Multi-strategy monarch butterfly optimization algorithm for discounted {0-1} knapsack problem

As an expanded classical 0-1 knapsack problem (0-1 KP), the discounted {0-1} knapsack problem (DKP) is proposed based on the concept of discount in the commercial world. The DKP contains a set of item groups where each group includes three items, whereas no more than one item in each group can be packed in the knapsack, which makes it more complex and challenging than 0-1 KP. At present, the main two algorithms for solving the DKP include exact algorithms and approximate algorithms. However, there are some topics which need to be further discussed, i.e., the improvement of the solution quality. In this paper, a novel multi-strategy monarch butterfly optimization (MMBO) algorithm for DKP is proposed. In MMBO, two effective strategies, neighborhood mutation with crowding and Gaussian perturbation, are introduced into MMBO. Experimental analyses show that the first strategy can enhance the global search ability, while the second strategy can strengthen local search ability and prevent premature convergence during the evolution process. Based on this, MBO is combined with each strategy, denoted as NCMBO and GMMBO, respectively. We compared MMBO with other six methods, including NCMBO, GMMBO, MBO, FirEGA, SecEGA and elephant herding optimization. The experimental results on three types of large-scale DKP instances show that NCMBO, GMMBO and MMBO are all suitable for solving DKP. In addition, MMBO outperforms other six methods and can achieve a good approximate solution with its approximation ratio close to 1 on almost all the DKP instances.