Mathematical programming approaches to pricing problems

To do so, the company must consider customers' reactions to these prices, as they may refuse to buy a given product or service if its price is too high. This is commonly known in literature as a pricing problem.

This class of problems, which is typically bilevel, was first studied in the 1990s and is NP-hard, although polynomial algorithms do exist for some particular cases. Many questions are still open on this subject.

The aim of this thesis is to investigate mathematical properties of pricing problems, in order to find structural properties, formulations and solution methods that are as efficient as possible. In particular, we focus our attention on pricing problems over a network. In this framework, an authority owns a subset of arcs and imposes tolls on them, in an attempt to maximise his/her revenue, while users travel on the network, seeking for their minimum cost path.

First, we provide a detailed review of the state of the art on bilevel pricing problems.

Then, we consider a particular case where the authority is using an unit toll scheme on his/her subset of arcs, imposing either the same toll on all of them, or a toll proportional to a given parameter particular to each arc (for instance a per kilometre toll). We show that if tolls are all equal then the complexity of the problem is polynomial, whereas in case of proportional tolls it is pseudo-polynomial.

We then address a robust approach taking into account uncertainty on parameters. We solve some polynomial cases of the pricing problem where uncertainty is considered using an interval representation.

Finally, we focus on another particular case where toll arcs are connected such that they constitute a path, as occurs on highways. We develop a Dantzig-Wolfe reformulation and present a Branch-and-Cut-and-Price algorithm to solve it. Several improvements are proposed, both for the column generation algorithm used to solve the linear relaxation and for the branching part used to find integer solutions. Numerical results are also presented to highlight the efficiency of the proposed strategies. This problem is proved to be APX-hard and a theoretical comparison between our model and another one from the literature is carried out.