A path based algorithm for solve the hazardous materials transportation bilevel problem

In this work we consider the problem of determining a set of optimal tolls on the arcs of a multicommodity transportation network. The problem is formulated as a bilevel mathematical program where the upper level consists in a government agency that regulate the traffic of the dangerous materials by imposing tolls on arcs of the network trying to minimize the risk for the population in the case when an accident occurs to the carriers, while the lower level is represented by a group of carriers traveling on shortest paths with respect to a generalized travel cost. So, the problem can be seen in a simplistic form as find the equilibrium between tolls that minimize the population exposure to the risk and tolls that are convenient for the shippers. The paper applies a path based algorithm to solve a bi-level multi-commodity optimal toll setting 'hazmat' problem. The algorithm consists in find upper bounds for the tolls considering the total cost and the risk associated to a particular path. We made several experiments and the results are shown in this work.