A smoothing approach for solving transportation problem with road toll pricing and capacity expansions

In this paper, we establish a bi-level optimization model for the equilibrium transportation problem concerning both capacity expansion and road toll pricing under the user equilibrium conditions. The bi-level optimization problem is reformulated as a mathematical programming problem with complementarity constraints (MPCC), which fails to satisfy the Mangasarian-Fromovitz constraint qualification (MFCQ). We adopt a smoothing approach to overcome the lack of constraint qualifications in the MPCC problem. Under mild conditions, it has been proven that the sequence of the global optimal solutions generated by solving corresponding smoothing subproblems converges to one optimal solution of the original MPCC problem. Numerical experiments show that the proposed method is practical in solving user equilibrium transportation problems with capacity expansion combining road toll pricing.