Online Routing with Nonlinear Disutility Functions with Arc Cost Dependencies

The importance of routing and shortest path problems in the study of transportation networks is well-known. For instance, these are used to describe user behavior, to find equilibrium states of large-scale networks, and to evaluate the impact of network improvements and broader transportation policies. Thus, the development of efficient, behaviorally realistic algorithms is of paramount interest. Although research in this area dates back over a half-century, continued refinements are being made to account for the specific nature of congested transportation systems. In particular, researchers have realized that accounting for the fundamentally uncertain day-to-day nature of transportation networks is crucial for correctly modeling travelers’ preferences. This operational uncertainty can be traced to a number of phenomena, both on the demand side (such as daily fluctuations in travel patterns), as well as the supply side (such as capacity reductions due to adverse weather or incidents). In turn, the presence of stochasticity affects user behavior in multiple ways. If the travel time (or cost) along a route cannot be predicted absolutely, the assumption that travelers choose the least-cost path is illdefined. Indeed, it may not even be enough to simply say that travelers choose the path with least average cost; rather, depending on the trip type and the traveler’s attitudes, one might expect some degree of risk aversion or other, more complicated preferences regarding trip costs. Additionally, uncertain conditions require assumptions to be made about the information or beliefs that travelers have regarding network conditions. The advent of real-time traffic information provision makes this question especially relevant, as travelers can change their chosen path en route in response to new information, a phenomenon known as recourse. Within this context, Boyles [2006] developed an algorithm for one specific type of information and cost dependency structure, but a number of alternate structure exist as well. Thus, the intent of this paper is to develop additional algorithms that simultaneously account for general nonlinear user preferences and recourse decisions in the presence of various assumptions on cost structure and information. To simplify the presentation, we will assume that users are only concerned with travel time, although extending these algorithms to account for any additive, nonnegative travel attributes is straightforward. These travel times are both dynamic and stochastic, and exhibit limited forms of dependency. That link travel times are dependent is natural, although introducing this mathematically leads to additional complications. To simplify, we consider three specific types of dependence and develop a shortest path algorithm for each, using the nomenclature from Waller and Ziliaskopoulos [2002].