Integer Programming Formulation of the Bilevel Knapsack Problem

R. Mansi; S. Hanafi; L. Brotcorne

Mathematical Modelling of Natural Phenomena (2010)

  • Volume: 5, Issue: 7, page 116-121
  • ISSN: 0973-5348

Abstract

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The Bilevel Knapsack Problem (BKP) is a hierarchical optimization problem in which the feasible set is determined by the set of optimal solutions of parametric Knapsack Problem. In this paper, we propose two stages exact method for solving the BKP. In the first stage, a dynamic programming algorithm is used to compute the set of reactions of the follower. The second stage consists in solving an integer program reformulation of BKP. We show that the integer program reformulation is equivalent to the BKP. Numerical results show the efficiency of our method compared with those obtained by the algorithm of Moore and Bard

How to cite

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Mansi, R., Hanafi, S., and Brotcorne, L.. Taik, A., ed. "Integer Programming Formulation of the Bilevel Knapsack Problem." Mathematical Modelling of Natural Phenomena 5.7 (2010): 116-121. <http://eudml.org/doc/197702>.

@article{Mansi2010,
abstract = {The Bilevel Knapsack Problem (BKP) is a hierarchical optimization problem in which the feasible set is determined by the set of optimal solutions of parametric Knapsack Problem. In this paper, we propose two stages exact method for solving the BKP. In the first stage, a dynamic programming algorithm is used to compute the set of reactions of the follower. The second stage consists in solving an integer program reformulation of BKP. We show that the integer program reformulation is equivalent to the BKP. Numerical results show the efficiency of our method compared with those obtained by the algorithm of Moore and Bard},
author = {Mansi, R., Hanafi, S., Brotcorne, L.},
editor = {Taik, A.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {Bilevel programming; Knapsack problem; dynamic programming; branch-and-bound; bilevel programming},
language = {eng},
month = {8},
number = {7},
pages = {116-121},
publisher = {EDP Sciences},
title = {Integer Programming Formulation of the Bilevel Knapsack Problem},
url = {http://eudml.org/doc/197702},
volume = {5},
year = {2010},
}

TY - JOUR
AU - Mansi, R.
AU - Hanafi, S.
AU - Brotcorne, L.
AU - Taik, A.
TI - Integer Programming Formulation of the Bilevel Knapsack Problem
JO - Mathematical Modelling of Natural Phenomena
DA - 2010/8//
PB - EDP Sciences
VL - 5
IS - 7
SP - 116
EP - 121
AB - The Bilevel Knapsack Problem (BKP) is a hierarchical optimization problem in which the feasible set is determined by the set of optimal solutions of parametric Knapsack Problem. In this paper, we propose two stages exact method for solving the BKP. In the first stage, a dynamic programming algorithm is used to compute the set of reactions of the follower. The second stage consists in solving an integer program reformulation of BKP. We show that the integer program reformulation is equivalent to the BKP. Numerical results show the efficiency of our method compared with those obtained by the algorithm of Moore and Bard
LA - eng
KW - Bilevel programming; Knapsack problem; dynamic programming; branch-and-bound; bilevel programming
UR - http://eudml.org/doc/197702
ER -

References

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  1. L. Brotcorne, S. Hanafi, R. Mansi. A dynamic programming algorithm for the bilevel knapsack problem. Operations Research Letters, 37 (2009), No. 3, 215–218. 
  2. P. Calamai, L. Vicente. Generating linear and linear-quadratic Bilevel programming problems. SIAM Journal on Scientific and Statistical Computing, 14 (1993), 770–782. 
  3. B. Colson, P. Marcotte, G. Savard. Bilevel programming, a survey. 4OR, 3 (2005), 87–107.  
  4. S. Dempe. Foundation of Bilevel programming. Kluwer academic publishers, 2002.  
  5. S. Dempe, K. Richter. Bilevel programming with Knapsack constraints. Central European Newspaper of Operations Research, 8 (2000), 93–107. 
  6. P. Hansen, B. Jaumard, G. Savard. New branch-and-bound rules for linear bilevel programming. SIAM Journal on Scientific and Statistical Computing, 13 (1992), 1194–1217. 
  7. H. Kellerer, U. Pferschy, D. Pisinger. Knapsack problems. Springer-Verlag, 2004.  
  8. J.T. Moore, J.F. Bard. The mixed integer linear Bilevel programming problem. Operations Research, 38 (1990), 911–921. 

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