A cross entropy algorithm for the Knapsack problem with setups

• Published in: Computers & operations research Vol. 35; no. 1; pp. 241 - 252
• Main Authors: Caserta, M., Quiñonez Rico, E., Márquez Uribe, A.
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 2008; Elsevier Science; Pergamon Press Inc
• Subjects: Algorithms; Applied sciences; Branch & bound algorithms; Combinatorial optimization; Cross entropy; Dynamic programming; Entropy; Exact sciences and technology; Flows in networks. Combinatorial problems; Heuristic; Knapsack Problem; Metaheuristics; Monte Carlo simulation; Operational research and scientific management; Operational research. Management science; Optimization algorithms; Portfolio theory; Scheduling; Studies; Cross entropy; Scheduling; Dynamic programming; Combinatorial optimization; Metaheuristics; Knapsack Problem; Branch and bound method; Economic optimization; Entropy; Knapsack problem; Heuristic method; Cost function; Step by step method; Robustness; Large scale; Objective function; Econometrics; Toll road
• ISSN: 0305-0548; 1873-765X; 0305-0548
• DOI: 10.1016/j.cor.2006.02.028

In this article we propose a new metaheuristic-based algorithm for the Integer Knapsack Problem with Setups. This problem is a generalization of the standard Integer Knapsack Problem, complicated by the presence of setup costs in the objective function as well as in the constraints. We propose a cross entropy based algorithm, where the metaheuristic scheme allows to relax the original problem to a series of well chosen standard Knapsack problems, solved through a dynamic programming algorithm. To increase the computational effectiveness of the proposed algorithm, we use a turnpike theorem, which sensibly reduces the number of iterations of the dynamic algorithm. Finally, to testify the robustness of the proposed scheme, we present extensive computational results. First, we illustrate the step-by-step behavior of the algorithm on a smaller, yet difficult, problem. Subsequently, to test the solution quality of the algorithm, we compare the results obtained on very large scale instances with the output of a branch and bound scheme. We conclude that the proposed algorithm is effective in terms of solution quality as well as computational time.