The multidimensional 0–1 knapsack problem – bounds and computational aspects

The multidimensional 0–1 knapsack problem (MKP) is a resource allocation model that is one of the most well-known integer programming problems. During the last few decades, an impressive amount of research on the 0–1 knapsack problem has been published in the literature, and efficient special-purpose methods have become available for solving very large-scale instances. However, the multidimensional case has received less attention from the operational research community. Although recent advances have made solving medium size instances possible, solving the NP-hard problem remains a very interesting challenge, especially when the number of constraints increases. This paper surveys the principal results published in the literature concerning both the problem's theoretical properties and its approximate or exact solutions. The paper focuses on the more recent results – for example, those relevant to surrogate and composite duality, new preprocessing approaches creating enhanced versions of leading commercial software, and efficient metaheuristic-based methods.