Computational Power of a Hybrid Algorithm for Solving the Multiple Knapsack Problem with Setup

Abstract

The Multiple Knapsack Problem with Setups (MKPS) belongs to the well-known NP-hard knapsack family, which represents an extended version of the binary multiple knapsack problem. In this paper, we study MKPS in which a set of families of items and a set of knapsacks are available. Each item is characterized by a knapsack-dependent profit and each family is associated with a knapsack-dependent cost. An item can be selected only if the corresponding family is activated and a family can only be setup in one knapsack. A key feature is that the activation of a family incurs a knapsack-dependent setup cost that should be considered both in the objective function and constraints. The setup cost varies with the knapsack and the goal consists in selecting appropriate items, from different families, to enter a knapsack while maximizing its value with respecting its capacity. We first propose a hybrid algorithm that combines the solution related to the mixed integer linear relaxation and a series of knapsack problems. The mixed linear relaxation can be viewed as the driving problem, where it is solved by using a special black-box solver while the series of knapsack try to build the solutions provided by calling the state-of-the-art Cplex solver. We then propose an enhanced version of the first method, where a series of valid constraints are added for providing a powerful method. The performance of the proposed methods are evaluated on benchmark instances of the literature, where their provided results are compared to those reached by the state-of-the-art Cplex solver and the best methods available in the literature. The computational power of the proposed method shows the importance of hybridization, especially for that type of problems.

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