An important element in creating a train timetable is the design of train services to meet a known travel demand within some fixed time interval. Demand is defined by the number of people who wish to travel between each pair of stations. The set of all possible origin-destination (OD) demands is described by an OD matrix (Borndorfer et al, 2004).
A key feature of each train service is the list of stations at which the train is required to stop. This is called a stopping pattern. Any collection of train services that meets the known demand can be regarded as a solution to the design problem. The solution may include many different stopping patterns. The best solutions could be those that maximise the number of express services, or those that minimise travel time, the number of passenger transfers or the number of unnecessary stops; or it could be those that minimise line operating costs (Borndorfer et al, 2004). At a higher level is the issue of how an OD matrix is obtained in the first place. For a rail network this may be a non-trivial task. Household surveys are costly and labour-intensive, and so indirect estimation methods based on traffic counts have been developed. See Cascetta & Postorino (2001) for an overview of methods. Although these methods require more complex mathematics, the data collection problem is convenient and relatively inexpensive.
We consider two related problems, which we now state informally.
Problem 1: For a given OD matrix and train services with prescribed stopping patterns and given capacities, find the number of services for each stopping pattern that best meets the demand.
Problem 2: On the basis of a collection of observed traffic counts, find the best estimate for the OD matrix.
The key results in this paper are Lemmas 1 and 2 in section 2.2, and the statement and solution of Problem 1 in section 2.4 using an integer program. It is equally important to understand the contextual relevance of our results. In this regard our assumption that demand is known underlines the estimation of demand as a critical practical problem for transport planners. The methods proposed by Van Zuylen & Willumsen (1980), which we describe in sections 3.4 and 3.5, are regarded as fundamental to the problem of demand estimation. For any serious discussion of Problem 1, we believe it is essential to emphasise the intrinsic connection to Problem 2 and to understand the mathematical implications.
The Australian Mathematical Sciences Institute (AMSI) industry internship project provides funding for a graduate or postdoctoral student to work on an industrial mathematics problem under the supervision of an academic mentor and an industry advisor. The problems discussed in this paper arose during an AMSI internship at the University of South Australia with industry partner TTG Transportation Technology, which provides technological support and consulting services to the rail industry.
In an undergraduate context, the problems described here are well-suited as a basis of a senior level project-based mathematics course, such as the Mathematics Clinic program conducted at Harvey Mudd College (HMC) in California (see Spanier (1977) for a description). A similar course modelled on the HMC program is run at the University of South Australia. These programs are designed around industrial problems that require students to build research skills and develop real-world technical experience. Projects are student-driven with the instructor acting as facilitator and advisor. The instructor may use the train service design problem to drive and motivate the study of deterministic mathematical programming. The demand estimation problem requires the study of stochastic optimisation.
2 SATISFYING DEMAND
An OD matrix specifies the number of people wishing to travel within a given time interval between each OD pair of a traffic network. …