## Abstract

In this article, a mixed integer bilevel problem having a probabilistic knapsack constraint in the first level is proposed. The problem formulation is mainly motivated by practical pricing and service provision problems as it can be interpreted as a model for the interaction between a service provider and customers. A discrete probability space is assumed which allows a reformulation of the problem as an equivalent deterministic bilevel problem. The problem is further transformed into a linear bilevel problem, which in turn yields a quadratic optimization problem, namely the global linear complementarity problem. Based on this quadratic problem, a procedure to compute upper bounds on the initial problem by using a Lagrangian relaxation and an iterative linear minmax scheme is proposed. Numerical experiments confirm that the scheme practically converges.

Original language | English |
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Pages (from-to) | 107-116 |

Number of pages | 10 |

Journal | Networks |

Volume | 59 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2012 |

Externally published | Yes |

## Keywords

- Bilevel
- Networks
- Optimization
- Pricing
- Stochastic