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Exact Solution of the Knapsack Problem

In: Knapsack Problems

Author

Listed:
  • Hans Kellerer

    (University of Graz, Department of Statistics and Operations Research)

  • Ulrich Pferschy

    (University of Graz, Department of Statistics and Operations Research)

  • David Pisinger

    (University of Copenhagen, DIKU, Department of Computer Science)

Abstract

Assume that a set of n items is given, each item j having an integer profit p j and an integer weight w j . The knapsack problem asks to choose a subset of the items such that their overall profit is maximized, while the overall weight does not exceed a given capacity c. Introducing binary variables x j to indicate whether item j is included in the knapsack or not the model may be defined: (5.1) $$ {\rm{(KP)}}\,{\rm{maximize}}\;\sum\limits_{j = 1}^n {{p_j}{x_j}} $$ (5.2) $$ {\rm{subject}}\;{\rm{to}}\;\sum\limits_{j = 1}^n {{w_j}{x_j}} \le c,$$ (5.3) $$ {x_j} \in \left\{ {0,1} \right\},\;j = 1,...,n.$$

Suggested Citation

  • Hans Kellerer & Ulrich Pferschy & David Pisinger, 2004. "Exact Solution of the Knapsack Problem," Springer Books, in: Knapsack Problems, chapter 5, pages 117-160, Springer.
    • Handle: RePEc:spr:sprchp:978-3-540-24777-7_5
    DOI: 10.1007/978-3-540-24777-7_5
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