Advanced Modeling for Transit Operations and Service Planning

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ADVANCED MODELING FOR TRANSIT OPERATIONS AND SERVICE PLANNING

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ADVANCED MODELING FOR TRANSIT OPERATIONS AND SERVICE PLANNING

edited by Professor William H.K. Lam Department of Civil and Structural Engineering The Hong Kong Polytechnic University Hong Kong, China

and

Professor Michael G.H. Bell Centre for Transport Studies Imperial College London United Kingdom

2003

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CONTENTS Preface

Chapter 1 - Initial Planning for Urban Transit Systems S.C. Wirasinghe

vii

1

Chapter 2 - Public Transport Timetabling and Vehicle Scheduling Avishai Ceder

31

Chapter 3 - Designing Public Transport Network and Routes Avishai Ceder

59

Chapter 4 - Transit Path Choice and Assignment Model Approaches Agostino Nuzzolo

93

Chapter 5 - Schedule-Based Transit Assignment Models Agostino Nuzzolo

125

Chapter 6 - Frequency Based Transit Route Choice Models Michael Florian

165

Chapter 7 - Capacity Constrained Transit Assignment Models and Reliability Analysis Michael G.H. Bell

181

Chapter 8 - Dynasmart-IP: Dynamic Traffic Assignment Meso-Simulator for Intermodal Networks Hani S. Mahmassani and Khaled F. Abdelghany

201

Chapter 9 - Modeling Competitive Multi-Modal Services HongK. Lo, C W. YipandK.H. Wan

231

Chapter 10 - Modeling Urban Taxi Services: A Literature Survey and An Analytical Example Hai Yang, Min Ye, Wilson H. Tang and S C. Wong

257

Chapter 11 - The Estimation of Origin-Destination Matrices in Transit Networks S.C. Wong and C.O. Tong

287

Chapter 12 - Models for Optimizing Transit Fares Jing Zhou and William H.K. Lam

315

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PREFACE The idea for this book grew out of the organization of the Advanced Study Institute (ASI), which was sponsored by the Croucher Foundation (http://www.croucher.org.hk/) for the dissemination of knowledge and the formation of international scientific contacts on advances in modelling transit systems. While public transport (or transit) systems have arguably been in existence much longer than road traffic systems, the mathematical analysis techniques so necessary for the proper planning of transit operations have lagged far behind those for road traffic systems. For example, the body of literature available on the design of schedules for urban rail lines is miniscule in comparison to the literature on the coordination of traffic signals along an urban road. On the other hand, transit professionals appear to have disregarded most of the wealth of insights that have been available in the literature for more than a decade. The literature on transit assignment is a good example. However, public transport operators, particularly in Hong Kong and Asia, are facing ever-greater pressure in competitive markets and transit systems are congested. The need to estimate passenger demand, to monitor the performance of individual services as well as the system as a whole, to support better planning and tighter operations management, and for external reporting has increased. The optimization of transit line frequencies and transit fares has become very important for operations and service planning. Reliability and control issues are also critical in making transit systems more efficient, supported by the introduction of Intelligent Transport Systems (ITS). As tightening constraints raise serous questions about the cost-effectiveness of existing public transport services, improvements which can be implemented in the short and long term are continuously sought. Collectively, these pressures have focused attention on advanced methods and new techniques for improving transit planning and operations. In Hong Kong and other major cities in Asia, over 90% of people are using transit facilities for their daily travel. The recent rapid development and deployment of ITS makes it possible to improve the efficiency of transit operations. This book addresses the important and timely problems of how to improve transit operations and service planning by making use of new technologies and advanced modeling techniques. It will provide important references for determining the outcomes of introducing these technologies and methods, and thus assist transit professionals and scientists in resolving practical issues arising from the implementation of ITS. This book appears to be the first devoted exclusively to the topic of advanced modeling for transit operation and service planning. This book consists of 12 chapters chosen to represent the broad base of contemporary themes in modeling transit systems. Scholars from America, Europe and Asia have contributed their knowledge to produce a unique compilation of recent developments in the field. Topics both in theory and innovative applications to real world problems are included. The book covers Transit Planning and Network Design, Transit Assignment Models and Solution Algorithms, Simulation of Passenger Behaviors, Effects of ITS on Passenger Choices and Transit Service Improvements, Modeling Multi-modal Transit and Urban Taxi Services. Outline of the book contents: Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6

- Initial Planning for Urban Transit Systems - Public Transport Timetabling and Vehicle Scheduling - Designing Public Transport Network and Routes - Transit Path Choice and Assignment Model Approaches - Schedule-Based Transit Assignment Models - Frequency Based Transit Route Choice Models

Chapter 7 - Capacity Constrained Transit Assignment Models and Reliability Analysis Chapter 8 - Dynasmart-IP: Dynamic Traffic Assignment Meso-Simulator for Intermodal Networks Chapter 9 - Modeling Competitive Multi-Modal Services Chapter 10 - Modeling Urban Taxi Services: A Literature Survey and an Analytical Example Chapter 11 - The Estimation of Origin-Destination Matrices in Transit Networks Chapter 12 - Models for Optimizing Transit Fares Special appreciation is extended to Elsevier Science Ltd. who made possible the publication of all the contributions in the form of the present book in time to be available to participants attending the ASI workshop from 9th to 13th December 2002 in Hong Kong. Professor Mike Bell of Imperial College of Science, Technology & Medicine (U.K.) provided valuable oversight and guidance in enhancing the quality of the book. His support during this effort has been remarkable. Finally, I am thankful for the patience, availability, and dedication of the editorial staff at Elsevier Science Ltd., particularly Julie Neden and Chris Pringle.

William H.K. Lam Professor Department of Civil and Structural Engineering The Hong Kong Polytechnic University Yuk Choi Road Hung Horn, Kowloon HONG KONG Tel: (852) 2766-6045; Fax : (852) 2334-6389 E-mail: [email protected]

CHAPTER 1

INITIAL PLANNING FOR URBAN TRANSIT SYSTEMS S. C. Wirasinghe, Department of Civil Engineering, University of Calgary, Calgary, Alberta T2N1N4 Canada

1.

BACKGROUND

The current state of initial transit planning in many transit agencies could be described at best as an art and at worst as a collection of ad-hoc rules. There are many reasons for this situation. The complexity of the problems involved, the non-catastrophic mode of functional failure associated with transit systems, the lack of trained planners, political interference in detailed planning and the failure of people with alternative planning tools to communicate their ideas to front-line planners have contributed to the problem. In the typical transit planning problem we are concerned with providing a good transit service, which has a minimal environmental impact, at a reasonable cost to the transit agency and to the users. A good level of service is provided by a transit service which is reliable, easily accessible in time and space and provides a safe, fast and comfortable ride at a reasonable price. The precise definition of the objectives to be satisfied in providing the transit service, let alone their attainment, is a difficult task. The problem is further complicated by the conflicting nature of the objectives.

2

Advanced Modeling for Transit Operations and Service Planning

A good estimate of the future demand for public transit is necessary to plan a transit system. However, the demand, in addition to being a random quantity at any given time, is also to some extent dependent on the type of transit system and its parameters. This is another dilemma faced by planners. A transit system is a failure if the objectives with which it was planned are not met to a large degree. However, this type of functional failure, as opposed to engineering failures, is not catastrophic since the system generally continues to function and to satisfy the objectives to some degree. Further, errors in planning cannot be easily pinpointed as the major cause of failure even when this is the case.

2.

THE GENERAL PROBLEM

Consider a city or a part thereof for which a public transit system is being planned. The goal of such an exercise could be stated simply as the choice of the mix of transit modes or technologies (e.g. Bus, Light Rail, etc.) and related optimal functional designs, (routes, dispatching policies, etc.) for various areas of the city for different time periods (peak, off-peak, etc.) that maximizes the expected utility to society. However, the practical realization of the goal is not a simple matter. In theory, the above problem could be converted into four related sub-problems: (i) (ii)

Determination of the set of relevant technologies or mixes of technologies, Estimation of the present and future demand for transit given each of the possible technologies (transit systems). (iii) Optimal functional planning of each transit system for the related demand, (iv) Choice of one of the transit systems as the 'best' one. However, the sub-division does not provide us with four simple problems. Rather, each one is in itself a complex problem. 2.1

Selection of Technologies

Various available technologies can be mixed in many ways for various areas of the city, for different trip purposes and for different time periods. However, the number of possible combinations is so large that it becomes prohibitive to carry out the rest of the analysis (sub-problems ii to iv) for each technology-mix. So, a smaller perhaps more relevant set has to be chosen based on speed and capacity considerations,

Initial Planning for Urban Transit Systems

3

compatibility with technology currently in use, environmental impact, geographical constraints, etc. The interested reader is referred to the Canadian Transit Handbook (1980), Gray and Hoel (1979), Vuchic (1981), and Parajuli & Wirasinghe (2001). 2.2

Demand Estimation

An extensive literature is available on the estimation of demand. A good introductory work is that of Ortuzar and Willumsen (1994) . For further treatment of the modalsplit of demand see Domencich and McFadden (1975) and Daganzo (1979). In general, one could proceed with the functional planning aspect under the assumption that the demand for transit is given. However, in selected instances that assumption can be released in favour of the one where the demand is a random quantity with known mean and variance. 2.3

The Best System

The choice of one transit-system for implementation, out of several possible, essentially boils down to a political decision. It is the planners' duty, however, to advise the decision-maker regarding the best choice. Bayesian Decision Theory offers a rational approach by which the planner can take into account the several options available, the uncertainty regarding demand, costs, etc. and a social utility function. The reader is referred to Raiffa (1970) for an introduction to Decision Theory and to Parajuli and Wirasinghe (2001) for an example of an application to transit planning.

3.

FUNCTIONAL PLANNING

We are concerned with improving the present state of the functional planning of transit and not with introducing a completely new methodology. An attempt can be made to make the planning exercise more consistent by using analytical models based on some of the more relevant and quantifiable factors that pertain to the problem at hand. An analytical model can be optimised to obtain a theoretically sound 'initialsolution' that can then be improved and 'fine-tuned' using all available hard and soft, quantifiable and non-quantifiable constraints and other information. The 'science' of transit planning can be considered to be the analytical modelling of a real system and its optimisation, while the 'art' is the conversion of an optimal analytical solution into a practical answer to a real, complex problem.

4


It should be emphasised from the beginning that several of the sociological, political and geographic factors that affect a problem cannot be included easily in a model. Thus the results obtained from optimising a model should not be taken too seriously or as the "truth". A model can, however, quantify some of the more relevant aspects, serve as an initial solution and thus prevent one from proposing or implementing the ridicule. The following interrelated factors have to be considered when a transit system with a specified mix of technologies is being planned for a given area. 3.1

Network

The network is the collection of routes for each technology for each time period. In some cases, e.g. a variable route dial-a-ride service, it is sufficient to specify a zoneof-operation. 3.2

Mode of Operation

The mode of operation is defined as the type of service offered on a particular route or a network. For fixed routes, the service may be, for example, all-stop (local), fewstops (express), non-stop, zone-stop (local or express in a zone and non-stop outside the zone), or stop-on-call (e.g. dial-a-ride). For variable routes the service may be non-stop, zone-stop, stop-on-call, etc. 3.3

Dispatching Policy and Fleet-Size

Given the mode(s) of operation, the rule according to which vehicles area dispatched on the network routes is the dispatching policy. Essentially, the dispatch rate of vehicles and the travel time on each route have to be specified. The network dispatching policies, vehicle sizes and the fleet-size of each type of vehicle are interrelated and should be considered more or less together. The detailed operational aspects include the assignment of crews to vehicles and vehicles to routes. 3.4

Location of Transfer-Facilities

A transfer-facility is a location at which passengers can transfer from one mode to another. Bus stops and rail stations are the most common examples. The location of transfer-facilities is intimately related to the mode-of-operation. A terminal is a transfer-facility at an end of a route.

