Evolutionary Algorithm for Chance Constrained Quadratic Multiple Knapsack Problem
Kokila Kasuni Perera, Aneta Neumann
TL;DR
This work addresses the stochastic Profit Chance Constrained Quadratic Multiple Knapsack Problem (QMKP) by formulating chance constraints at a confidence level $\alpha$ and estimating profits via Chebyshev bounds. It proposes a hybrid optimization scheme that couples a simple or population-based evolutionary algorithm with a locally focused multi-factorial optimisation (MFO) component, implemented through a localised model that partitions the problem into knapsack-specific subproblems guided by item preferences. The local optimiser uses a preference-based mutation and an MFO-based transfer mechanism to refine solutions around a reference, enabling effective knowledge sharing between related knapsack tasks. Experimental results show the hybrid approach, especially with MFO, yields robust performance when capacity bounds are tight and profit uncertainty is high, while pure EAs perform better on easier, looser instances; the study demonstrates the viability of a multi-tasking local optimisation paradigm for uncertain combinatorial problems and outlines directions for extending this framework to other domains.
Abstract
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.