Transportation Resource Management

In this chapter, we describe a variety of modeling and solution approaches for transportation resource management problems. The solution approaches that we propose either build on deterministic linear programming formulations, or formulate the problem as a dynamic program and use tractable approximations to the value functions. We describe two classes of methods to construct approximations to the value functions. The first class of methods relax certain constraints in the dynamic programming formulation of the problem by associating Lagrange multipliers with them, in which case, we can solve the relaxed dynamic program by concentrating on one location at a time. The second class of methods use a stochastic approximation idea along with sampled trajectories of the system to iteratively update and improve the value function approximations. After describing several solution approaches, we provide a flexible paradigm that is useful for modeling and communicating complex resource management problems. Our numerical experiments demonstrate the benefits from using models that explicitly address the randomness in resource management problems. Many problems in transportation logistics involve managing a set of resources to satisfy the service requests that arrive randomly over time. For example, truckload carriers decide which load requests they should accept and to which locations they should reposition the empty vehicles. Railroad companies deploy the empty railcars to different stations in the network to serve the random customer demand in the most effective manner. Irrespective of the application setting, transportation resource management problems pose significant challenges. To begin with, these problems tend to be large in scale, spanning transportation networks with hundreds of locations and planning horizons with hundreds of time periods. Furthermore, the service requests arrive randomly, and we may not have full information about the future service requests. When making the decisions for the current time period, we need to keep a balance between maximizing the immediate profits and getting the vehicles into favorable positions to serve the potential future service requests. Finally, the decisions that we make for different locations and for different time periods interact in a nontrivial fashion. For example, the decision to serve a particular load from location i to j at time period t affects what other loads we can serve at the future time periods, since the vehicle that is used to serve the load at time period t becomes available at location j at a future time period. In this chapter, we describe a variety of modeling and algorithmic approaches for transportation resource management problems under uncertainty. We begin with a problem that takes place over two time stages. At the first stage, we decide to which locations we should reposition the resources to serve the demands that arrive later. At the second stage, we observe the demand arrivals and decide which of these demands we should serve. The first algorithmic strategy that we discuss for the twostage problem is based on a deterministic linear programming formulation. This formulation is simple to implement, but it does not explicitly address the randomness in the demand arrivals. Motivated by this observation, we move on to other algorithmic strategies that address the randomness in the demand arrivals by using a dynamic programming formulation. The dynamic programming formulation of the two-stage problem turns out to be computationally difficult as it involves a high dimensional state variable, and we resort to approximation strategies. In particular, the first two approximation strategies that we describe decompose the dynamic programming formulation by the locations and obtain good solutions by focusing on one location at a time. We also discuss a third approximation strategy that constructs separable value function approximations by using sampled trajectories of the system. After the discussion of two-stage problems, we move on to resource management problems that take place over multiple time periods. We give a dynamic programming formulation for a multi-stage resource management problem and demonstrate that the algorithmic concepts that we develop for two-stage problems can be extended to multi-stage problems without too much difficulty. In addition to providing algorithmic tools for two-stage and multi-stage problems, we also dwell on modeling issues. Most transportation resource management problems involve complex resources. For example, a driver resource in a driver scheduling application may require keeping track of its inbound location, time to reach the inbound location, duty time within the shift, days away from home, vehicle type and home domicile. It would be cumbersome to model such complex entities through traditional mathematical programming tools, and we provide a flexible paradigm that is useful for modeling and communicating complex resource management problems. We conclude the chapter with a series of