Initial Planning for Urban Transit Systems 3.5

5

Location of Vehicles Garages

Normally, each mode will have one or more garages at which vehicles are parked and maintenance is carried out. Occasionally, a terminal can be shared by modes with some similarities and also serve as a terminal. The location of terminals is dependent on fleet size, the network and to a large degree on the availability of land.

4.

APPROACH TO AN ACCEPTABLE APPROXIMATE METHODOLOGY

Obviously, the five factors described above are interrelated and cannot be tackled in complete isolation. However, each is a complex problem and their combination can only be described as a 'mess'. The reason for the use of ad-hoc methods and rules-of-thumb in the functional planning of transit is now fairly obvious. The fact of the matter is that the overall problem is too complex to be formulated properly as a whole for solution even by computer-based iterative methods. In any case, it would be naive to write a complex non-linear objective function with many non-linear constraints and optimise it with respect to several variables using a massive computer algorithm, since even the output from the most complex and largest possible computer program, would have to be significantly altered by planners to allow for still unmodellable factors. The basic premise of this chapter is that the 'human' planner reins supreme over computing 'machines'. The 'block-box' approach where a planner's function is simply to run local data with a 'canned' program and to accept the output as inviolable is rejected. The methodology proposed here is to break up the functional planning problem into more or less independent sub-problems and to obtain approximately optimal analytical solutions to the sub-problems. It is conceded that optimising parts of the problem does not necessarily lead to the optimal solution to the combined problem. The exercise of analytical modelling forces us to think formally about the problem at hand and to isolate the critical factors. Thus it enhances our understanding of the problem.

6


The analytical solutions highlight the basic interrelationships among the relevant parameters which could easily be clouded by enumerative type techniques. These solutions are also easily subjected to sensitivity analysis. Analytical solutions are ideal for planners who can use them as 'initial-solutions' to complex problems since they are easily manipulable and modifiable. 4.1

The Objective Function

To avoid issues such as the additivity of individual utilities, etc. [see Jordaan (1985)] we shall express the total disutility of a system as a total cost expressed in monetary units. The objective function in all instances unless specified otherwise will be the total cost function. The objective is to choose the relevant system parameters so as to minimize the total cost. Certain other commonly used objectives such as the minimization of the total travel time and minimization of the operating cost can be shown to be special cases of the objective chosen here. The above philosophy is illustrated in the following through the applications to a single bus route. Aspects of the dispatching policy and scheduling of travel time are discussed. They are provided for illustrative purposes and should not be considered to be sufficiently detailed for use in practice.

5.

DISPATCHING POLICIES FOR A Bus ROUTE

Most bus routes serve multiple passenger origins and destinations with a limited number of buses. The capacity of the buses, while being uniform in most cases, can vary from bus to bus. Each bus may be used for several round-trips on the route. The policy that is followed with respect to the dispatching of buses on a route is of fundamental importance since it has a direct bearing on the level of service provided to the passengers and the cost of providing service on the route. We shall explore dispatching policies for several different situations. 5.1

Newell's Policy

Newell's dispatching policy for a (public) transportation route with time-varying many to one demand [Newell (1971)], simply stated, is that the dispatch rate of vehicles is proportional to the square root of the arrival rate of passengers if the vehicles are sufficiently large, and to the arrival rate if, otherwise. The objective of


1

the policy is to minimize total passenger waiting time subject to a fixed number of dispatches (not less than the minimum required to transport all the passengers in vehicles that are filled.) It can be written in the form

where g(t) is the dispatch rate of vehicles at time t in vehicles per unit time, p(t) is the arrival rate of passengers at time t in passengers per unit time, c is the capacity of the vehicles and K is a constant which is chosen to satisfy the constraint of a fixed number of dispatches. Newell (1973) has also pointed out that the identical policy applies if the objective is to minimize total passenger waiting time costs and dispatching costs, and how AT is a function of yw , the average value of a unit of waiting time per passenger and yD , the cost of dispatching a vehicle. The latter formulation of the policy is given by

(2)

p(f)lc The original constraint of a fixed number of dispatches is identical to fixing yw at a particular value, given y D . The dispatching policy can be in a sense unconstrained and the level of service (frequency) provided can be increased by increasing yw . If somehow the "real" value of yw were known, the related dispatch rate and hence the number of dispatches could be determined. On the other hand, one could interpret yw as the amount that the bus company is willing to pay to save one passenger one unit of waiting time. There is no evidence to date that this policy has been adopted by public transit systems, and it is not mentioned in transit manuals [Gray & Hoel (1979), Soberman & Hazard (1980), Pushkarev, et al. (1982)] and even in some reviews of dispatching policies [Furth & Wilson (1981), Chua (1984)].

8 5.2

Advanced Modeling for Transit Operations and Service Planning Uniform Dispatch Rate During Off-Peak Periods

It is common for bus companies to adopt a uniform dispatch rate during (off-peak) periods when the demand for travel is low and capacity constraints do not apply. If a uniform dispatch rate g is to be adopted during a period of duration T, in response to a time-varying demand ofpft) passengers per unit time, the sum of the waiting time cost and dispatching cost per unit time is given by

dt

(3)

if a passenger on average waits one half of a headway. This is minimized when

(4) where p is the mean demand, in passengers per unit time, during the period T. Thus the uniform dispatch rate is predicated on the square-root of the mean demand. The mean total cost per unit time during the period T under a uniform dispatch policy is obtained by substituting for g from Equation (4) in Equation (3): -

(5) The mean total cost per unit time if Newell' s Policy of a variable dispatch rate is adopted, (6)

is always lesser than the total cost given in Equation (5). However, the difference can be quite small when the demand is not highly variable with /, which is the likely situation during an off-peak period. Since there are intangible benefits in adopting a uniform dispatch rate policy for an off-period, the practice can be made consistent with Newell' s Policy if the dispatch rate in Equation (4) is used.

Initial Planning for Urban Transit Systems 5.3

9

Policy Headways

Bus companies commonly use a "policy headway" - a maximum acceptable headway, hp, as an integral part of their dispatching policy. The policy headway (usually in the range 30-60 minutes) is chosen subjectively. Newell's Policy and subjective judgement can be amalgamated by using the maximum of l/hp and, the dispatch rate given by Equation (2), as the actual dispatch rate. 5.4

Average Waiting Time

A "wait" is defined as the (absolute) deviation between the actual and desired departure times of a passenger. Consequently, the wait caused by the bus schedule could take place not only at the origin bus stop but also at the origin or destination of the trip depending on the type of trip. For example, morning peak commuters, who presumably have a specific time before which they should be at work, will likely spend the wait time at the destination (schedule-delay), subsequent to taking the last possible bus that will transport them to the destination on or before the required arrival time. Thus, we can consider the desired departure time to be independent of the schedule, and interpret p(t) to be the desired departure rate of passengers at time /. If p(t) is (approximately) uniform within a headway, then the average wait is (approximately) one half of a headway. It has been assumed implicitly in the above discussion that buses run as scheduled. 5.5

Value of Waiting Time

The Policy is based on the dual assumptions that the average wait is one half a headway and that the value of a unit of waiting time per passenger, yw, is invariant with the elapsed waiting time. The latter is an unstated assumption that is routinely made in the literature on dispatching policies and even in books on travel time [see, for example, Tarski (1987)]. Horowitz (1981), using data from a laboratory psychological scaling experiment, found that the subjective value of a unit of waiting time did not change significantly during a period of waiting for the range 5-15 minutes of waiting. However, it was higher in the range 0-5 minutes. Given that the experiments were conducted at discrete 5 minute intervals, he concluded that there is a fixed penalty attached to having to wait, which is independent of the amount of time spent waiting, and that the value of a unit of waiting time is a constant. This penalty has no effect on Newell's Policy since it is independent of the time spent waiting.

10


Walther (1975), using data from field studies for the range 0-6 minutes of waiting, found that the riding time equivalent to a wait is a non-linear increasing function of the wait. This is equivalent to the value of waiting time per unit time being an increasing function of the elapsed waiting time. Newell's Policy will result in nonoptimal headways if Walther's results extend to the range of higher waiting times. In the face of conflicting evidence regarding the relationship between the value of a unit of waiting time and the elapsed waiting time, it is worthwhile to investigate the relationship between headway and yw further. If we assume that the value of a unit of waiting time is a linear function Y*w+PwY (7) of the elapsed waiting time T, the total cost of passenger waiting time in a headway h(t) is

(8) where p(t) is the arrival rate of passengers and (x-y) is the elapsed waiting time up to a time * of a passenger who arrived at time y. Minimizing the sum of, the total cost of passenger waiting time/per unit time obtained by dividing the Expression (8) by h(t), and the cost of dispatching buses per unit time, yD /h(t) , we find the minimum total cost headway to be given by (l/2)p(t)y*w -yD/h2(t)-(l/3)p(t)pwh(t)=0

(9)

A unique solution for h(t) exists in h(t)>0 if fiw >0, i.e. the value of a unit of waiting time increases with elapsed waiting time, and Newell's square-root headway is an upper bound. Similar results are likely when the value of a unit of waiting time is a nonlinear increasing function of the elapsed waiting time. The use of policy-headways can be interpreted as an attempt to keep headways within reasonable limits, given that the optimal headways under the assumption of a uniform value of yw can give high headways, particularly when the demand is low. However, if the values of y*w can be used instead.

an
y' when passengers have similar trip purposes. Thus, if the vehicle is on time or early (t > S), the cost per trip is y'S and if the vehicle is delayed (t > S), it is y'S + y(t - S). Then the expected cost of travel time is

(19) 9.4

Delay Penalty

The cost related to a vehicle being behind schedule can, in many cases, have another component that is unrelated to the actual amount of delay. For example, if a transfer is missed, the related cost to passengers is independent of the amount of time by which it was missed. This component, the delay penalty, which includes the


23

inconvenience of missed appointments, etc. may be modelled as a cost equal to zero for on time vehicles and to y for delayed vehicles. Therefore, the expected penalty

9.5

Expected Total Cost

The expected total cost for the trip, E[Z/(S)J, is the sum of the expected travel time cost (Equation 19) and the expected penalty cost (Equation 20):

10.

OPTIMAL SCHEDULED TRAVEL TIME

10.1

Optimisation

We define the optimal scheduled travel time as that which minimizes the expected total cost given by Equation 21. The respective first and second order conditions for the optimum are:

F(s}-(yp/y}f(s}+y'/y

=i

(22a)

and

f(s)-(yp/y)f'(s)>0

(22b)

respectively. The solutions to Equation 22a depend on the exact form of the probability density function (f(t) and the various cost parameters [Figure 6]. The slope of £'[z(5')] isy'/y'1 (tM. Thus any root of Equation 22a that is greater than IM will be a relative minimum of ^[z^)]. However, under certain conditions -£"[z(s)] can also have a relative minimum in S < IM- The function, /?[z(,S)] has only one minimum and no internal maxima in all cases with one or two roots. In the case

24


of three roots, there are two relative minima and the minimum has to be selected by enumeration.

