A Stochastic and Dynamic Vehicle Routing Problem in the Euclidean Plane

We propose and analyze a generic mathematical model for dynamic, stochastic vehicle routing problems which we call the dynamic traveling repairman problem (DTRP). The model is motivated by applications in which the objective is minimizing the wait for service in a stochastic and dynamically changing environment. This is a departure from traditional vehicle routing problems which seek to minimize total travel time in a static, deterministic environment. Potential areas of application include repair, inventory, emergency service and scheduling models. The DTRP is defined as follows: Demands for service arrive according to a Poisson process in a Euclidean region A and, upon arrival, are independently and uniformly assigned a location in A. Each demand requires an independent and identically distributed service by a vehicle that travels at unit velocity. The problem is to find a policy that minimizes the average time a demand spends in the system. We propose several policies for the DTRP and analyze their behavior. Using approaches from queueing theory, geometrical probability, combinatorial optimization and'simulation, we find a provably optimal policy in light traffic and several policies that have system times within a constant factor of the optimal policy in heavy traffic. We also show that the waiting time grows much faster than in traditional queues as the traffic intensity increases, yet the stability condition does not depend on the system geometry. · Dimitris Bertsimas, Sloan School of Management, MIT, Rm E53-359, Cambridge, MA 02139. tGarrett van Ryzin, Operations Research Center, MIT, Cambridge, MA 02139. $The research of both authors was partially supported by the National Science Foundation under grant DDM-9014751 and by a grant from Draper Laboratory.