On the Dispatching Problem

1 Introduction On the Dispatching Problem Federico Di Miele Giorgio Gallo October 18, 2000 Most often, space is the scarce resource in Bus Depots located in congested urban areas: vehicles arriving at the end of their trips are packed together in a rather small space. That implies that, when a vehicle has to leave to start a new trip, most often, other vehicles have to be moved to clear the way. Here a new model, based on “Min Non Crossing Matching” and “Generalized Assignment”, is presented to allocate parking spaces to vehicles in a Depot in order to minimize the shunting cost. The results obtained applying our approach to a real life case are discussed. Among the many different problems which arise in the day-by-day operations of urban transportation companies, most of them leading to the formulation of challenging Mathematical Programming problems, one which, at least so far, has attracted rather little attention is the so called “Bus Dispatching Problem”. This is the problem of optimally managing the parking space in a vehicle depot. The dispatching problem is a trivial one when enough space exists in the depot, and parking slots are located in such a way that no parked vehicle can obstacle the movement of any arriving or leaving vehicle. Unfortunately this is not the case in many European towns where depots are small and quite crowded. As an example, in …g. 1 the layout of a real depot located in Florence is shown. Of course, also in this case the problem would be easy if all the vehicles were equal, but unfortunately this is not the case. Busses are different in size: small busses are required on the lines passing through the old center of many medieval cities, while large busses are needed in rush hours on heavy traffic lines. Some busses may have special lifting devices for disabled people, which are required on certain lines at given hours. Further, busses might carry ads on their sides, and the contracts with the advertisers determine the lines and the hours in which the vehicles should run. Clearly, in this situation, a poor assignment of vehicles to slots at arrival may lead to time consuming maneuvering at departure to clear the way out. 1 queue stack 2 The Problem Figure 1: A Depot In this paper, after a short description of the problem, we present the approach to its solution proposed by Winter and Zimmermann [1], which is based on a Quadratic Assignment model. This model is the only one we know which fully captures the main characteristics of our problem. Then we present a new model which has as main components a set of Generalized Assignment models and a set of NonCrossing Matching models. This new model, which is in our opinion the main contribution of this paper, is well suited for a decomposition approach. An algorithm of this type is presented together with some computational results on real life problems. An extension of our model to take in consideration the case in which arrivals and departure overlap is described before the …nal section. Here we give a formal description of the Bus Dispatching Problem (BDP). First, we assume the parking space in the depot to be partitioned into a given number of columns of known length; vehicles enter a column at one end and leave it from the other to service the scheduled vehicles duties. busses are parked in the columns one behind the other. Here we have assumed that the operations in the columns mimic the operations in a . Of course we could have chosen a -like type of operations; the changes to the models 2