NEW FORMULATIONS AND VALID INEQUALITIES FOR THE TOLL SETTING PROBLEM G. Heilporn, M. Labb´ e, ∗ P. Marcotte ∗∗ G. Savard ∗∗∗ ∗ Service Graphes et Optimisation Math´ ematique, D´ epartement d’Informatique, Universit´ e Libre de Bruxelles, Bruxelles, Belgique. ∗∗ D´ epartement d’Informatique et de recherche op´ erationnelle, Universit´ e de Montr´ eal, Canada. ∗∗∗ D´ epartement de math´ ematique et g´ enie industriel, ´ Ecole Polytechnique de Montr´ eal, Canada. Abstract: Consider the problem of maximizing the revenue generated by tolls set on a subset of arcs of a transportation network, and where origin-destination flows are assigned to shortest paths with respect to the sum of tolls and initial costs. This work is concerned with two new combinatorial formulations of this problem and provides a framework for deriving valid inequalities. Copyright c 2006 IFAC Keywords: Combinatorial mathematics. Optimization problems. 1. INTRODUCTION In this paper, we construct valid inequalities based on novel formulations of a toll setting problem (TP), initially proposed by Labb´ e et al. (Labb´ e et al., 1998). It is structured as follows: Section 2 introduces the basic model; new formulations are presented in Section 3; valid inequalities are con- structed in Sections 4 and 5; extensions of the analysis are considered in Section 6. 2. THE TOLL SETTING PROBLEM Let us consider a multicommodity network defined by a node set N , an arc set ¯ A and a set of origin- destination pairs K = {(o k ,d k ): k ∈ K}, each one endowed with a demand η k . Let A ⊂ ¯ A be a subset of arcs a upon which tolls t a can be added to the original fixed cost vector c and B = ¯ A − A the complementary subset of toll-free arcs, for which the cost vector d is fixed. Assuming that, for a given toll policy t =(t a ) a∈A , network users travel on shortest paths with respect to t, the TP consists in devising a revenue maximizing toll policy. Upon the introduction of Boolean vectors x and y that specify whether an arc belongs to the shortest path traveled by a given commodity, i.e., x k ij =  1 if commodity k uses arc (i, j ) ∈ A, 0 otherwise, y k ij =  1 if commodity k uses arc (i, j ) ∈ B, 0 otherwise, we formulate the TP as the bilevel program (TP1) max t,x,y  k∈K  (i,j)∈A η k t ij x k ij where (x, y) solves the shortest path problem min x,y  k∈K    (i,j)∈A (c ij + t ij )x k ij +  (i,j)∈B d ij y k ij   s.t. :  i:(i,j)∈A x k ij +  i:(i,j)∈B y k ij −  l:(j,l)∈A x k jl −  l:(j,l)∈B y k jl