An Operational Planning Model for the Dynamic Vehicle Allocation Problem with Uncertain Demands?

The dynamic vehicle allocation problem arises when a motor carrier must simultaneously and in real time coordinate the dispatching of vehicles from one region to the next all over the country. The decision to send a vehicle loaded or empty from one region to the next, arriving in the destination region at some point in the future, must anticipate the downstream impacts of the decision. The consequences of another load in a particuiar region at some point in the future, however, are highly uncertain. A simple methodology is proposed which calculates approximately the marginal value of an additional vehicle in each region in the future. This information is then used to generate a standard pure network which can be efficiently optimized to give dispatching decisions for today. The dynamic vehicle allocation problem arises when a carrier must manage the positioning of a fleet of vehicles over time. The application motivating this paper arises from truckload motor carriers where a shipper might request a trailer to carry a load of freight from one city to the next on a given day. The shipper pays for the entire tractor-trailer combination and there is no issue of consolidating shipments on one trailer. The problem the carrier faces is one of anticipating the needs of shippers by ensuring that trailers are in the right place at the right time. The principal difficulty is that a carrier typically has very little advance notice regarding future needs of a shipper. At the beginning of a day, a truckload carrier might only know 40% of the loads that will be carried that day (the remaining 60% are called in during the morning and early afternoon) and less than 20% of the loads that need to be carried the following day. Just the same, the carrier must be able to anticipate the future cost and revenue implications of decisions made now in order to make intelligent decisions regarding which loads the carrier should accept and where to move empty trailers. The vehicle allocation problem was originally formulated as a transportation problem by considering current supplies of and demands for vehicles, and ignoring downstream impacts of decisions (see, for example, Misra, 1972). White and Bomberault (1969) and White (1972) indicated how the problem might be approached as a dynamic problem by setting up a network where each node represented a particular region at a particular point in time. Future demands for vehicles between regions would be forecasted over a specified planning horizon. If each forecast is assumed to be known with certainty, the problem reduces to a simple network transshipment problem. Such an approach is currently in use at one of the major truckload motor carrier (Crowe, 1983). A major drawback of the deterministic model, however, is that the forecasts must be integer in order to guarantee an integer solution. In many instances, and at almost every truckload motor carrier, demand forecasts are fractions in the range .05 to .5, necessitating the use of heuristic rounding rules. Recognizing the frequently large uncertainty associated with the demand forecasts, Powell er al. ( 1984) reformulated the problem as a nonlinear network by representing forecasts as random variables with known means and variances. The decision variables are still the number of vehicles moving between pairs of regions over time, but now the number of vehicles moving full is random, and the expected number of full vehicles, and hence the expected net revenue generated between every pair of regions, becomes a nonlinear function of the flow. A key assumption made that significantly simplifies the problem is that the number of vehicles moving from region i to region j on a given day is determined before the number of loads available is I’This research was supported in part by the National Science Foundation under Grant ECE-8408044.