Application Of Graph Theory And Linear Algebra In Railway Planning
Railway planning uses graph theory and linear algebra in:
- Network planning – deciding the connections of stations and different tracks.
- Line planning – determining the route of lines, origin/destination termini and their frequent usages so as to provide sufficient capacity for the passenger trips to be served.
- Timetabling – determining the despatch times of trains for each line; ensuring minimal waiting times and transit times for all passengers is particularly difficult.
- Vehicle scheduling – assignment of locomotives and coaches to trips
- Crew scheduling – rosters and duties for the drivers are determined, work rules and skills compatibilities make the assignment of operators to trips a difficult problem.
Network and line planning
Line planning is planning with the existing railway networks and still fulfil changing travel demands and to reorganise the routes of the trains. In every railway network, each possible path is a possible line. Generally, there are too many paths to consider, so only a subset of such paths are candidates for lines. The line-planning problem is to choose the frequency for each line. The capacity of the line is given by where is the capacity of the train used on the line. The set of lines and frequencies should be chosen so the anticipated passenger demand is covered. The main objective of line planning is to maximise the number of direct trips for all passengers because passengers would not prefer to change trains during their trips.
Timetabling
Given the set of lines and overall frequencies, the timetabling problem determines the exact time of despatch from each terminus for each line. For regional trains of lower frequency, the preference of passengers is for periodic timetables (easier to remember), although often the frequencies are different for rush hour and/or weekends. One useful mathematical model for designing periodic schedules is the periodic event scheduling problem (PESP) introduced by Serafini and Ukovich. It can be used to model key aspects of railway operations, such as the sequence of arrivals/departures at stations of each line, matching timings for interchanges between lines, safety distances between trains, and sharing of tracks among lines. For high-frequency inner-city metro lines, aperiodic timetables are sometimes used and can be tuned to offer higher service levels than periodic timetables. Most urban transit rail networks are complex and passengers may need to make several interchanges between lines to reach their destination. Balancing the arrival/departure times of trains at all the interchange stations to try to minimise the waiting times for all interchanging passengers is a very difficult task. Holding back one train at a station to accommodate connecting passengers may lead to other passengers missing their connections further down the line.
Equipment and crew scheduling
A given timetable can be considered as a number of trips (each with a given originating location and time, given duration and ending location and time). The determination of which equipment (locomotives and coaches) and crew to operate these trips is the equipment and crew scheduling problem. It can be modelled using graph. Since the locomotives and trains are expensive equipment, a typical objective is to minimise the number of trains needed to operate the given timetable. Thus the problem is to find the minimum number of Start –Terminate paths such that every trip node is incident to exactly one path. After the train configurations have been assigned, crew must be assigned to the operation of trips. The framework for assignment of crew to trips is similar to that of equipment scheduling, but there are many more constraints. There are many labour rules to be considered regarding shift lengths, meal breaks and rest breaks, required days off during a week/month, rest period after night-shift work, etc. Unlike trains/locomotives that need not return to the depot until many days later, it is desirable that a crew start and end each day in the same depot; this means that the trips obtained from the equipment scheduling problem might be split into segments to allow for a change of crew between segments. Moreover, not all crew have the skills to operate all types of trains. This often leads to an explosion in the size and complexity of the problem. The crew scheduling problem is typically solved as a set-covering problem using a column-generation approach, where each column represents a feasible duty for the crew, and each row (constraint) represents a trip to be covered. A duty is a sequence of trips worked by the crew that respects the skills compatibility of the crew and equipment, and obeys labour rules.
Disruption management
Vehicles break down, crew may be absent, signalling or power systems malfunction, and weather and traffic conditions also cause delays. Recovery of the system after a disruption is a key aspect of railway operations. In case of a disruption where a part of the railway network becomes inoperable, decisions and adjustments have to be made in real time, often with limited information as to the extent and duration of the disruption. If a part of the tracks are blocked or inoperable, some lines would need to be re-routed or cut short, trains may be cancelled and/or the timetable adjusted. Vehicles and crew will need to be redeployed to operate the trips for the adjusted timetable.
Passengers react to news of the disruption by choosing alternative routes or modes of transport, thus estimating the demand for the disrupted network and ensuring sufficient capacity for the modified demand is particularly difficult. A key aspect of the disruption management of railway systems is the real-time reallocation of tracks to accommodate the modified timetable and routings of the trains. The carrying capacity of the system is reduced, yet safety distances and time gaps must be maintained. Both passengers and crew have to make modifications to their habitual routine, often leading to confusion and further dela
