Statistical analyses of solution methods for the multiple-choice knapsack problem with setups: Implications for OR practitioners

Abstract

An interesting extension of the classic Knapsack Problem (KP) is the Multiple-Choice Knapsack Problem with Setups (MCKS) which is focused on solving practical applications that involve both multiple periods and setups. Sophisticated solution methods for the MCKS that are presented in the operations research (OR) literature are not readily available for use by OR practitioners. Using MCKS test instances that appear in the literature, we demonstrate that the general-purpose integer programming software Gurobi sometimes used in an iterative manner can efficiently solve these MCKS instances using all default parameter values on a standard PC. It is shown both empirically and statistically that these Gurobi solutions are competitive with solution approaches from the literature. Hence, our approach using Gurobi is both easy for the OR practitioner to use and gives results competitive with the best specialized MCKS solution methods in the literature without the need to generate algorithm-specific code. Furthermore, this paper presents significant concerns regarding the solutions stated in the literature by the approximate solution method that reports the best results on 120 MCKS test instances. Specifically, 26% of this method’s solutions violate Gurobi upper bounds and an additional 33% of its solutions, on average, exceed the known guaranteed optimums by a value of 12,510.