Routing Optimization with Deadlines under Uncertainty

We consider a class of routing optimization problems on networks with deadlines imposed at a subset of nodes, and with uncertain arc travel times. The problems are static in the sense that routing decisions are made prior to the realization of uncertain travel times. The goal is to find optimal routing policies such that arrival times at nodes respect deadlines “as much as possible”. We propose a precise mathematical framework for defining and solving such routing problems. We first introduce a performance measure, called lateness index, to evaluate the deadline violation level of a given policy for the network with multiple deadlines. The criterion can handle risk, where probability distributions of the travel times are considered known, and ambiguity, where these distributions are partially characterized through descriptive statistics, such as means and bounded supports. We show that for the special case in which there is only one node with a deadline requirement, the corresponding shortest path problem with deadline can be solved in polynomial time under the assumption of stochastic independence between arc travel times. For the general case, we provide two mathematical programming formulations: a linear decision rule formulation, and a multi-commodity flow formulation. We develop practically “efficient” algorithms involving Lagrangian relaxation and Benders decomposition to find the exact optimal routing policy, and give numerical results from several computational studies, showing the attractive performance of lateness index policies, and the practicality associated with the computation time of the solution methodology.