A two-phase kernel search variant for the multidimensional multiple-choice knapsack problem

Abstract

The Multidimensional Multiple-choice Knapsack Problem (MMKP) is a complex combinatorial optimization problem for which finding high quality feasible solutions is very challenging. Recently, several heuristic approaches and a few exact algorithms have been proposed for its solution. These methods have been able to provide new best-known values for benchmark instances although many of them still remain unclosed to optimality. In this paper, we provide a new variant of the heuristic framework Kernel Search and apply it to the MMKP. The proposed variant keeps the method’s main idea of solving a sequence of restricted mixed-integer subproblems but innovates by partitioning the solution process into two different phases with complementary goals. The first phase strives for feasibility and collects important information to dynamically adapt subproblems’ dimension and solution time in the second phase that is focused on getting high quality solutions. This makes the global approach more scalable and efficient.Computational results on different sets of benchmark instances demonstrate that our method is extremely effective outperforming all state-of-the-art heuristics for the MMKP. The method compares extremely well also with respect to exact approaches running for five hours. The proposed algorithm is able to improve the best-known value of 185 out of 276 open benchmark instances and gets 8 out of 17 optimal solutions for the closed ones. The average deviation from optimality is always negligible.