Figure 6. Solutions to Equation (22a) The optimal scheduled travel time, S*, is dependent only on the probability density function of T and on the two unit cost ratios: y /y, the unit cost of budgeted travel time as a fraction of the unit cost of delayed travel time. We now consider some special situations to investigate the properties of S*. 10.2

No Delay Penalty (yp = o)/ Recreational Trips

Consider a recreational trip in which there is no specific penalty in arriving late (y = 0) and no advantage in arriving early. However, a delay means that time that was planned to be spent in recreational activities is pent travelling in a vehicle. Setting y = 0 in Equation 22a gives:

(23) For S* to be the median value of T, y' /y- 0.5, i.e. a unit of delayed travel time is worth two units of budgeted travel time. It is common, however, for S* to be set by adding k (> 0) standard deviations (a) of "slack time" to the mean. If T is assumed to be normally distributed, setting k = Vi, for example, is equivalent to the unlikely situation where a unit of delayed travel time is valued at 3.25 units of budgeted travel time. The equivalent probability of being on time (reliability) is about 0.7. A rather


25

unconventional insight is that S* can be set at less than the median value of T for trips with no penalty if y' I y > 1A. For example, if y' / y - 2/3, S* = F "7(0.33). 10.3

Delay Penalty Only (y' = y)

Consider a feeder trip where any delay; regardless of the amount, results in a penalty equivalent to a missed transfer. If all disutilities related to the delay are absorbed in yp and y' = y, Equation 24 gives

(24) For example, S* = E[T] + 0.5 a when T is normally distributed and the penalty is equivalent to the cost of 1.96 0 due to work related penalties of being late. For example, some workers may loose a half hour of pay for being late by less than or equal to one half hour. Consequently, both y'/y and yp /y will be factors in the choice of S*. It is unlikely that y' /y < Y2. Hence from two sub-sections above, it is apparent that S* will be greater than the median value of T, mostly when yp also exists in addition to y>y'. For example, if

y'/y

= 2/3,

yp/y^

1.75 y , if passengers in a vehicle have different trip purposes. For example, consider a bus in which half the passengers are transfers and the other half are workers at the destination. The value of y' will be calculated based on all the passengers, while y will be based only on the group of passengers who work at the destination, causing y' > y in some cases. However, the penalty cost now includes the sum of the cost of delayed travel time spent in the bus by transferring passengers and the cost of additional time spent out of the bus while waiting to transfer. Consequently, a minimum total cost solution for S is still available [Figure 6] and is given by S*(2) which is the higher one of two roots.

11.

CONCLUDING COMMENTS ON SCHEDULING TRAVEL TIME

An idealized situation (a single link) was analysed with respect to the scheduled travel time estimation problem. However, the cost based approach is insufficiently developed for application to most real routes with many time-points. It was shown that an optimal scheduled travel time that minimizes the sum of the expected costs of budgeted and delayed travel time and expected penalty costs exists for a link. The scheduled travel time (and the related reliability) are dependent to a large extent on the trip type. For example some recreational trips can have negative


27

optimal slack times and reliabilities < 0.5 while work trips have positive optimal slack times and reliabilities around 0.75. Further investigation is required regarding the assertion that delayed travel time "costs" more per unit time than budgeted travel time, because of the inconvenience and stress associated with delays. The Equation (22a) for S* cannot be easily used in practice until y/y', the factor by which the cost of (budgeted) travel time per passenger per unit time is increased for delayed travel time, is estimated for various trip purposes using techniques such as disaggregate mode choice analysis. The model proposed here can be extended to allow for the case where early arrival of a vehicle allows passengers to use the extra time (S-t). However, it is then necessary to estimate the cost of "early time' per passenger per unit time which is presumably less than the equivalent cost of budgeted in-vehicle travel time. When required, the proposed method can be used solely with either operator costs or passenger costs. The estimation of the probability distribution of the travel time of unscheduled vehicles, f(t), is also problematic. Since most existing routes operate under some control strategy, the above travel times cannot be easily observed. In a scheduled single link route, all travel times will be appropriate observations if the drivers have not intentionally "killed" time enroute. In the case of multiple time-points, observations must be made at each time-point and any holding time enroute deducted from the total, again with the stipulation that time not be killed between time-points. Thus cooperation of the vehicle operators is required in running without killing time enroute, in not leaving time-points early, and in leaving time points as soon as passenger processing is complete, if running late. Regardless of the above, and except in pre-booked or large headway routes, the observations are strictly valid only if the vehicle is following a vehicle that was run exactly on schedule. Application of the above methodology to a normal many to many demand bus route with several time points is given in Wirasinghe and Liu (1995) and extended in Liu and Wirasinghe (2001).

ACKNOWLEDGEMENTS This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under grant number A4711. The sections on Dispatching Policies

28


and on Scheduling Travel Time were essentially taken from Wirasinghe (1990) and Wirasinghe (1993) respectively.

REFERENCES Abkowitz, M. and I. Engelstein (1984). Methods for maintaining transit service regularity. Transportation Research Record, 961, 1-8. Bruzelius, N. (1978). The Value of Travel Time — Theory and Measurement. Nationalekonomiska Institution, Stockholm University, Sweden, Skrift No. 3. Canadian Transit Handbook (1980). (R. M. Soberman and H. A. Hazard, eds.), University of Toronto-York University Joint Programme in Transportation, Toronto, 767. Chua, T. A. (1984). The planning of urban bus routes and frequencies: A survey. Transportation., 12, 147-172. Daganzo, C. (1979). Multinomial Probit. Academic Press, New York, 222 . Domencich, T. A. and D. McFadden (1975). Urban Travel Demand. North Holland, Amsterdam, 215 . Furth, P. G. and N. H. M. Wilson (1981). Setting frequencies on bus routes: Theory and practise. Transportation Research Record, 818, 1-7. Gray, G. E. and L. A. Hoel (1979). Public Transportation: Planning, Operations and Management. Prentice-Hall, New Jersey, 749. Gray, G. E. (1992). Perceptions of public transportation. In: Public Transportation 2nd Edition (G.E. Gray and L.A. Hoel, eds.), pp. 617-635. Horowitz, A. J. (1981). Subjective value of time in bus transit travel. Transportation, 10, 149-164. Jordaan, I. J. (1985). Probability and Decision for Civil Engineers. Ellis-Horwood, U.K. Lesley, L. J. S. (1975). The role of the timetable in maintaining bus service reliability. Proceedings of the Symposium on Operating Public Transport, University of Newcastle Upon Tyne, U.K., 36-53. Lindley, D. B. (1965). Introduction to Probability and Statistics. Cambridge University Press, Cambridge. Liu, G. and S. C. Wirasinghe (2001). A simulation model of reliable schedule design for a fixed transit route. Journal of Advanced Transportation, 35(2), 145-174. Newell, G. F. and R. B. Potts (1964). Maintaining a bus schedule. Proceedings of the Second Conference, Australian Road Research Board, 2(1), 388-393.


29

Newell, G. F. (1971). Dispatching policies for a transportation route. Transportation Science, 5, 91-105. Newell, G. F. (1973). Scheduling, location, transportation and continuous mechanics: Some simple approximations to optimization problems. SIAM Journal of Applied Mathematics, 25(3), 346-360. Ortuzar, J. de D. and L. G. Willumsen (1994). Modelling Transportation. Wiley, 439. Parajuli, P. and S. C. Wirasinghe (2001). A line haul transit technology selection model. Transportation Planning and Technology, 24, 271-308. Pushkarev, B. S., et al. (1982). Urban Rail in America - An Exploration of Criteria for Fixed Guideway Transit. Indiana University Press, Bloomington. Raiffa, H. (1970). Decision Analysis. Adison Wesley, Mass., 309. Soberman, R. M. and H. A. Hazard (eds.) (1980). Canadian Transit Handbook, (University of Toronto - York University Joint Program in Transportation, Toronto. Tarski, P. (1987). The Time Factor in Transportation Processes. Elsevier, Amsterdam. Vuchic, V. R. (1981). Urban Public Transportation: Systems and Technology. Prentice Hall, 673. Walther, K. (1975). Die fahrzeitaquivalente reisezeit im offentlichen personennahverkehr. Verkehr und Technik, 7, 271-274. Wirasinghe, S. C. and R. Roose (1987). Modified DISPOL Program and Users Manual. Department of Civil Engineering, the University of Calgary, Calgary. Wirasinghe, S. C. (1990). Re-examination of NewelFs dispatching policy and extension to a public transportation route with many to many time varying demand. In: Transportation and Traffic Theory, (M. Koshi, ed.), pp. 363-378. Elsevier. Wirasinghe, S. C. (1993). Cost based approach to scheduling travel time on a public transportation route. In: Transportation and Traffic Theory, (C.F. Daganzo, ed.), pp. 205-216. Elsevier. Wirasinghe, S. C. and G. Liu (1995). Determination of number and locations of time, points in transit schedule design - Case of a single run. Annals of Operations Research, 60, 161-191.


CHAPTER 2

PUBLIC TRANSPORT TIMETABLING AND VEHICLE SCHEDULING Avishai Ceder, Cn il Engineering Department, Transportation Research Institute, Technion-Israel Institute of Technology, Haifa, Israel 32000.

1.

INTRODUCTION

The bus, railway and passenger ferry operational planning process includes four basic components performed usually in sequence: (1) network route design, (2) setting timetables, (3) scheduling vehicles to trips, and (4) assignment of drivers (crew). The framework of this process appears in Figure 1. It is desirable for all the four components to be planned simultaneously to exploit the system's capability to the greatest extent and maximize the system's productivity and efficiency. However this planning process is extremely cumbersome and complex, and therefore seems to require separate treatment of each component, with the outcome of one fed as an input to the next component. In the last twenty years, a considerable amount of effort has been invested in the computerization of the four components mentioned above, in order to provide more efficient controllable and responsive schedules. The best summary as well as the accumulative knowledge of this effort was presented in the second through the eight International Conferences on Public Transport Scheduling, and appear in the books edited by Wren (1981), Rousseau (1985), Daduna and Wren

32


Figure 1. Framework of public transport operational planning process with emphasis on this chapter's components

Public Transport Timetabling and Vehicle Scheduling

33

(1988), Desrochers and Rousseau (1992), Daduna et al. (1995), Wilson (1999) and Voss and Daduna (2001). This chapter focuses on two scheduling components: timetabling, and vehicle scheduling while assuming that the public transport network is unchanged. The first component in Figure 1 deals with the establishment of public transport interchanges (meeting point of various public transport modes), terminals (trip start 'and end points), and routes. This first component is covered in the next chapter. The fourth scheduling component is usually divided into creating crew duties and crew rosters (rotation of duties among the drivers) and will not be dealt with in this book. The timetable component in Figure 1 is aimed to meet the general public transportation demand. The demand varies during the hours of the day, the day of the week, from one season to another, and even from one year to another. This demand reflects the business, industrial, cultural, educational, social and recreational transportation needs of the community. It is the purpose of this component to set appropriate timetables for each transit route to meet the variation in the public demand. Determination of timetables is performed on the basis of passenger counts, and must comply with service frequency constraints. The vehicle scheduling component appearing in Figure 1 is to schedule vehicles to trips according to given timetables. A public transport trip can be either planned to transport passengers along its route or to make a dead-heading trip in order to connect efficiently two service trips. The scheduler's task is to list all daily chains of trips (some dead-heading) for a vehicle, ensuring the fulfillment of the timetable requirements and the operator requirements (refueling, maintenance, etc.). The major objective of his task is to minimize the number of vehicles required. All the components in Figure 1 are very sensitive to internal and external factors, sensitivity which could easily lead toward an inefficient solution. The complexity involved in the public transport operational planning process challenges researches to develop automated computerized procedures which led to number of software packages available in the market. It is worth mentioning that the evaluation module of such a software package should be based on an external input related to cost coefficients, and performance criteria. The cost coefficients include: vehicle cost (fixed and variable), crew cost (fixed and variable), service benefit, and other costs. The performance criteria include: measures of passenger service, measures for vehicle and crew schedules, and measures for duty rosters, and other criteria. This chapter includes two main parts. The first covers the determination of vehicle frequencies and the construction of alternative public timetables for both cases of

34


even headway and even average load. The second part provides an overview on exact solutions to the vehicle scheduling problem and describes a graphical heuristic procedure for the determination of minimum fleet size and its lower bound. The chapter ends with concluding remarks and list of references.

