An exact algorithm for the budget-constrained multiple knapsack problem

This paper is concerned with a variation of the multiple knapsack problem (MKP) [5, 6], where we are given a set of n items N = {1, 2, . . . , n} to be packed into m possible knapsacks M = {1, 2, . . . ,m}. As in ordinary MKP, by w j and p j we denote the weight and profit of item j ∈ N respectively, and the capacity of knapsack i ∈ M is ci. However, a fixed cost fi is imposed if we use knapsack i, and knapsacks are freely available within a fixed budget B. The problem is to determine the set of knapsacks to be employed as well as to fill the adopted knapsacks with items such that the capacity constraints are all satisfied and the total profit is maximized. Let yi and xi j be the decision variables such that yi = 1 if we use knapsack i, and yi = 0 otherwise. Also, xi j = 1 if item j is put into knapsack i, and xi j = 0 otherwise. Then, the problem is formulated as the following budget-constrained multiple knapsack problem. BCMKP: