Two-stage stochastic bilevel programming over a transportation network

• A two-stage stochastic extension of a generic pricing model.
• Reformulation as a single-stage bilevel program, and next as a mixed integer program.
• Analysis of the theoretical properties of the model.
• Sensitivity analysis with respect to a parameter that limits the difference between first and second-stage tariffs.

We consider a two-stage stochastic extension of the bilevel pricing model introduced by Labbé et al. (1998). In the first stage, the leader sets tariffs on a subset of arcs of a transportation network, with the aim of maximizing profits while, at the lower level, flows are assigned to cheapest paths of a multicommodity transportation network. In the second stage, the situation repeats itself under the constraint that tariffs should not differ too widely from those set at the first stage, a condition that frequently arises in practice. We analyze properties of the model, provide numerical illustrations, and open avenues for further research into this area.