2.

CONSTRUCTION OF ALTERNATIVE PUBLIC TIMETABLES

2.1

Background

Mathematical programming methods for determining frequencies and timetables have been proposed by Furth and Wilson (1981), Koutsopoulos et al. (1985) and Ceder and Stern (1984). The objective in Furth and Wilson (1981) is to maximize the net social benefit, consisting of ridership benefit and wait time saving, subject to constraints on total subsidy, fleet size and passenger loading levels. Koutsopoulos et al. (1985) extended this formulation by incorporating crowding discomfort costs in the objective function and treating the time dependent character of transit demand and performance. Their initial problem comprises a non-linear optimization program relaxed by linear approximations. Ceder and Stern (1984) addressed the problem with an integer programming formulation and heuristic person-computer interactive procedure. The latter approach focuses on reconstructing timetables when the available vehicle fleet is restricted. Public transport timetable is commonly constructed for given sets of derived frequencies. The basic criteria for the determination of frequencies are: (a) to provide adequate vehicle's space to meet passenger demand, and (b) to assure a minimum frequency (maximum-policy headway) of service. Ceder (1984) described four different methods for calculating the frequencies. Two are based on point-check (counting the passengers on-board the public transport vehicle at certain point(s)), and two - on ride-check (counting the passengers along the entire public transport route). In the point-check methods the frequency is the division between passenger load at the maximum (max) load point (either the one across the day or in each hour) and the desired occupancy or load factor. In the ride-check methods the frequency is the division between the average or restricted-average passenger load and the desired occupancy. The average load is determined by the area under the load profile (in passenger-km) divided by the route length (km), and the restricted average is a higher value than the average one, in order to assure that in certain percentage of the route length the load does not exceed the desired occupancy. This desired occupancy (or load factor) is the desired level of passenger load on each vehicle, in each time period (e.g. number of seats). Commonly, across almost all the public transport agencies, the


35

frequency is determined by the maximum load procedure. This max load procedure is established to ensure adequate space to accommodate the maximum number of onboard passengers, along the entire route, for a given time period (e.g. one hour). That is,

where lj is the average (over days) maximum number of passenger (max load) observed on-board in period j, dj is the desired occupancy (load factor) in period j, and Fmj is the minimum required frequency (number of vehicles) in period j. In a follow-up study Ceder (1987) analyzed optional ways for generating public timetables. This analysis allows for establishing a spectrum of alternative timetables, based on three categories of options: (a) selection of type of headway, (b) selection of frequency determination method for each period, and (c) selection of special requests. In category (a) the headway (time interval between adjacent departures) can be equal or balanced. Equal headway refers to the case of evenly spaced headways and balanced headway - to the case of unevenly spaced headways while treating each vehicle separately at the hourly maximum load point. These cases are being extended in this chapter. In category (b) it is possible to select for each time period one of the four frequency determination methods (two point-check, and two ride-check) mentioned above, or a given frequency by the scheduler. In category (c) it is possible to request clock headways (departure times that repeat themselves in each hour, easyto-memorize) and/or certain number of departures (usually for cases with limited resources). The outcome of these analyses is a set of optional timetables in terms of vehicle's departure times at all specified timepoints, using passenger load data. Each timetable is accompanied by two comparison measure which are used as an evaluation indicator in conjunction with resource saving. The first measure is the total required vehicle runs (departures) and the second is an estimate for the minimum required fleet size at the route level only. Public transport timetable is perhaps the main reference for defining unreliable public transport service. The assumption that passengers will adjust themselves to given timetables (with headways of, say, longer that 10 minutes) instead of adjusting the timetables to the passenger demand is one of the largest sources of unreliable service. When passenger demand is not met, the vehicles are slowing down (increased dwell time), behind the schedule and entering the inevitable process of further slow down. This will eventually lead to the known bunching phenomenon with the vehicles behind. Opposite to that is the situation of overestimating the demand which may result in vehicles running ahead of time. Both situations are not observed when the

36


service is highly frequent and characterized by low variance of the headway distribution. 2.2

Three Procedures

In this section of the chapter three different procedures are proposed and analyzed for better matching the passenger demand with a given timetable while attempting to minimize the number of departures (leads to reduce the number of vehicles which is one of the main resources). This will result in a more reliable and comfortable service. Procedure 1 produces departure times with evenly spaced headways while considering a smooth transition between adjacent hours. This procedure is based on the given standards dj and Fmj for each hour j and on the j-th hourly max load, lj . Procedure 2 determines departure times such that, in average sense, vehicles will carry on even dj loads at the hourly max load point. This procedure 2 is based on d j , Fmj and on individual vehicle loads at the hourly max load point where lj is observed. Procedure 3 derives the departure times such that, in average sense, the on-board passenger load will not exceed d j , and will be equal to dj at each individual vehicle max load point (as opposed to the lj points in procedure 2). Example Problem and Initial Analysis The example problem is used as an explanatory device for three procedures. Table 1 contains the necessary information and data for a 2-hour example j = 1, 2 of a bus route from A to C with one stop at B. There are 5 departures observed. For each departure Table 1 contains the average observed on-board passenger at both boarding points A and B. The minimum required frequency is Fmj = 2, j - 1, 2. The desired occupancy is di = 50 and d2 = 55 passenger per bus, and the average travel time from A to B is 18 minutes. The estimated hourly demand is also included in Table 1. It is based on two basic assumptions: (a) the average load observed is a representative value of the actual demand and it is independent of the exact setting of departure times; (b) the passengers observed on-board are accumulated at a uniform rate. The first assumption can be realized when using the vast amount of data anticipated from equipment like APC (Automated Passenger Counters), or when the schedulers have reliable sources of information provided by road inspectors and supervisors. The second assumption usually holds when the observed headways are relatively small. For headways greater than 30 minutes, part of the passengers may time their arrival and, if data is available, this second assumption may not be needed.

37

Public Transport Timetabling and Vehicle Scheduling Table 1. Input and observed data of the example Route:

G>

Average Travel Time:

18 min

Average Observed Number of Passengers on-board the Bus Dept Time at A at Time A 7-8 a.m. 8-9 a.m.

7:15 7:45

30

8:10 8:30 8:50

25 94

80

Derived Values Headway Frequency (bus/hr) (min) H H I I A M A M M M L L

at B

Hourly Average Max. Hourly Demand Ind. Load at at Max. (HAML) A Point B

65 35

125

148

193

50

2.96

3.86

20

16

80 72 67

192

177

214

55

3.49

3.89

17

15

Desired Occ. (pass/ bus)

Minimum Frequency: 2 buses per hour Referring to the example in Table 1, the hourly demand for the first hour, between 7 8 a.m., j = 1, is based not only on the average loads observed, but also on part of the load observed on the first bus in the second hour, j = 2. That is, the average load on the first bus in j = 2 is divided proportionally in order to reflect the demand at the end to the period j = 1. Therefore, at point A and B the loads of 25 and 80 associated with the 8:10 departure, are divided into 3/5 and 2/5 where the 3/5 portion is related to the j = 1 demand. This proportion is stem from the 25 minutes difference between the last departure of the period] = 1 (7:45) and 8:10 where 15 minutes of this time difference belongs to j = 1, and 10 minutes to j = 2. The hourly demands at A and B are 125 and 148 passengers, respectively, for j = 1, and 192 and 171, respectively, for j = 2. This means that for j = 1 the hourly max load point is B with 1, = 148, and - point A for j = 2 with 12 = 192. hi addition, the third column under hourly demand, in Table 1, includes a newly element called individual max hourly demand. This demand reflects the sum of the max on-board loads observed on each bus, in each hour, while considering also the proportion of max demand associated with the first bus of the next hour. It results are 193 and 214 passenger demand for j = 1, 2, respectively, where, for the example of j=l, one

38


obtains 193 = 65+80+(3/5)x80. The interpretation of this element is clarified under the description of procedure 3. Finally in the last four columns in Table 1 there are the derived frequencies and headways based on equation (1). The headway Hj for hour j, is simply the inverse of the frequency, and in minutes:

Therefore, for j = 1, 2 the frequencies based on the hourly max load points are 2.96 and 3.49, respectively, and are 3.86 and 3.89 buses per hour for the individual max load hourly demand, respectively. Procedure 1 One characteristic of existing transit timetables is the repetition of the same headway in each time period. The scheduler, using Hj, is facing, however, a problem on how to set the departure times in the transition segments between adjacent time periods. In addition, the scheduler (or existing software) usually rounds-up the frequencies F, to the next integer, prior the use of equation (2). It is shown here that in order to save resources there is no need to round-up Fj and moreover the transition between hours (or any other time periods) can be carried out in a simple and accurate manner. The Underlining Principle of Procedure 1 The simple way, used by many bus agencies, to smooth the headways during the transition time is to consider an average headway between two adjacent hours. This average rule may result in either undesirable overcrowding or underutilization. For example, using equations (1) and (2) one obtains HI - 25 and H2 = 9 minutes with average of 17 minutes. Thus, a timetable can be set to 7:00, 7:25, 7:50, 8:07, 8:16, .... By assuming uniform arrival rate with di = 50 and d2 = 60, j - 1 contributes for the 8:07 departure (10/25) x 50 = 20 passengers, for the remaining 10 minutes between 7:50 and 8:00, and j - 2 contributes (7/7) x 60 = 60 passengers. The total is 20 + 60 = 80 average passengers, on the 8:07 departure, representing overcrowding. In order to overcome this undesirable situation the following principle is employed. Principle 1: establish a curve representing the accumulative frequency versus the time (adding the non-integer value of the frequency determined with respect to time). Moving horizontally, for each departure, until intersecting the accumulative curve, and then vertically, results with the required departure times.


