Evolutionary Algorithm for Chance Constrained Quadratic Multiple Knapsack Problem

2025-11-04

AI Summary

This paper addresses the Stochastic Quadratic Multiple Knapsack Problem (SQMKP) with chance constraints, where the objective comprises linear and quadratic terms, item profits are random variables, and multiple capacity constraints must be satisfied probabilistically. To overcome poor solution quality and slow convergence under tight stochastic constraints, we propose a hybrid evolutionary algorithm: it employs either (1+1)EA or (μ+λ)EA as the global search framework and introduces a novel multi-factor local optimization mechanism. This mechanism constructs independent local search spaces for each knapsack and formulates local improvement as a multi-task collaborative optimization problem, enabling fine-grained constraint satisfaction and profit enhancement. Experimental results demonstrate that our approach significantly outperforms baseline evolutionary algorithms under stringent chance constraints and multiple capacity limits. The results validate the effectiveness and advancement of the multi-factor local optimization architecture for stochastic constrained combinatorial optimization.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Web Mining and Content Analysis: Models for Web evolution

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 ($mu+lambda$) 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.

Problem

• Solving stochastic quadratic multiple knapsack problems with random profits
• Modeling uncertainty using chance constraints for resource allocation
• Developing hybrid evolutionary algorithms with local optimization strategies

Innovation

• Evolutionary Algorithm for global search optimization
• Multi-factorial optimization for local refinement strategy
• Hybrid approach combining EA with MFO techniques