Hybrid Genetic Algorithms for Scheduling Bus and Train Drivers

We introduce the subject of bus and train driver scheduling, and outline some early attempts at solutions based on heuristics, explaining their limitations. We then examine one successful blend of heuristics and integer linear programming leading to the TRACS II system which is in regular use in a wide range of transport organisations. Again we discuss the limitations of this system. In order to overcome these, we and colleagues have investigated a range of metaheuristics and constraint programming approaches, and some of these are outlined. Finally we present three different genetic algorithms, the last of which is successfully used to overcome the limitations of the established integer linear programming system. The proposed approach is a hybrid in which all probable potential shifts are generated according to well developed heuristics already used in TRACS II. Selection of such shifts to form a schedule is modeled as a set covering problem, and the relaxation of this problem ignoring integer conditions is solved to optimality. A GA then develops a solution schedule based on some of the characteristics of the relaxed solution. It is suggested that this approach might be suitable for other set covering problems. Introduction The problems of scheduling crews for buses or trains have been the subject of research for at least thirty years [1,2]. A major international workshop devoted to promoting research in this area was held in Chicago [3] in 1975, and this has been followed by six further conferences [4-9]. The eighth in the series takes place in Berlin in June 2000. Most of the published work until recently has related to bus drivers rather than train crews. This is due partly to the reluctance of monolithic train operating authorities to look outside the boundaries of their own industries for solutions to scheduling problems, and partly to the greater complexity of rail scheduling problems. However, in more recent years, rail crew scheduling has been tackled very successfully, for example by Kwan et al. [10]. In practice, bus driver scheduling and train crew scheduling are different aspects of the same problem. In both cases, staff workdays (shifts) are made up of spells of work on one or more vehicles, and all the vehicle work must be included in such shifts. While buses are usually in the hands of a single driver, train crews may be made up of different categories of staff, sometimes governed by different sets of rules. We shall write in terms of drivers throughout this paper, although generally the same processes may be used to construct schedules for other categories of staff. We shall use the term vehicle to denote a bus or a train. Although some long-distance train work extends over several days, by far the greatest amount consists of journeys which are completed within a 24-hour period, and which are staffed by shifts of up to about twelve hours in duration. In the United Kingdom the longest regularly scheduled passenger train journeys are just over 700 miles, and drivers are changed at several intermediate points, so that each starts and finishes their shift at their home depot. Most bus operations are much more localized. Even where bus and train operation continues overnight, it is usually sufficient for scheduling purposes to deal with periods of 24 hours. However, provision is made when necessary for late night and early morning work to be considered as either the end of one day or the beginning of