39

Proposition 1: Principle 1 provides the required evenly spaced headways with a transition load approaching the average of du and d u +i , where du and d u +i are the desired occupancies for two consecutive time periods. Proof; Figure 2 illustrates Principle 1 using the information in Table 1. Since the slopes of the lines are 2.96 and 3.49 for j = 1 and j = 2, respectively, the resultant headways are those required. The transition load is the one determined for the 8:01 departure, and is comprised of 20 minutes arrivals for j = 1, and 1 minute arrival for j = 2. Therefore (20/20) x 50 +(1/17) x 55 = 53 approximately. This transition load is not the exact average between d] = 50 and d2 = 55 since departures are made in integer minutes. That is, the exact determined departure after 8:00 is (3-2.96) x 60/3.49 = 0.688 minutes, and inserting this value instead of 1 minutes in the above calculation yields a closer value to the exact average. Basically, the proportions considered satisfy the proof-by-construction of Proposition 1. Figure 2 exhibits the resultant six departures for procedure 1 where the determined frequencies are kept non-integer. Principle 1, therefore, allows for saving some unnecessary bus runs and also stabilizes the average load during the transition segment between time periods. Procedure 2 While arriving with procedure 1 to a satisfactory timetable, with even headways, it is still unclear if the loads on individual buses will not exceed d j , for all j. It is wellknown that passenger demand varies even within a single time period, reflecting the business, industrial, educational, cultural, social and recreational transit needs of the community. This dynamic behavior provides a basis for the scheduler to adjust the departure times. These adjustments are not done frequently unless there is a clear cut information (e.g. from the road supervisions) to support it. Nonetheless with the anticipated vast amount of passenger load data (e.g. from APCs) it is possible to construct procedures to better match the timetables (departure times) with the variable demand. This and the next section provide such procedures. The Undergoing Principle of Procedure 2 The results of procedure 1 starts with the 7:20 and 7:40 departures for j = 1. The frequency required, based on the hourly max load point, is 2.96 for j = 1 (as is shown in Table 1). This frequency aimed at 50 passengers per bus while considering the entire hourly max demand. However, the assumption of uniform passenger arrival rate, between the observed departures, results in 65/15 = 4.3 passengers/minute between 7:00 and 7:15 and 35/30 = 1.2 pass/min between 7:15 and 7:45 at the hourly max load point B. Therefore, the 7:20 departure (by procedure 1) may result in 65 +

40


1.2 x 5 = 71 passengers; significantly above the desired 50 passengers. In order to avoid this imbalanced situation the following principle is exploited.

7:00

Figure 2. Determination of the example departure times (at A) for evenly spaced headways with a smoothing process between time periods 71 passengers; significantly above the desired 50 passengers. In order to avoid this imbalanced situation the following principle is exploited. Principle 2: construct a curve representing the accumulative loads observed on individual buses at the hourly max load points. Moving horizontally per each dj for all j, until intersecting the accumulative curve, and then vertically, results with the required departure times.


41

Proposition 2: Principle 2 results in departure times such that the average max load on individual buses, at the hourly j max load point, approaches the desired occupancy Proof; Figure 3 illustrates Principle 2 for the example problem appearing in Table 1. The derived departure times are unevenly spaced to obtain even loads at points B for j = 1 , and point A for j = 2. These even loads are constructed on the accumulative curve to approach d\ and da for j = 1, j = 2, respectively. Assuming uniform passenger arrival rate between each two observed departures shows that the load (at B) of the 7:45 departure (at A), for example, is comprised of the arrival rate between 7:12 and 7:15 (65/15 = 4.3) and the rate between 7:15 and 7:45 (35/30 = 1.2). Thus, 4.3 x 3 + 1.2 x 30 = 49 which is approaching di = 50. Moreover, in the transition between j = 1 and j = 2, the value of da = 55 is considered since the resultant departure is after 8:00. The load of the bus departing A on 8:16 at its hourly max load point A, is comprised of (25/25) x 25 + (94/20) x 6= 53.2 which is approaching d2 = 55. The exact value of da can be obtained only for departures with non-integer minutes. This completes the proof-by-construction of Proposition 2. Figure 3 includes the results of procedure 2 with six departures. The last departure at 8:52 is determined using a slight extrapolation of the uniform passenger arrival rate between 8:30 and 8:50. Procedure 3 While procedure 2 ensures even average loads of dj at the j-th max load point, it does not guarantee that in other bus stops the average load will not exceed dj and, therefore, may result in overcrowding. The purpose of the procedure presented below is to derive the bus timetable provided that in an average sense all buses will have even loads (equal to the desired occupancy) at the max load stop of each bus. That is, for a given time period each bus may have a different max load point across the entire bus route with a different observed average load. The objective set forth is to change the departure times such that all observed average max loads will be same and equal to dj during all j. Certainly the adjustments in the timetable are not intended for highly frequent urban services where the headway is less than say, 10 minutes, or an hourly frequency of about 6 vehicles or more. Behind this procedure is the notion that passenger overcrowding situations (loads greater than dj) should be avoided. The Underling Principle of Procedure 3 The results of procedure 2 are exhibited in Figure 3. Considering in that Figure, for example, the resultant departure at 7:45 with 50 average passengers on-board at point B. From Table 1 it is clear that point B is the j=l max load point. However, one does not know what is the average load in the other stops, and in the example problem, it is referred to point A. Since the first departure is at 7:12, the accumulative load at A

42


between 7:12 and 7:45 is of interest. For that purpose the data in Table 1 is used while constructing an accumulative curve of the observed loads at A. The average load at A for the 7:11 departure results in 22 passengers (30/15 = 2 pass/min arrival rate). The average load at 7:45 is combined from the remaining passengers between the observed 30 at 7:15 and 22, and those observed on the 7:45 departure. That is (30 22) +80 = 88 passengers. No doubt that the 7:45 departure faces, in an average sense, overcrowding at A while complying with di = 50 at B. In order to overcome this undesirable possible overcrowding the following principle is employed.

Figure 3. Determination of the example departure times (at A) with even loads at the hourly maximum load point


43

Principle 3: construct an accumulating passenger load curve at each stop (except the arrival point). Moving horizontally per each dj , for all j, on each curve, until intersecting each of the accumulative curves, and then vertically to establish a departure time for each curve. The required departure time is the minimum one across all curves. Using the last determined departure time, set the loads across all the curves and add the considered or next dj. Repeat until the end of the time span. Proposition 3: Principle 3 results in departure times such that the average max load observed on individual buses approaches the desired occupancy dj. Proof: Figure 4 illustrates Principle 3 for the first three departure of the example problem in Table 1. Figure 4 shows the accumulative load curves of the three buses where the curve at B is shifted by 18 minutes to allow for an equal time basis (at the

122 r-72

Figure 4. Determination of the first three departure times (at A) considering even loads at the individual bus maximum load point

44


route's departure point) in the analysis. At the initialization the value of 50 is coordinated with the two accumulative curves to obtain: 7:11 at B and 7:22.5 at A. According to Principle 3 one selects the minimum time between the two to be the first departure at 7:11 (emphasized in Figure 4). It means that the first bus is shifted backward by 4 minutes to have at B, in an average sense, 50 instead of 65 passengers. Then one adds di = 50 to 50 at stop B curve, and to 22 at stop A curve. This results in 7:31 and 7:45 departures. Hence, 7:31 is the next departure, and the procedure continues and results in 7:56 as the last departure at the period [7:00 - 8:10]. Adding dj =50 to 122 (at A) or to 134 (at B) results in departures beyond 8:10. The bus of 7:11 has its d]=50 passengers at B and the bus of 7:31 -at A. This completes the proof-by-construction of Proposition 3. Figure 4 includes at its bottom the complete set of departure times of the example problem. If extrapolating the accumulative curve, another departure can be set at approximately 9:00. Comparison The comparison between the observed data of the example problem and the results of the three procedures is summarized in Table 2. In this table the associated individual average max load and its corresponding stop appear in brackets under each departure. It can be seen, as expected, that only procedure 3 complies with balanced loads at the critical individual max load points. Table 2. Departure times and loads of the observed data and for the three procedures 5th 6th Departure 1st 2nd 3rd 4th 7th Characteristic Observed 7:15 7:45 8:30 8:50 Observed 8:10 (65,B)* (80,A) (803) (94,A) (88,A) Procedure 1 7:20 7:40 8:35 8:52 Even Headways 8:01 8:18 (81,A) (75,A) (583) (723) (54,A) (573) Procedure 2 7:11 7:45 8:28 8:40 8:53 Even Load at 8:16 (88,A) (993) (79,A) (52,B) (61,A) _ Hourly Max (503) Load Point 8:24 Procedure 3 7:11 7:31 7:56 8:12 8:36 8:48 Even Load at (50,A) (50,B) (55,B) (55,A) (55,A) (5 5, A) Individual Max (503) Load Point "(i, j) in bracket means: i = average individual max load associated with the cell's departure time, j = the stop where i is observed or determined


3.

OPTIMAL DETERMINATION OF VEHICLE SCHEDULES

3.1

Background

45

Figure 1 presents the public transport operations planning framework as a multistep process. Due to the complexity of this process each step is normally conducted separately, and sequentially fed into the other. In order for this process to be costeffective and efficient, it should embody a compromise between passenger comfort and cost of service. For example, a good match between vehicle supply and passenger demand occurs when vehicle schedules are constructed so that the observed passenger demand is accommodated while the number of vehicles in use is minimized. Following the construction of an adequate public timetable in section 2 above, the next step is to determine vehicle schedules or chains of trips carried out by individual vehicles so as to reach the minimum number of vehicles required to cover the entire timetables. It is assumed that each vehicle has the same number of seats and same capacity (seats plus standees). This section provides an overview on exact solutions to the vehicle scheduling problem and describes a graphical heuristic procedure for the determination of minimum fleet size and its lower bound. 3.2

Exact solutions

The problem of scheduling vehicles in a multi-depot scenario is known as the MultiDepot Vehicle Scheduling Problem (MDVSP). This problem is complex (NP-hard) and considerable effort is devoted to solve it in an exact way. Review and description of some exact solutions can be found in Desrosiers et al. (1995), Daduna and Paixao (1995), Lobel (1999), and Mesquita and Paixao (1999). An example formulation of the MDVSP is as follows: [„ + !„ + ,

objective function: wm^VVc..^.. ^^ ^^ J J

(3)

where / is the event of-ending of a trip at time ai , j is the event of-start of a trip at time b • , and y;j -

ending is connecting to start! 0, otherwise

For i = n + 1 then yn+lj -I if a depot supplies a vehicle for the j'-th trip. For / = n + 1 then yi>n+l - 1 if after the i-th trip end, the vehicle returns to a depot, and yn+i,n+\ ~ No. of vehicles remain unused at a depot.

