Arc-Routing Models for Small-Package Local Routing

This paper studies the arc-routing problem that arises in small-package delivery. In practice, each service provider is encouraged to follow a master route---a predesigned sequence of street addresses---over an extended planning horizon (more than one day). The objective here is to construct efficient master routes. The focus on arc routing offers two advantages. First, real-world vehicle routing takes place over a street network, rather than in Euclidean space. Second, there are, typically, many fewer streets than customer locations. Currently, a deterministic arc-routing problem (DARP) model is used to solve the problem. However, this approach ignores the uncertainty in the street segment presence probability---the probability that a street segment requires (i.e., there is a demand for) a visit on a particular day. We introduce two new models, namely, the probabilistic arc-routing problem (PARP) model and the multiday arc-routing problem (MARP) model, which take into account the street segment presence probabilities. PARP attempts to minimize the expected length of the master route. It assumes that the street segment presence probabilities are independent. This model can require excessive amounts of computation time. On the other hand, MARP tries to minimize average length of the master route over prespecified days. This model can also be viewed as a Monte Carlo simulation approximation of the PARP. This approximation significantly reduces the computational burden. Additionally, by utilizing historical data, MARP incorporates real-world correlations among the street segment presence probabilities. Our computational results show that PARP and MARP may produce more efficient master routes than DARP by taking demand uncertainty into account.