# On a Traveling Salesman based Bilevel Programming Problem

### Pablo Adasme, Rafael Andrade, Janny Leung, Abdel Lisser

#### Abstract

In this paper, we consider a linear bilevel programming problem where both the leader and the follower maximize their profits subject to budget constraints. Additionally, we impose a Hamiltonian cycle topology constraint in the leader problem. In particular, models of this type can be motivated by telecommunication companies when dealing with traffic network flows from one server to another one within a ring topology framework. We transform the bilevel programming problem into an equivalent single level optimization problem that we further linearize in order to derive mixed integer linear programming (MILP) formulations. This is achieved by replacing the follower problem with the equivalent Karush Kuhn Tucker conditions and with a linearization approach to deal with the complementarity constraints. The topology constraint is handled by the means of two compact formulations and an exponential one from the classic traveling salesman problem. Thus, we compute optimal solutions and upper bounds with linear programs. One of the compact models allows to solve instances with up to 250 nodes to optimality. Finally, we propose an iterative procedure that allows to compute optimal solutions in remarkably less computational effort when compared to the compact models.

#### References

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#### Paper Citation

#### in Harvard Style

Adasme P., Andrade R., Leung J. and Lisser A. (2017). **On a Traveling Salesman based Bilevel Programming Problem** . In *Proceedings of the 6th International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,* ISBN 978-989-758-218-9, pages 329-336. DOI: 10.5220/0006190503290336

#### in Bibtex Style

@conference{icores17,

author={Pablo Adasme and Rafael Andrade and Janny Leung and Abdel Lisser},

title={On a Traveling Salesman based Bilevel Programming Problem},

booktitle={Proceedings of the 6th International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,},

year={2017},

pages={329-336},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0006190503290336},

isbn={978-989-758-218-9},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 6th International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,

TI - On a Traveling Salesman based Bilevel Programming Problem

SN - 978-989-758-218-9

AU - Adasme P.

AU - Andrade R.

AU - Leung J.

AU - Lisser A.

PY - 2017

SP - 329

EP - 336

DO - 10.5220/0006190503290336