The price of discretizing time: a study in service network design

Researchers and practitioners have long recognized that many transportation problems can be naturally and conveniently modeled using time-expanded networks. In such models, nodes represent locations at distinct points in time and arcs represent possible actions, e.g., moving from one location to another at a particular point of time, or staying in the same location for a period of time. To use a time-expanded network, time must be discretized, i.e., the planning horizon is partitioned into discrete time intervals. The length of these intervals, therefore, must be chosen, and the parameters involving time, e.g., travel duration and due times, need to be mapped to these discrete intervals. Short intervals yield a high-quality approximation to the continuous-time problem, but typically induce a computationally intractable model; whereas long intervals can yield a computationally tractable, but low-quality model. The loss of quality is due to the approximation introduced by the mapping of parameters involving time. To guide researchers and practitioners in their use of time-expanded networks, we explore the choice of time discretization and its impact, by means of an extensive computational study on the service network design problem. The empirical results show that in some cases the loss of quality, i.e., the relative gap between the discretized and continuous-time optimal values, can be greater than 20%. We also investigate metrics that characterize and help identify instances that are likely to be sensitive to discretization and could incur a large loss of solution quality.