Introduction Giorgio Gallo Federico Di Miele Dispatching Buses in Parking Depots

Introduction Giorgio Gallo Federico Di Miele Dispatching Buses in Parking Depots Department of Computer Science, University of Pisa, Corso Italia 40, 56125 Pisa, Italy gallo@di.unipi. it Among the many different problems arising in the day-by-day operations of urban transportation companies, the problem of optimally managing the parking space in a vehicle depot has attracted little attention. This problem 1 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. This 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 this approach to a real life case are discussed. dispactching problem is particularly relevant in congested urban areas, where space is a scarce resource: vehicles arriving at the end of their trips are packed together in a rather small space, which 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. The problem of optimally allocating arriving vehicles to the parking spaces in bus depots will be referred to as the . The dispatching problem is trivial when sufficient space exists in the depot, and parking slots are located in such a way that no parked vehicle can obstruct 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. Even in this case the problem would be easy if all the vehicles were equal, but unfortunately this is not so. Buses are different in size: small buses are required on the lines passing through old city centers, while large buses are needed in rush hours on heavy traffic lines. Some buses may have special lifting devices for disabled people, which are required on certain lines at given hours. Further, buses 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. In this paper, after a short description of the problem, we present the solution approach proposed by Winter and Zimmermann (2000), based on a 2 1 The problem Quadratic Assignment model. Their model is the only one we know in the literature that captures the main characteristics of our problem. Then we present a new model which, unlike that of Winter and Zimmermann, takes into account the fact that the vehicles have different lengths. This allows a more exible use of the available parking space, without the need of it being partitioned into slots of equal size as in Winter and Zimmermanns model. This new model has as main components a set of Generalized Assignment models and a set of Non-Crossing Matching models, and is well suited for a decomposition approach. A decomposition heuristic algorithm is presented together with some computational results on real life problems. An extension of our model that considers the case in which arrivals and departure overlap is also presented. The paper is organized as follows. Section 1 contains a brief description of the problem. In Section 2 the model by Winter and Zimmermann is described. Sections 3 and 4 contain the decription of our model and of the decomposition approach. In Section 5 the algorithm together with some computational results are presented. Section 6 contains a relevant extension of the model. Final remarks follow in the last section. Here we give a formal description of the Bus Dispatching Problem (BD). First, we assume that the parking space in the depot is partitioned into a 3