An exact algorithm for the budget-constrained multiple knapsack problem

This paper is concerned with a variant of the multiple knapsack problem (MKP), where knapsacks are available by paying certain 'costs', and we have a fixed budget to buy these knapsacks. Then, the problem is to determine the set of knapsacks to be purchased, as well as to allocate items into the accepted knapsacks in such a way as to maximize the net total profit. We call this the budget-constrained MKP and present a branch-and-bound algorithm to solve this problem to optimality. We employ the Lagrangian relaxation approach to obtain an upper bound. Together with the lower bound obtained by a greedy heuristic, we apply the pegging test to reduce the problem size. Next, in the branch-and-bound framework, we make use of the Lagrangian multipliers obtained above for pruning subproblems, and at each terminal subproblem, we solve MKP exactly by calling the MULKNAP code [ D. Pisinger, An exact algorithm for large multiple knapsack problem, European J. Oper. Res. 114 (1999), pp. 528-541]. Thus, we were able to solve test problems with up to 160,000 items and 150 knapsacks within a few minutes in our computing environment. However, solving instances with relatively large number of knapsacks, when compared with the number of items, still remains hard. This is due to the weakness of MULKNAP to this type of problems, and our algorithm inherits this weakness as well.