Advertisement

Max-Min Optimization of the Multiple Knapsack Problem: an Implicit Enumeration Approach

  • Takeo Yamada
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 43)

Abstract

The binary knapsack problem is fundamental in combinatorial optimization, and algorithms to solve problems with nearly a million items are now available through the Internet. This paper is concerned with a variation of the problem, where there are n items to be packed into m knapsacks. Our problem is to find the assignment of items into knapsacks such that the minimum of the knapsack profits is maximized. This problem is referred to as the max-min multiple knapsack problem (M 3 KP). First, some upper bounds and a heuristic algorithm are presented, and based on these, we explore algorithms to solve the problem to optimally. Then, we make use of a novel pruning method to develop an implicit enumeration algorithm that can solve M 3 KPs with up to a few hundred items exactly.

Keywords

Multiple knapsack problem max-min optimization implicit enumeration 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aarts, E. and Lenstra, J. K. ed. Local Search in Combinatorial Optimization. Chichester:John Wiley & Sons, 1997.Google Scholar
  2. Dash Optimization Inc. XPRESS-MP Release 11. 2000.Google Scholar
  3. Du D.-Z. and Pardalos, P.M. Minimax and Applications. Boston:Kluwer Academic Publishers, 1995.CrossRefGoogle Scholar
  4. Fourer. R., Software survey: linear programming. OR/MS Today 1999, 26:64–71.Google Scholar
  5. Garey, M. R. and Johnson, D. S. Computers and Intractability: A Guide to the Theory of NP-Completeness. New York:Freeman and Company, 1979.Google Scholar
  6. Gill, P. E., Murray, W., Wright, M. H. Practical Optimization. New York:Academic Press, 1981.Google Scholar
  7. Martello, S. and Toth, P. Knapsack Problems: Algorithms and Computer Implementations. Chichester:John Wiley & Sons, 1990.Google Scholar
  8. Martello, S. and Toth, P. Solution of the zero-one multiple knapsack problem. European Journal of Operational Research 1980; 4:276–283.CrossRefGoogle Scholar
  9. Sedgewick, R. Algorithms in C, 3rd Edition. Reading:Addison-Wesley, 1998.Google Scholar
  10. Wolsey, L. A. Integer Programming. Chichester:John Wiley & Sons, 1998.Google Scholar
  11. Yamada, T., Takahashi, H. and Kataoka, S A branch-and-bound algorithm for the mini-max spanning forest problem. European Journal of Operational Research, 1997; 101:93–103.CrossRefGoogle Scholar
  12. Yamada, T., Futakawa, M. and Kataoka, S. Some exact algorithms for the knapsack sharing problem. European Journal of Operational Research, 1998; 106:177–183.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Takeo Yamada
    • 1
  1. 1.Department of Computer ScienceThe National Defense AcademyYokosuka, KanagawaJapan

Personalised recommendations