Multi-objective Evolutionary Approaches for the Knapsack Problem with Stochastic Profits

Kokila Kasuni Perera, Frank Neumann, Aneta Neumann

Abstract: Uncertainties in real-world problems impose a challenge in finding reliable solutions. If mishandled, they can lead to suboptimal or infeasible solutions. Chance constraints are a natural way to capture uncertain problem parameters. They model probabilistic constraints involving the stochastic parameters and an upper bound of probability that mimics the confidence level of the solution. We focus on the knapsack problem with stochastic profits to guarantee a certain level of confidence in the profit of the solutions. We present a bi-objective fitness formulation that uses expected profit and standard deviation to capture the chance constraints. This formulation enables optimising the problem independent of a specific confidence level. We evaluate the proposed fitness formulation using well-known evolutionary algorithms GSEMO, NSGA-II and MOEA/D. Moreover, we introduce a filtering method that refines the interim populations based on the confidence levels of its solutions. We evaluate this method by applying it along with GSEMO to improve the quality of its population during optimisation. We conduct extensive experiments to show the effectiveness of these approaches using several benchmarks and present a detailed analysis of the results.