Dynamic Control of Logistics Queueing Networks for Large-Scale Fleet Management

Dynamic eet management problems are normally formulated as networks over dynamic networks. Additional realism usually implies the inclusion of complicating constraints, typically producing exceptionally large integer programs. In this paper, we present for the rst time the formulation of dynamic eet management problems in an optimal control setting, using a novel formulation called a Logistics Queueing Network (LQN). This formulation replaces a single, large optimization problem with a series of very small problems that involve little more than solving a single sort at each point in space and time. We show that this approach can produce solutions that are within a few percent of a global optimum, but providing for considerably more exibility than standard linear programs. We consider the problem of managing a homogeneous eet of vehicles over time to serve a set of loads, each with a known origin and destination, and a speci ed time window in which they must be served. The problem has been widely studied. The most common formulation is a dynamic network with three types of arcs: revenue generating arcs representing the market demand between two cities at a point in time, empty repositioning arcs, which represent the movement of capacity from one location to the next at a point in time (at a positive cost), and inventory arcs, capturing the cost of holding capacity at the same location over time. The rst published formulation of this problem as a linear network appears to be by White & Bomberault (1969) and White (1972), although the basic formulation had been well known prior to this time (see, for example, Dantzig & Fulkerson (1954)). The work of White and Bomberault focused on presenting specialized algorithms for this model, and said little about the model itself. Magnanti & Simpson (1978), in an unpublished technical report, give a series of linear programming models, extending the basic dynamic network formulation to handle multiple eet types and time windows for task arcs. An extensive review of these models can be found in Powell, Jaillet & Odoni (1995a). The formulation and solution of dynamic networks for eet assignment models has received attention in an airline context, but has not proved e ective in large scale eet management problems arising in rail, containers and trucking. These problems involve the routing of tens of thousands of vehicles, serving thousands of tasks per day. The di culty is that tasks (representing the movement of freight over space and time) can be moved within time windows that can range from very narrow to exceptionally wide. We are not aware of any research demonstrating the feasibility of linear programming-based models for these large eet assignment problems with time windows. Jordan & Turnquist (1983) approach the eet management problem for rail as a nonlinear inventory distribution problem for empty freight cars, and allow for backlogging of unsatis ed demands. Powell (1988), Frantzeskakis & Powell (1990), and Powell & Cheung (1994), show how the dynamic eet management problem can be modi ed to handle uncertainty in demand forecasts, but these models require demands to be served in a speci c time period. Powell (1996) shows how eet management problems with time windows can be solved on a rolling-horizon basis, but does not explicitly solve the eet assignment problem with time windows. Recently, Powell, Carvalho, Godfrey & Simao (1995b) presented a new formulation of the eet management problem, called a logistics queueing network. The solution approach starts with the