46


The cost function cfj takes the form c, =

(4)

where: K = the saving incurred by reducing the fleet size by one vehicle, Ly- direct dead-heading cost from event i toy , and Ey = cost of idle time of a driver between / andy. This formulation which appears in a similar form in Gavish et al. (1978) covers the chaining of vehicles in a sequential order from the depot to the transit routes alternating with idle time and dead-heading trips, and back to the depot. This is a zero-one integer programming problem that can be converted to a large scale assignment problem. In addition, the assignment of vehicles from the depots to the vehicle schedule generated in the above chaining process can be formulated as a "transportation problem" known in every operations research literature. Lobel (1999) is using a branch-and-cut method for MDVSP with the generation of upper bounds and the use of Lagrangean relaxations and pricing. Mesquita and Paxiao (1999) are comparing in this problem the linear relaxation based on multicommodity network flow approach. This review covers exact solutions to the vehicle scheduling component while realizing the complexity of the problem. This leads to look for some heuristics, hi the next section a heuristic solution is described. 3.3

Deficit Function (DF) approach

Following is a description of a step function approach described by Ceder and Stern (1981), for assigning the minimum number of vehicles to allocate for a given timetable. The step function is called Deficit Function (DF) as it represents the deficit number of vehicles required at a particular terminal in question in a multiterminal transit system. That is, DF is a step function that increases by one at the time of each trip departure and decreases by one at the time of each trip arrival. To construct a set of deficit functions, the only information needed is a timetable of required trips. The main advantage of the DF is its visual nature. Let d(k,t,S) denote the DF for the terminal k at the time t for the schedule S. The value of d(k,t,S) represents the total number of departures minus the total number of trip arrivals at terminal k, up to


47

and including time t . The maximal value of d(k,t,S) over the schedule horizon [T} , T2 ] is designated D(k, S) . Let t's and t'e denote the start and end times of trip i, i e S . It is possible to partition the schedule horizon of d(k,t,S) into sequence of alternating hollow and maximal intervals. The maximal intervals [sf ,ef ]/' = l,...,n(k) define the interval of time over which d(k,i} takes on its maximum value. Note that the S will be deleted when it is clear which underlying schedule is being considered. Index / represents the z'th maximal intervals from the left and n(k) represents the total number of maximal intervals in d(k,t}. A hollow interval Hf , l=Q,\,2,...,n(k) is defined as the interval between two maximal intervals. Hollows may consist of only one point, and if this case is not on the schedule horizon boundaries (7^ or 7^) , the graphical representation of d(k,t} is emphasized by clear dot. If the set of all terminals is denoted as T, the sum of D(k) for all k e T is equal to the minimum number of vehicles required to service the set T. This is known as the fleet size formula. Mathematically, for a given fixed schedule 5: max < / M

(5)

Where D(S) is the minimum number of buses to service the set T. When Deadheading (DH) trips are allowed, the fleet size may be reduced below the level described in Equation 5. Ceder and Stern (1981) described a procedure based on the construction of a Unit Reduction DH Chain (URDHC), which, when inserted into the schedule, allows a unit reduction in the fleet size. The procedure continues inserting URDHCs until no more can be included or a lower boundary on the minimum fleet is reached. The lower boundary G(S) is determined from the overall deficit function defined as g(t,S) = ^d(k,t,S) keT

where G(S) = max g(t , S) . This >el \' 2\

function represents the number of trips simultaneously in operation. Initially, the lower bound was determined to be the maximum number of trips in a given timetable that are in simultaneous operation over the schedule horizon. Stem and Ceder (1983) improved this lower bound, to G(S') > G (S) based on the construction of a temporary

48


timetable, S', in which each trips is extended to include potential linkages reflected by DH time consideration in S. This lower bound is further improved in this work. The algorithms of the deficit function theory are described in detail by Ceder and Stern (1981). However, it is worth mentioning the next terminal (NT) selection rule and the URDHC routines. The selection of the NT in attempting to reduce its maximal deficit function may rely on the basis of garage capacity violation, or on a terminal whose first hollow is the longest, or on a terminal whose overall maximal region (from the start of the first maximal interval to the end of the last one) is the shortest. The rationale here is to try to open up the greatest opportunity for the insertion of the DH trip. In the URDHC routines there are four rules: R=0 for inserting the DH trip manually in a conversational mode, R=l for inserting the candidate DH trip that has the minimum travel time, R=2 for inserting a candidate DH trip whose hollow starts farthest to the right, and R.-3 for inserting a candidate DH trip whose hollow ends farthest to the right. In the automatic mode (R-1,2,3), if a DH trip cannot be inserted and the completion of a URDHC is blocked, the algorithm backs up to a DH candidate list and selects the next DH candidate on that list. Figure 5 presents an example with 9 trips and four terminals (a, b, c, and d). In its upper part the 9 trips are shown with respect to time with departure and arrival terminals. Note that trip 4 starts and ends in the same terminal. Four DFs are constructed along with the overall DF. The maximal areas of the DFs are emphasized with a heavy line. Based on the NT procedure terminal a (whose maximal region is the shortest) is selected for possible reduction in D(a). Given that all DH times are 3 units of time, and using R=2, a DH trip is inserted from terminal c to a, DH]. This will increase d(c,i) at t=3 from -1 to 0, reduce d(a,f) at t=6 from 3 to 2, but will also increase d(c,t) at t=10 from 0 to 1. In order to eliminate the increase ofD(c) from 0 to 1 another DH trip is inserted, DH2 from a to c. The result is that D(a) is reduced from 3 to 2, and the DFs of a and c are updated with d(a,f)=2 between t=6 and t=7, and d(c,i)=Q between t=3 and t=10. One can see that no more DH trips (with trip time of 3 units) can be further inserted to reduce D(k), k=a,b,c,d. Hence D(S)=5. The sum of all the DFs, g(t), is illustrated at the bottom of Figure 5 and has G=3 (maximal number of vehicles in simultaneous operation). It will be used in a following section for the lower bound improvement. Finally, all of the trips, including the DH trips, are chained together for constructing the vehicle schedule (blocks). Two rules can be applied for creating the chains: first in-first out (FIFO), and a chain-extraction procedure described by Gertsbach and Gurevich (1977). The FIFO rule simply links the arrival time of a trip to the nearest


49

departure time of another trip (at the same location), and continues to create a schedule until no connection can be made. The trips considered are deleted and the process continues. The chain-extraction procedure allows an arrival-departure connection for any pair within a given hollow (on each deficit function). The pairs considered are deleted and the procedure continues. Figure 6 illustrates for clarity one hollow (between two peaks of the deficit function) with arrivals of trips 1 , 2 , 3 and departures of trips 4, 5, 6. Below the figure there is the FIFO chain (within this hollow) as well as other alternatives, where in all the minimum the fleet size is maintained. 1 2 3 4 I I I I

Figure 5. Nine-trip example with deadheading trip insertion for reducing the fleet size

50


An example of creating chains of trips within a hollow using FIFO rule and all other possibilities 3.4

Lower Bound

The initial lower bound on the fleet size with DH trip insertions was proved by Ceder and Stern (1981) to be G.(S). An improved lower bound of this problem was established and proved later by Stem and Ceder (1983), and Ceder (2002) using the following procedure: 1. extend each trip's arrival time to the time of the first feasible departure time of a trip with which it may be linked to T2 (the ending time of the finite time horizon). 2. given that the extended schedule is S', construct the overall DF, g'(t,S'), and determine its maximum value as G'(S'). While creating 5" it is possible that several trips' arrival points will be extended forward to the same departure point being their first feasible connection. Nonetheless


51

in the final solution of the minimum fleet size problem only one of these extensions will be linked to the single departure point. This observation opens an opportunity to look into further artificial extensions of certain trips' arrival points without violating the generalization of all possible combinations needed to prove that the resultant boundary on the fleet size is its lower bound. A stronger lower bound than G'(S') is found and proved in Ceder (2002). However it is beyond the scope of this chapter and hence not presented here. The stronger the lower bound is, the closer it is to the minimum fleet size required. Also, the stronger the lower bound is the better it serves the public transport decision makers on how far the fleet size can be reduced via DH trip insertions. Figure 7 presents the schedule of Figure 5 with 5" in its upper part and two overall deficit functions: g(t) and g '(t). All trips in S' are extended either to their first feasible connection (with all DH times are 3 units of time) or to the time horizon, t=18. The improved lower bound is therefore G'(Sr) = 4.

3

•o 0>

I

Figure 7. The example of Figure 5 with artificial extensions of each trip to its first feasible connection which results with the improved lower bound, G'=4

52 3.5

Advanced Modeling for Transit Operations and Service Planning Shifting Departure Times with Given Tolerances

Another factor considered in a manually produced public transport schedule is related to the shifting of trip departure times. A general description of a technique to reduce the fleet size for a variable departure time scheduling problem can be found in Gerstbach and Stern (1978). This technique for job schedule utilizes the deficit function representation as a guide for local minimization in maximal intervals, M"VueT . However when considering variable departure times along with a possible insertion of DH trips, the problem becomes more complex. The scheduler who performed shifting in trip departure times is not always aware of the consequences which could arise from these shifts. Ceder (2002) describes a formal algorithm to handle the complexities of shifting departure times. The algorithm is intended for both automatic and man-computer conversational modes. Figure 8 illustrates an example of two terminals and seven trips using the DH representation in part (i). Part (ii) shows how to reduce the fleet size using shifting tolerances of % time unit (forward or backward) where the shifts are shown with small arrows and the update DF is marked by a dashed line. Part (iii) shows how to apply only the URDHC procedure with DH times of 2 time units, and part(iv) presents a modified URDHC (mixed with the shifting) procedure. As can be seen in Figure 8(i) the fixed schedule without DH considerations requires 5 vehicles. Using shifting allows for reducing the number of vehicles to 3. The use of URDHC allows for reducing it to 4, and the use of a combined approach requires 3 vehicles. The viewpoint of the public transport operator will lead to use first the shifting procedure while wishing to minimize the operational cost (reducing DH mileage). However there is also the issue of passenger comfort while trying to accommodate the observed demand. Changes in departure times may result in imbalance passenger loads and reduction in the service reliability. Past experience in applying the DF approach at several bus properties shows that best is to first identify small shifts in departure times, enabling the reduction of the fleet size, without noticeable changes in the timetable. Second is to apply the combined approach of URDHC and shifting departure times.


Figure 8. An example with seven trips and two terminals using three procedures

53

54


4.

CONCLUDING REMARKS

4.1

Timetabling

Different public transport agencies use different scheduling strategies based primarily on their own schedulers' experience, and secondarily on their scheduling software (if any). As the result, it is unlikely that two independent bus agencies will use exactly the same scheduling procedures, at the detailed level. In addition, even at the same public transport agency, the schedulers may use different scheduling procedures for different groups of routes. Consequently, there is a need when developing computerized procedures to supply the schedulers with alternative schedule options along with interpretation and explanation of each alternative. Three such alternatives are presented in this chapter. Also, undoubtedly, it is desirable that one of the alternatives will coincide with the scheduler manual procedure. In this way, the scheduler will be in a position not only to expedite manual tasks but also to compare the different procedures regarding the trade-off between passenger's comfort and operating cost. The first part of this chapter presents the creation of public transport timetables with even headways and even average passenger loads on individual vehicles. Average even loads on individual vehicles can be approached by relaxing the evenly spaced headways pattern (rearrangement of departure times). It is known that passenger demand varies even within one hour, reflecting the business, industrial, educational, cultural, social and recreational public transport needs of the community. This dynamic behavior can be detected through passenger load counts, and information provided by road supervisors. The adjustments of departure times, made in this chapter by three procedures, form the basis to improve the correspondence of vehicle departure times with the fluctuated passenger demand. These adjustments, resulting in a balanced load timetables, are based on a given vehicle desired occupancy at the maximum load point of each vehicle. The keyword here is to be able to control the loading instead of being exposed repeatedly to an unreliable service resulted from imbalance loading situations. With the growing problems of cultural, social and recreational public transport needs of the community. This dynamic behavior can be detected through passenger load counts. With advance in the technology of passenger information system the importance of even and clock headways is reduced. This allows for introducing optional timetables with the consideration of even average loads on individual vehicles. The construction of such timetables takes into account, in essence, the


55

passenger perspective. The controlled procedures for adjusting the timetable, will eventually reduce one of the major sources of unreliable service, resulting also in the reduction of wait and travel times. 4.2

Vehicle Scheduling

This chapter, in its second part, describes a highly informative graphical technique for the problem of finding the least number of vehicles required to service a given timetable of trips. The technique used is a step function, called a deficit function, which was introduced in the last 20 years as an optimization tool for minimizing the number of vehicles in a fixed trip schedule. The step function is called Deficit Function (DF) as it represents the deficit number of vehicles required at a particular terminal in question in a multiterminal public transport system. That is, DF is a step function that increases by one at the time of each trip departure and decreases by one at the time of each trip arrival. The second part of this chapter presents both fixed and variable trip schedules, where in the latter possible shifts in departure times are allowed. Also this part describes an improved lower bound to the fixed schedule fleet size problem. The use both the fixed and variable schedules allows for the combination of deadheading trip insertions and shifts in departure times in the fleet size minimization problem. It is achieved by the exploitation of the fixed-and variable scheduling characteristics. That is, a possible deadheading trip can be inserted and reduce the fleet size only if a certain shift in departure time is made, and a time shift can reduce the fleet size only if a deadheading trip is inserted. It is worth mentioning that there are several example applications of the deficit function use in designing better public transport services. These example applications are: network route design, short-turn design of individual and group of routes, the design of operational public transport parking spaces, vehicle scheduling with different vehicle types, and the task of crew scheduling. Overall the DF approach provides immediate feedback on the value of shifting departure times, within given tolerances, as well as combining these shifts with the insertion of deadheading trips for reducing the fleet size. The value of embarking on such a technique is to achieve the greatest vehicle saving while complying with passenger demand. This saving is attained through a procedure incorporating a man/computer interface which would allow the inclusion of practical considerations that experienced public transport schedulers may wish to introduce in the schedule.

