## Date of Award

May 2014

## Degree Type

Dissertation

## Degree Name

Doctor of Philosophy

## Department

Engineering

## First Advisor

Matthew E. H. Petering

## Committee Members

Jaejin Jang, Wilkistar Otieno, Alan Horowitz, Gabriella Pinter

## Keywords

Bus Routing, Cyclic Timetabling, Integer Programming, Railway Systems, Train Platforming, Train Timetabling

## Abstract

Public transportation or mass transit involves the movement of large numbers of people between a given numbers of locations. The services provided by this system can be classified into three groups: (i) short haul: a low-speed service within small areas with high population; (ii) city transit: transporting people within a city; and (iii) long haul: a service with long trips, few stops, and high speed (Khisty and Lall, 2003). It can be also classified based on local and express services. The public transportation planning includes five consecutive steps: (i) the network design and route design; (ii) the setting frequencies or line plan; (iii) the timetabling; (iv) the vehicle scheduling; and (v) the crew scheduling and rostering (Guihaire and Hao, 2008; Schöbel, 2012).

The first part of this dissertation considers three problems in passenger railway transportation. It has been observed that the demand for rail travel has grown rapidly over the last decades and it is expected that the growth continues in the future. High quality railway services are needed to accommodate increasing numbers of passengers and goods. This is one of the key factors for economic growth. The high costs of railway infrastructure ask for an increased utilization of the existing infrastructure. Attractive railway services can only be offered with more reliable rolling stock and a more reliable infrastructure. However, to keep a high quality standard of operations, smarter methods of timetable construction are indispensable, since existing methods have major shortcomings.

The first part of this dissertation, comprising Chapters 1-6, aims at developing a cyclic (or periodic) timetable for a passenger railway system. Three different scenarios are considered and three mixed integer linear programs, combined with heuristics for calculating upper and lower bounds on the optimal value for each scenario, will be developed. The reason of considering a periodic timetable is that it is easy to remember for passengers. The main inputs are the line plan and travel time between and minimum dwell time at each station. The output of each model is an optimal periodic timetable.

We try to optimize the quality of service for the railway system by minimizing the length of cycle by which trains are dispatched from their origin. Hence, we consider the cycle length as the primary objective function. Since minimizing travel time is a key factor in measuring service quality, another criterion--total dwell time of the trains--is considered and added to the objective function.

The first problem, presented in Chapter 3, has already been published in a scholarly journal (Heydar et al., 2013). This chapter is an extension of the work of Bergmann (1975) and is the simplest part of this research. In this problem, we consider a single-track unidirectional railway line between two major stations with a number of stations in between. Two train types--express and local--are dispatched from the first station in an alternate fashion. The express train stops at no intermediate station, while the local train should make a stop at every intermediate station for a minimum amount of dwell time. A mixed integer linear program is developed in order to minimize the length of the dispatching cycle and minimize the total dwell time of the local train at all stations combined. Constraints include a minimum dwell time for the local train at each station, a maximum total dwell time for the local train, and headway considerations on the main line an in stations. Hundreds of randomly generated problem instances with up to 70 stations are considered and solved to optimality in a reasonable amount of time. Instances of this problem typically have multiple optimal solutions, so we develop a procedure for finding all optimal solutions of this problem.

In the second problem, presented in Chapter 4, we present the literature's first mixed integer linear programming model of a cyclic, combined train timetabling and platforming problem which is an extension of the model presented in Chapter 3 and Heydar et al. (2013). The work on this problem has been submitted to a leading transportation journal (Petering et al., 2012). From another perspective, this work can be seen as investigating the capacity of a single track, unidirectional rail line that adheres to a cyclic timetable. In this problem, a set of intermediate stations lies between an origin and destination with one or more parallel sidings at each station. A total of T train types--each with a given starting and finishing point and a set of known intermediate station stops--are dispatched from their respective starting points in cyclic fashion, with one train of each type dispatched per cycle. A mixed integer linear program is developed in order to schedule the train arrivals and departures at the stations and assign trains to tracks (platforms) in the stations so as to minimize the length of the dispatching cycle and/or minimize the total stopping (dwell) time of all train types at all stations combined. Constraints include a minimum dwell time for each train type in each of the stations in which it stops, a maximum total dwell time for each train type, and headway considerations on the main line and on the tracks in the stations. This problem belongs to the class of NP-hard problems. Hundreds of randomly generated and real-world problem instances with 4-35 intermediate stations and 2-11 train types are considered and solved to optimality in a reasonable amount of time using IBM ILOG CPLEX.

Chapter 5 expands upon the work in Chapter 4. Here, we present a mixed integer linear program for cyclic train timetabling and routing on a single track, bi-directional rail line. There are T train types and one train of each type is dispatched per cycle. The decisions include the sequencing of the train types on the main line and the assignment of train types to station platforms. Two conflicting objectives--(1) minimizing cycle length (primary objective) and (2) minimizing total train journey time (secondary objective)--are combined into a single weighted sum objective to generate Pareto optimal solutions. Constraints include a minimum stopping time for each train type in each station, a maximum allowed journey time for each train type, and a minimum headway on the main line and on platforms in stations. The MILP considers five aspects of the railway system: (1) bi-directional train travel between stations, (2) trains moving at different speeds on the main line, (3) trains having the option to stop at stations even if they are not required to, (4) more than one siding or platform at a station, and (5) any number of train types. In order to solve large scale instances, various heuristics and exact methods are employed for computing secondary parameters and for finding lower and upper bounds on the primary objective. These heuristics and exact methods are combined with the math model to allow CPLEX 12.4 to find optimal solutions to large problem instances in a reasonable amount of time. The results show that it is sometimes necessary for (1) a train type to stop at a station where stopping is not required or (2) a train type to travel slower than its normal speed in order to minimize timetable cycle time.

In the second part of this dissertation, comprising Chapters 7-9, we study a transit-based evacuation problem which is an extension of bus routing problem. This work has been already submitted to a leading transportation journal (Heydar et al., 2014). This paper presents a mathematical model to plan emergencies in a highly populated urban zone where a certain numbers of pedestrians depend on transit for evacuation. The proposed model features a two-level operational framework. The first level operation guides evacuees through urban streets and crosswalks (referred to as "the pedestrian network") to designated pick-up points (e.g., bus stops), and the second level operation properly dispatches and routes a fleet of buses at different depots to those pick-up points and transports evacuees to their destinations or safe places. In this level, the buses are routed through the so-called "vehicular network." An integrated mixed integer linear program that can effectively take into account the interactions between the aforementioned two networks is formulated to find the maximal evacuation efficiency in the two networks. Since the large instances of the proposed model are mathematically difficult to solve to optimality, a two-stage heuristic is developed to solve larger instances of the model. Over one hundred numerical examples and runs solved by the heuristic illustrate the effectiveness of the proposed solution method in handling large-scale real-world instances.

## Recommended Citation

Heydar, Mojtaba, "Operations Research Modeling of Cyclic Train Timetabling, Cyclic Train Platforming, and Bus Routing Problems" (2014). *Theses and Dissertations*. 694.

https://dc.uwm.edu/etd/694