algorithm (EA) with a local optimisation strategy inspired by multi-factorial optimisation (MFO). EAs are used for global search due to their effectiveness in handling large, complex solution spaces. In the hybrid approach, EA periodically passes interim solutions to the local optimiser for refinement. The local optimiser applies MFO principles, which are typically used in multi-tasking problems. The local optimiser models the local problem as a multi-tasking problem by constructing disjoint search spaces for each knapsack based on an input solution. For each item, its assignment across all knapsacks is considered to determine the preferred knapsack. Items are then divided into disjoint groups corresponding to each knapsack, allowing each knapsack to be treated as a separate optimisation task. This structure enables effective application of MFO-based local refinements. We consider two EAs for the problem, (1+1) EA and ($μ+λ$) EA. We conduct experiments to explore the effectiveness of these EAs on their own and also with the proposed local optimiser. Experimental results suggest that hybrid approaches, particularly those incorporating MFO, perform well on instances where chance constraints and capacity constraints are tight.

翻译：二次多重背包问题（QMKP）是一种组合优化问题，其特征在于多重重量容量约束以及结合了线性与二次利润的利润函数。本文研究该问题的随机变体，其中利润被视为随机变量。该问题反映了现实场景中固有随机性的复杂资源分配问题。我们采用机会约束对该问题进行建模，以捕捉随机利润。针对此问题，我们提出了一种混合方法，该方法将进化算法（EA）与受多因子优化（MFO）启发的局部优化策略相结合。进化算法因其在处理大规模复杂解空间方面的有效性而被用于全局搜索。在混合方法中，进化算法周期性地将中间解传递给局部优化器进行精炼。局部优化器应用通常用于多任务问题的多因子优化原则。局部优化器基于输入解为每个背包构建互不相交的搜索空间，从而将局部问题建模为多任务问题。对于每个物品，通过考虑其在所有背包中的分配来确定首选背包。随后，物品被划分为对应于每个背包的互不相交的组，使得每个背包可被视为独立的优化任务。此结构使得基于多因子优化的局部精炼能够有效应用。我们针对该问题考虑了两种进化算法：（1+1）EA和（μ+λ）EA。我们通过实验探究这些进化算法单独使用以及与所提出的局部优化器结合时的有效性。实验结果表明，混合方法，特别是那些融入多因子优化的方法，在机会约束和容量约束较为严格的问题实例上表现良好。