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REFERENCES Ceder, A. and H. I. Stern (1981). Deficit function bus scheduling with deadheading trip insertion for fleet size reduction. Transportation Science, 15(4), 338-363. Ceder, A. and H. I. Stern (1984). Optimal transit timetables for a fixed vehicle fleet. Proceedings of the ICf International Symposium on Transportation and Traffic Theory. UNU Science Press, Holland, 331-355. Ceder, A. (1984). Bus frequency determination using passenger count data. Transportation Research, ISA (5/6), 439-453. Ceder, A. (1987). Methods for creating bus timetables. Transportation Research, 21A (1), 59-83. Ceder, A. (2002). A step function for improving transit operations planning using fixed and variable scheduling. Transportation & Traffic Theory, (15th ISTTT), (M. A. P.Taylor, ed.) Elsevier Science & Pergamon Pub, 1-21. Daduna, J. R. and A. Wren (Eds.) (1988). Computer-Aided Transit Scheduling. Lecture Notes in Economics and Mathematical Systems, 308, Springer-Verlag, Berlin. Daduna, J. R., I. Branco and J. M. P. Paixao (Eds.) (1995). Computer-Aided Transit Scheduling. Lecture Notes in Economics and Mathematical Systems, 410, Springer-Verlag, Berlin. Daduna, J. R and J. M. P. Paixao (1995). Vehicle scheduling for public mass transitand overview. (J. R. Daduna, I. Branco and J. M. P. Paixao, Eds.). ComputerAided Transit Scheduling, Springer-Verlag, Berlin. Desrosiers, J., Y. Dumas, M. M. Solomon and F. Soumis (1995). Time constrained routing and scheduling. (M. O. Ball, T. L. Magnanati, C. L. Monma, G. L. Nemhauser, Eds.). Network Routing Volume 8 of Handbooks in Operations Research and Management Science. Elsevier Science B.V., 35-39. Desrochers, M. and J. M. Rousseau (Eds.) (1992). Computer-Aided Transit Scheduling. Lecture Notes in Economics and Mathematical Systems, 386, Springer-Verlag, Berlin. Furth, P. G. and N. H. M. Wilson (1981). Setting frequencies on bus routes: Theory and practice. Transportation Research Board, 818, 1-7. Gavish, B., P. Schweitzer and E. Shlifer (1978). Assigning buses to schedules in a metropolitan area. Computers and Operations Research, 5, 129-138. Gertsbach, I. and H. I. Stem (1978). Minimal resources for fixed and variable job schedules. Operations Research, 26, 68-85. Gertsbach, J. and Y. Gurevich (1977). Construction an optimal fleet for a transportation schedule. Transportation Science, 11, 20-36.


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Koutsopoulos, H. N., A. Odoni and N. H. M. Wilson (1985). Determination of headways as function of time varying characteristics on a transit network. (J. M.Rousseau, Ed.). Computer Scheduling of Public Transport 2. North-Holland, Amsterdam, 391-414. Lobel, A. (1999). Solving large scale multiple-depot vehicle scheduling problems. (N. H. M. Wilson, ed.). Computer-Aided Scheduling of Public Transport. Lecture Notes in Economics and Mathematical Systems, 471, Springer-Verlag, Berlin, 192-220. Mesquita, M. and J. M. P. Paixao (1999). Exact algorithms for the multi-depot vehicle scheduling problem based on multicommodity network flow type formulations. (N. H. M. Wilson, ed.). Computer-Aided Scheduling of Public Transport. Springer-Verlag, Berlin, 221-243. Rousseau, J. M. (Ed.) (1985). Computer Scheduling of Public Transport 2. NorthHolland, Amsterdam. Stern, H. I. and A. Ceder (1983). An improved lower bound to the minimum fleet size problem. Transportation Science, 17(4), 471-477. Voss, S. and J. R. Daduna (Eds.) (2001). Computer Scheduling of Public Transport. Lectures Notes in Economics and Mathematical Systems, 505, Springer-Verlag, Berlin. Wilson, N. H. M. (Ed.) (1999). Computer-Aided Scheduling of Public Transport. Lecture Notes in Economics and Mathematical Systems, 471, Springer-Verlag, Berlin. Wren, A. (Ed.) (1981). Computer Scheduling of Public Transport: Urban Passenger Vehicle and Crew Scheduling. North Holland Publishing Co.


CHAPTER 3

DESIGNING PUBLIC TRANSPORT NETWORK AND ROUTES Avishai Ceder, Civil Engineering Department, Transportation Research Institute, Technion-Israel Institute of Technology, Haifa, Israel 32000.

1.

INTRODUCTION

In the previous chapter it is shown that the public transport operational planning process includes four basic components performed usually in sequence: (1) network route design, (2) setting timetables, (3) scheduling vehicles to trips, and (4) assignment of drivers (crew). The framework of this process appears in Figure 1 with an emphasize on the first component which is the interest of this chapter. The problem addressed here is a general one of how to design a new public transport network or redesign an existing network, given no a-priori specifications as to the desired network structure. From a practical perspective it is desirable that the route design procedures include interaction and feedback loops between the selection of effective routes and the operational scheduling components: setting frequencies and timetables and schedule vehicles to the established trips with a special attention to the fleet size required. This is the reason for firstly describing the scheduling components

60


Figure 1. Framework of public transport operational planning process with emphasis on this chapter's component

Designing Public Transport Network and Routes

61

in the previous chapter, and secondly - the network route design component in this chapter. This chapter focuses on two public transport route design parts: (a) design of routes at the network level, and (b) creating efficient operations of each route using the shortturn strategy. In part (a) the presented method generates all feasible routes and transfers which connect every place (node) in the network to all others. From this vast pool of possible routes and transfers it then generates smaller subsets which maintain network connectivity. For each subset thus generated the analysis meets transportation demands by calculating the appropriate frequency for each route. Next, it calculates pre-specified optimization parameters for each subset. Based on the specific optimization parameter desired by the user, it is then possible to select the most suitable subset. The method has been designed as a tool for the planning of future public transport networks as well as the maintenance of existing ones. The presented method ensures flexibility by allowing the user to either input his own data or to run the analysis automatically. Part (b) of this chapter presents a set of procedures to design efficiently individual public transport routes with trips that are initiated beyond the route departure point and/or terminated before the route arrival point. These trips are called short-turn trips, hi practice, transit frequency is determined at the heaviest load route segment, whereas at other segments, the operation may be inefficient due to situations characterized by empty seats. The public transport schedulers attempt to overcome this problem by constructing manually short-turn trips with the objective to reduce the number of vehicles required to carry on the timetable. The purpose of this second part is to show methods on how to improve and automate this task. These methods identify feasible short-turn points; derive the minimum fleet size required to carry on a given schedule; adjust the number of departures in each short-turn point to that required by the load data, minimize the number of short-turn trips while ensuring that the minimum fleet size is preserved; and create vehicle schedules.

2.

PUBLIC TRANSPORT NETWORK DESIGN

2.1

Background

Generally speaking only a few researchers have studied the interrelationship between the scheduling components and the network design element. The interrelationship

62


exists in two directions: (i) each set of routes yields, based on the demand, a different set of frequencies and timetables, and ultimately, the required fleet size, and (ii) the operational cost derived from the scheduling components and the passenger level of service affect the search for the optimal route design while relying on a compromise between the operator and the user. Practical network design focuses almost entirely on individual routes which have been identified as candidates for change. However, it may be possible that the overall public transport network could be improved through restructuring of the entire network. For many public transport properties which have not been reappraised in this respect since the 1950s, it is high time to consider precisely such an undertaking. Such considerations motivate to seek an efficient network route design method, based on certain objective functions and a set of constraints. The main purpose of the method presented in this part of the chapter is to transport a given origin-destination demand through the public transport network in the most cost-effective way. The special characteristics of route design problems are: (a) passenger demand is spread throughout the entire network where it is generated and terminated at many points along the network's links, and can be grouped in terms of an origin-destination matrix; (b) the demand is to be transported simultaneously; (c) over a given planning horizon, it is impossible to reconstruct the routes, i.e., once the route network is designed, it will remain as it is over an entire planning period. Prior approaches to the public transport network design problem can be grouped into those which simulate passenger flows, those which deal with ideal networks and those based on mathematical programming. Simulation models are presented in Dial and Bunyan (1968), Heathington et al. (1968) and Vandebona and Richardson (1985). These models require a considerable amount of data, and their proximity to optimality is uncertain. Ideal network methods are based on a broad range of design parameters and a choice of objectives reflecting user and operator interest. Such methods appear in Kocur and Henrickson (1982), Tsao and Schonfeld (1984) and Kuah and Perl (1988). These methods are adequate for screening or policy analyses in which approximate design parameters are to be determined rather than a complete design. Thus these methods cannot represent real situations. Mathematical programming models are divided to generalized network design models and PT specific networks models. Known generalized network models are well summarized and review in Kim andBarnhart (1999) and as an example one can look at the heuristics developed by Farovlden and Powell (1994). The public transport network design models are inevitably heuristic due to the extremely high computational effort required. These


63

partial optimization approaches appear in Hasselstrom (1981), Dubois et al. (1979), Lampkin and Saalmans (1967), Silman et al. (1974), Rea (1971), Mandl (1979), Marwah et al. (1984), Sharp (1974) and Keudel (1988). Apart from Hasselstrom's model, which is included in the Volvo public transport planning package, all other models have not been actually applied. However, the Hasselstrom model is quite complex, non-user oriented and expensive, both in terms of the data required and the direct cost and staff time needed for the process. The disadvantages of the existing mathematical programming models can be summarized in five points: (i) cannot handle large size public transport networks; (ii) do not consider optimal objectives functions and constraints; (iii) vehicle frequency determination is based on economic parameters rather than on passenger counts (as in done in practice, and described by Ceder (1987) and in the previuos chapter); (iv) cannot incorporate simultaneously three out of the four planning components: network design, setting timetables and vehicle scheduling. In particular, the models cannot evaluate the network without defining the vehicle requirements for each route and thus lacks precision in evaluating the cost effectiveness of design; (v) cannot incorporate non-quantitative constraints such as imposing certain links to be included in public transport routes, and considering operational strategies. Another stream of articles related to the public transport network design problem is the area of public transport assignment problem. A good review of the literature of this assignment problem appears in Spiess and Florian (1989) and Wu et al. (1994). The public transport assignment problem solutions consider only the user perspective while assuming that passenger behavior reflects the minimization of walk, wait and in-vehicle times, or a weighted sum of these times. The problem input is a given public transport network of routes, and the question is how to distribute a given public transport demand on this given network of routes. Spiess and Florian (1989) developed a two-part algorithm to assign the passenger demand. Their algorithm's first part computes the total travel times between origin and destination while considering possible transfers. In the second part, the passenger demand is assigned based on a defined optimal strategy. Wu et al. (1994) propose a model for the public transport equilibrium assignment problem. Their approach considers the effect of traffic congestion on the passenger route choice decision with the modeling of flow dependent waiting times. These assignment procedures could be especially useful to

