Advertisement

Greedy algorithm for the general multidimensional knapsack problem

  • Yalçın Akçay
  • Haijun Li
  • Susan H. Xu
Article

Abstract

In this paper, we propose a new greedy-like heuristic method, which is primarily intended for the general MDKP, but proves itself effective also for the 0-1 MDKP. Our heuristic differs from the existing greedy-like heuristics in two aspects. First, existing heuristics rely on each item’s aggregate consumption of resources to make item selection decisions, whereas our heuristic uses the effective capacity, defined as the maximum number of copies of an item that can be accepted if the entire knapsack were to be used for that item alone, as the criterion to make item selection decisions. Second, other methods increment the value of each decision variable only by one unit, whereas our heuristic adds decision variables to the solution in batches and consequently improves computational efficiency significantly for large-scale problems. We demonstrate that the new heuristic significantly improves computational efficiency of the existing methods and generates robust and near-optimal solutions. The new heuristic proves especially efficient for high dimensional knapsack problems with small-to-moderate numbers of decision variables, usually considered as “hard” MDKP and no computationally efficient heuristic is available to treat such problems.

Keywords

Integer programming Multidimensional knapsack problems Heuristics 

References

  1. Akcay, Y. and S.H. Xu. (2004). “Joint Inventory Replenishment and Component Allocation Optimization in An Assemble-to-Order System.” Management Science, 50(1), 99–116.Google Scholar
  2. Caprara, A., H. Kellerer, U. Pferschy, and D. Pisinger. (2000). “Approximation Algorithms for Knapsack Problems with Cardinality Constraints.” European Journal of Operational Research, 123, 333–345.CrossRefGoogle Scholar
  3. Chu, P.C. and J.E. Beasley. (1998). “A Genetic Algorithm for the Multidimensional Knapsack Problem.” Journal of Heuristics, 4, 63–86.CrossRefGoogle Scholar
  4. de Vries, S. and R.V. Vohra. (2003). “Combinatorial Auctions: A Survey.” INFORMS Journal on Computing, 15, 284–310.CrossRefGoogle Scholar
  5. Everett, H. (1963). “Generalized Langrange Multiplier Method for Solving Problems of Optimum Allocation of Resources.” Operations Research, 2, 399–417.Google Scholar
  6. Freville, A. (2004). “The Multidimensional 0-1 Knapsack Problem: An Overview.” European Journal of Operational Research, 155, 1–21.CrossRefGoogle Scholar
  7. Garey, M.R. and D.S. Johnson. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. San Francisco: W. H. Freeman.Google Scholar
  8. Kellerer, H., U. Pferschy, and D. Pisinger. (2004). Knapsack Problems. Springer.Google Scholar
  9. Kleywegt, A. and J.D. Papastavrou. (2001). “The dynamic and Stochastic Knapsack Problem with Random Sized Items.” Operations Research, 49, 26–41.CrossRefGoogle Scholar
  10. Kochenberger, G.A., B.A. McCarl, and F.P. Wyman. (1974). “A Heuristic for General Integer Programming.” Decision Sciences, 5, 36–44.Google Scholar
  11. Lin, E.Y.-H. (1998). “A Bibliographical Survey on Some Well-Known Non-Standard Knapsack Problems.” INFOR, 36(4), 274–317.Google Scholar
  12. Loulou, R. and E. Michaelides. (1979). “New Greedy-like Heuristics for the Multidimensional 0-1 Knapsack Problem.” Operations Research, 27, 1101–1114.Google Scholar
  13. Lu, L.L., S.Y. Chiu, and L.A. Cox. (1999). “Optimal Project Selection: Stochastic Knapsack with Finite Time Horizon.” Operations Research, 50, 645–650.CrossRefGoogle Scholar
  14. Magazine, M.J. and O. Oguz. (1984). “A Heuristic Algorithm for the Multidimensional Zero-One Knapsack Problem.” European Journal of Operational Research, 16, 319–326.CrossRefGoogle Scholar
  15. Petersen, C.C. (1967). “Computational Experience with Variants of the Balas Algorithm Applied to the Selection of R&D Projects.” Management Science, 13, 736–750.Google Scholar
  16. Pirkul, H. (1987). “A Heuristic Solution Procedure for the Multiconstraint Zero-One Knapsack Problem.” Naval Research Logistics, 34, 161–172.Google Scholar
  17. Pirkul, H. and S. Narasimhan. (1986). “Efficient Algorithms for the Multiconstraint General Knapsack Problem.” IIE Transactions, pp. 195–203.Google Scholar
  18. Senju, S. and Y. Toyoda. (1968). “An Approach to Linear Programming with 0-1 Variables.” Management Science, 15(4), B196–B207.Google Scholar
  19. Shih, W. (1979). “A Branch and Bound Method for the Multiconstraint Zero-One Knapsack Problem.” Journal of Operational Research Society, 30, 369–378.CrossRefGoogle Scholar
  20. Toyoda, Y. (1975). “A Simplified Algorithm for Obtaining Approximate Solutions to Zero-One Programming Problems.” Management Science, 21(12), 1417–1427.Google Scholar
  21. Volgenant, A. and J.A. Zoon. (1990). “An Improved Heuristic for Multidimensional 0-1 Knapsack Problems.” Journal of Operational Research Society, 41, 963–970.CrossRefGoogle Scholar
  22. Weingartner, H.M. and D.N. Ness. (1967). “Methods for the Solution of the Multi-Dimensional 0/1 Knapsack Problem.” Operations Research, 15, 83–103.Google Scholar
  23. Zanakis, S.H. (1977). “Heuristic 0-1 Linear Programming: Comparisons of Three Methods.” Management Science, 24, 91–103.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  1. 1.College of Administrative Sciences and EconomicsKoç UniversityIstanbulTurkey
  2. 2.Department of MathematicsWashington State UniversityPullmanUSA
  3. 3.Department of Supply Chain and Information Systems, The Smeal College of Business AdministrationThe Pennsylvania State UniversityPennsylvaniaUSA

Personalised recommendations