  In this paper we propose a dynamic Stackelberg game-theoretic model for urban
freight transportation planning which is able to characterize the interaction
between freight and personal transportation in an urban area. The problem is
formulated as a bi-level dynamic mathematical program with equilibrium
constraints (MPEC) which belongs to a class of computationally challenging
problems. The lower level is dynamic user equilibrium (DUE) with inhomogeneous
traffic that characterizes traffic system optimum (SO) freight transportation
planning problem which aims at minimizing the total cost to a truck company. A
mathematical program with complementarity constraints (MPCC) reformulation is
derived and a projected gradient algorithm is designed to solve this
computationally challenging problem. Numerical experiments are conducted to
show that when planning freight transportation the background traffic is
nonnegligible, even though the amount of trucks compared to other vehicles
traveling on the same network is relatively small. What's more, in our proposed
bi-level model for urban freight transportation planning, we find a dynamic
case of a Braess-like Paradox which can provide managerial insights to a
metropolitan planning organization (MPO) in increasing social welfare by
restricting freight movement.
