Profit optimization of public transit operators: examining both interior and boundary solutions

We concern the modal choice of commuters in a transport system comprising a highway and two transit lines. For the operation of the transit lines, two market structures are considered: monopolistic and duopolistic. The problem of optimizing the profit of each transit operator is formulated as an optimization model with equilibrium constraints. We theoretically prove that, to obtain both the interior and boundary solutions of the optimization model, it is sufficient to solve an alternative optimization model with equality constraints. For each market structure, we propose a period-to-period transit fare and auto toll scheme to maximize the profit of each transit operator and to simultaneously make the profit of each transit operator more than a certain value. Finally, by numerical examples, we show the effectiveness of the scheme in each market and the necessity of examining both the interior and boundary solutions of the optimization model.