# 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

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topMansi, 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

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

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