The Dynamic Traveling Repairman Problem

We propose and analyze a generic mathematical model for dynamic 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 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.