Sensitivity analysis and relaxation of the Static Traffic Assignment Problem with Capacity Constraints

This article introduces sensitivity analysis, reduction of the feasible set around the optimal solution, and LP and QP relaxations on convex, capacity-constrained network flow problems to evaluate the impact of a change in link capacity on the optimal flow allocation. This is done in the context of the static traffic assignment problem under capacity constraints [TAP-C], to understand the impact of traffic incidents on traffic flow in road networks, though the results also apply beyond transportation.The dual formulation of the convex [TAP-C] using gener-alized travel costs is exploited to show that dual variables associated to the link capacity constraints can be used to understand sensitivity of link flows to perturbation. To avoid expensive computations of the [TAP-C] optimal flow solution, the impact of a perturbation of link capacities on the optimal flow allocation is shown to be lower-bounded by a dual sensitivity analysis, and upper-bounded by LP and QP relaxations of the [TAP-C] on a reduced set of feasible flow allocations around the optimal solution.Simulations are conducted on a benchmark city network (the Sioux Falls network with 75 links) to assess the impact of an incident on the traffic flow allocation. When one road (2 links) is closed, the total vehicle hours of travel (VHT) doubles in the network. For this network, the computation of the solution through relaxation is shown to be 300 times faster than recomputing the [TAP-C].