The Dynamic Assignment Problem

There has been considerable recent interest in the dynamic vehicle routing problem, but the complexities of this problem class have generally restricted research to myopic models. In this paper, we address the simpler dynamic assignment problem, where a resource (container, vehicle or driver) can only serve one task at a time. We propose a very general class of dynamic assignment models, and propose an adaptive, nonmyopic algorithm that involves iteratively solving sequences of assignment problems no larger than what would be required of a myopic model. We consider problems where the attribute space of resources and tasks in the future is small enough to be enumerated, and propose a hierarchical aggregation strategy for problems where the attribute spaces are too large to be enumerated. Finally, we use the formulation to also test the value of advance information, which offers a more realistic estimate over studies which use purely myopic models. The problem of dynamically assigning resources to tasks over time arises in a number of applications in transportation. In freight transportation, truckload motor carriers, railroads and shipping companies all have to manage fleets of containers (trucks, boxcars and intermodal containers) which move one load at a time, with orders arriving continuously over time. In the passenger arena, taxi companies and companies that manage fleets of business jets have to assign vehicles (taxicabs or jets) to move customers from one location to the next. It is common to assume that the arrival of customer demands is random (e.g., known only through a probability distribution) over time, but it may also be the case that the vehicles also become available in a random way. Finally, each assignment of a resource to a task generates a contribution to profits which may also be random. We refer to the problem of dynamically assigning resources to tasks as a dynamic assignment problem. In general, it may be possible to assign a resource to a sequence of two or more tasks at the same time, but we focus on problems where we assign a resource to one task at a time. We assume that resources and tasks are each characterized by a set of attributes (that may be unique), where the contribution generated by an assignment will depend on the attributes of the resource and task. Resources do not have to be used and tasks do not all have to be covererd (although there can be a cost for holding either one). The dynamic assignment problem is a fundamental problem in routing and scheduling. It is a special case of the dynamic vehicle routing problem, without the complexities of in-vehicle consolidation. For this reason, it provides a natural framework for modeling the dynamic information processes and comparing myopic models with those which exploit distributional information about the future. It is common practice, for example, to model dynamic vehicle routing problems using myopic models (which ignore any forecasts of the future based on currently available data). These problems are themselves quite difficult because it is necessary to solve vehicle routing problems (where static versions are already quite difficult) very quickly to respond to new information. In our problem, a static instance of an assignment problem is quite easy, allowing us to focus on the challenge of modeling the informational dynamics more carefully, and to study policies which consider the impact of decisions now on the future.