Robust efficiency measures for linear knapsack problem variants

Abstract

Numerous combinatorial optimization applications, such as the mobile retailer product mix problem, utilize the multidemand, multidimensional knapsack problem variant in which there are multiple knapsack constraints requiring a weighted summation of the variables to be less than or equal to a nonnegative value and multiple demand constraints requiring a weighted summation of the variables to be greater than or equal to a nonnegative value. The purpose of this paper is to demonstrate that core variables and efficiency measures, concepts used in the most efficient solvers to-date for binary knapsack problems and some of its variants, can be extended to the multidemand, multidimensional knapsack problem variant. Specifically, new efficiency measure calculations are provided and their properties are mathematically proven and experimentally demonstrated. The contribution of such measures to knapsack problem research is that these measures are applicable to all knapsack problem variants with a single linear objective function and linear constraints of any quantity. The applicability of these new measures is demonstrated through the development of three heuristic procedures: a Fixed-Core heuristic, a Genetic Algorithm, and a Kernel Search heuristic. The results from these tests are compared with the results from a commercial solver and an existing heuristic. The findings from these tests demonstrate that the Fixed-Core and Kernel Search heuristics developed for this paper are the most efficient solvers to-date for hard multidemand, multidimensional knapsack problems.