Constrained optimization for bottleneck coarse tolling

We study the optimal single-step coarse toll design problem for the bottleneck model where the toll level and tolling duration have maximum acceptable upper bounds and the unconstrained optimal solution exceeds the upper bounds. We consider proportional user heterogeneity where users’ values of time and schedule delay vary in fixed proportions. Three classic coarse tolling models are studied, the ADL, Laih and braking models. In the ADL model, untolled users form a mass arrival at the bottleneck following the last tolled user. In the Laih model, there is a separated waiting facility for untolled users to wait until the toll ends. In the braking model, untolled users can choose to defer their arrival at the bottleneck to avoid paying the toll. We find that, in the ADL and the Laih models, the optimal solution chooses the maximum acceptable toll level and tolling duration. The ADL model further requires the tolling period to be started as late as possible to eliminate the queue at the toll ending moment. In the braking model, if the upper bound of the tolling duration is too small, no toll should be charged. Otherwise the optimal solution chooses the maximum acceptable tolling duration and may choose a toll price less than the maximum acceptable level.