64


describe passenger behavior for a given network of routes while evaluating different traffic control and road management strategies. However the examination of an extensive amount of networks of routes, by these procedures, is time consuming. 2.2

Methodology

The overall methodology or system comprises six elements shown in Figure 2. Its entire formulation and interpretation appear in Ceder and Israeli (1992), Israeli and Ceder (1995, 1996) and Ceder (2001). In the first element, the system generates every feasible route and transfer (throughout the entire network), from all terminals including shortest path computation. Initially the network contains average travel times covering a time window, which is usually the peak period. These measured average travel times are then input into the calculation of shortest path between each origin-destination (O-D). Each determined candidate route meets the route length factor constraint. In other words, one procedure in this element screens out routes according to given boundaries on the route length, hi addition, there is a limit on the public transport route average travel time between each origin-destination (O-D) pair. That is, a given public transport demand, usually during peak hours, cannot be assigned to a candidate route if its average travel time exceeds the shortest path travel time by more than a given percentage. The feasible transfers are based on the following method: establishing additional direct routes between O-D pairs characterized by high O-D demands (predetermined O-D). These direct routes are actually initiated and/or terminated at non-terminal nodes and consequently, deadheading trips are responsible for their connection to the terminals. Also, a low OD demand, without a direct route, is not considered for obtaining service. The transfers are created using a mapping algorithm (branching routing possibilities along with constraints check). This applies to the disconnected O-D pairs, as well as to all the O-D pairs. hi the second element the system creates minimal set(s) of routes and their related transfers, such that connectivity between nodes is maintained and their total deviation from the shortest path is minimized. This problem is defined as a Set Covering Problem (SCP), which is hard to solve (Minieka, 1978). The SCP can determine the minimal set of routes from the matrix of the feasible routes, hi this matrix each row represents either a feasible route or a feasible transfer, hi the third element the entire O-D demand is assigned to the chosen set of routes. The assignment algorithm which

Designing Public Transport Network and Routes Input: Network nodes and arcs ; Average travel time on each arc ; public transport Origin -Destination demand ; set of terminals

Maximal deviation from shortest path ; Maximum route length

Additional routes and transfers given by user

^w

Element 1: Creation of public transport routes and determination of transfers

Element 2: Creation of minimal set of routes and transfers (Set Covering Problem)

Policy headway (inverse of min. frequency); Vehicle desired occupancy ; Set of routes and transfers given by user (e.g. existing); weights of passenger wait, transfer and in-vehicle times Monetary weights for the optimization criteria elements

Constraints on the effectiveness of each set of solutions

Element 3: Demand assignment procedure for frequency calculation

Element 4: Calculation of the optimization criteria, Z j and Z 2 (in terms of passenger hours, passenger hours difference from shortest path, waiting time, empty passenger hours, fleet size)

Element 5: Generation of more sub-optimal solutions (Z i, Z2)

Element 6: Multi-objective user decision Figure 2. A methodology for designing public transport routes

65

66


has been developed (Israeli and Ceder, 1996) includes steps which are related to a route-choice decision investigation, i.e. the algorithm includes a probabilistic function for passengers who are able to select the public transport vehicle that arrives first, or alternatively, wait for a faster vehicle. The passengers' strategy is to minimize the total weights of wait, transfer and in-vehicle times. The methodology used is similar to that developed by Marguier and Ceder (1984) but with a different probabilistic function. The approaches used in Spiess and Florian (1989) and Wu et al. (1994) can be inserted in this element as well though they require extensive computation time and do not intend to be executed on a vast amount of alternative public transport networks. The fourth element represents the optimization criteria from the passengers, operator, and community perspectives. It is detailed below in the next section where Z, is the criteria in passenger-hours units and Z2 comes in terms of the fleet size required. The fifth element is responsible for constructing alternative sets of routes in order to search for additional (Z\, 7.2) values in the vicinity of their optimal setting. The procedure for this search is based on incremental changes in the set of routes, much like the reduced gradient methods. Given the set of routes associated with the minimum Z\ value, the single route which is the worst contributor to Z\ is deleted and then the SCP is solved in the second component, followed by the execution of the third and fourth elements. This process could continue, but there is no guarantee that a previous alternative will not be repeated. In order to overcome this problem, a new matrix is constructed with the idea of finding the minimal and worst set of candidate routes for possible deletion in each iteration, i.e., a new SCP matrix is constructed in which the candidate routes are the columns and each row represents a previous set of routes which was already identified in the vicinity of the optimal (Z\, Z2) setting. The solution to this new SCP matrix is a set of rejected routes so as not to repeat a previous alternative solution. During this process, a number of unique collections of routes are termed "prohibited columns" as they are the only ones which can transport a certain demand. These prohibited columns are assigned an artificially high cost value, so as not to be included in the solution. This process also involves some bounds in order to converge on a desired number of iterations, or number of (Z\, Z2) solutions. The sixth and final element of the system involves multi-objective programming of the two objective functions Zj and Z2- Given the alternative sets of routes derived in the fifth element, the purpose is to investigate the various alternatives regarding the most efficient (Z\, Z2) solution. The method selected in this element is called the


67

compromise set method (Zeleny, 1973, 1974). It fits linear objective functions (Duckstein and Opricovic, 1980) for discrete variables. This multi-objective method can be also found in a recent book by Coello et al. (2002). The outcome of this method is the theoretical point in which (Z\, 7.2) attains its relatively minimal value. The results can be presented in a table or a two-dimensional graph which shows the trade-off between Z\ and Z2- These results also indicate the optimal zone or the socalled pareto front. The decision-maker can then decide whether or not to accept the proposed solution. In the latter case, for example, the decision-maker can see how much Zj is increased by decreasing Z2 to a certain value and vice versa. 2.3

Formulation

The two objective functions Z\ and Z2 mentioned above can be formulated as follows:

where, APH(i,j) = Passenger Hours difference, for the demand between nodes i and j, i, j e N, between the public transport route and the shortest (automobile) path (defined as passengers' riding time in a public transport vehicle on an hourly basis minus the total passengers' time if riding the shortest route. It measures how much excessive time is spent by passengers on public transport vehicles between the two nodes); WH(ij) = Waiting Time between nodes i and j, i, j e N, (defined as the amount of wait time at public transport stops and transfer points, on an hourly basis. It measures how much wait time is spent by passengers between the two nodes); EHr = Empty Space-Hours on route r (defined as the unused seats in a public transport vehicle on an hourly basis. Empty Space-Hours measures to what capacity public transport vehicle are used); FS — Fleet Size (number of public transport vehicles required to provide all trips along the chosen set of routes); • r line 6

D

£

c

)= = = = =

Hne5

J -fr.~< >-===-"=*"""^ line 5 jj line 5

ine 7

•s

O

line 7

£ stop pedestrian nodes • stop nodes • line nodes

0

lineS

ine 8

-A- centroids O pedestrian nodes

Jk

•= * () *

^

.

line 7 line 8

pedestrian links boarding/alighting links ''ne linlcs

Figure 3. Line-based network representation for high-frequency transit services

Transit Path Choice and Assignment Model Approaches

97

TIMETABLE run

Terminal A

Terminal B

arr.

dep.

08.30

10.10

10.15

11.00

12.35

12.37

arr.

dep.

IC634

08.55

09.00

IC640

08.25

IC741

10.58

Terminal C

arr.

dep.

11.15

11.18

12.00

12.05

14.00

14.02

Figure 4. The run-based representation of transit services The service sub-graph Qg consists, in turn, of different sub-graphs ngr (one for each run of the transit services). Referring to the generic run r, the relative sub-graph Qgif consists of nodes representing the arrival and departure times at stops, and links representing travel from one stop to another (run section) or the dwelling of the vehicle at a given stop. Other nodes represent the time in which users board or alight from each run at the stop. These nodes are connected to the nodes representing run arrival and departure through boarding and alighting links. Finally, the whole subgraph Qg is built by connecting all the sub-graphs ngr through links representing the users' transfer from one run to the next one at the same stop (stop axis).

98


The demand sub-graph Qd representing temporal demand segmentation is made up, in turn, by the same number of sub-graphs Qj.c as centroids. For each spatial centroid c, the sub-graph Od.c consists of temporal centroids, that are nodes located spatially in the position of the spatial centroid c and temporally according to the user origin departure times TotThe access/egress sub-graph Qae is made up of: - links connecting origin temporal centroids to nodes on the first boarding stop axis, to represent the access to the services; - links connecting alighting nodes of the stop axis to the destination spatial centroids, to represent the egress from the transit system; - links connecting stops, to represent possible interchanges between different stops, too. Finally, the global diachronic graph Q is obtained connecting the three sub-graphs by using proper links. The diachronic graph is very useful as it allows efficient use of standard network algorithms (as least-cost paths) and allows more straightforward treatment of congestion, when it is considered. Note that, in the line-based approach, all runs belonging to the same line are represented by a single sequence of links (see Figure 4) to which average values of run attributes belonging to the line are associated with. This representation is simpler than the run-based one but allows us to obtain only line average values (e.g. average on-board loads) as assignment results. The dual graph was initially developed to model turn prohibitions/restrictions in road networks and then was extended to represent public transport networks. The extension to public transport networks is done by replacing each transit segment (which represents a line or a single run according to the supply representation approach) with a node. Nodes are connected by links considering time congruence relative to arrival/departure times of transit vehicles (runs) at stops. An example of the dual graph representation of a stop with multiple transit routes and runs is reported in Figure 6. Note that, for simplicity, Figure 6 only shows the dual link for one particular link/route, but all other combinations can be created in the same way according to the arrival/departure times of scheduled runs. This implies that for large transit networks we have to consider an high number of links to represent all possible connections in


99

terms of arrival/departure runs at stops, in addition to the fact that the transformation from the original network to the dual network and vice versa have to be considered at least at the starting and ending phases of the simulation procedure.

temp oral centroids

stop axis s

Figure 5. The diachronic run-based representation of transit services

100


Figure 6. The dual graph representation of transit services A different approach is the mixed line-based/database one, which uses a database to consider the timetable together with a traditional line-based network model. It uses a line-based approach (see section 2.1) to describe spatially the service network topology in terms of routes, lines and stops, while data associated with nodes and links define the clock-dependent movements of transit vehicles (runs) on the network. This approach is more complex with respect to the above described ones, even if considering some simplifications made by expanding on-line the strictly necessary part of the network, in order to define the time-dependent congruence of services on the line-based representation. The disadvantage of this approach is that the direct use of traditional network algorithms to carry out path search and network loading is not allowed, as clockdependent variables have to be considered through the database (e.g. using modified Dijkstra type algorithms).

3.

USER PATH CHOICE MODELS FOR TRANSIT SERVICES

User decisions in path choice can be classified according to two types of choice behaviour:

Transit Path Choice and Assignment Model Approaches -

-

101

pre-trip choice behaviour, which underlies user choices before departure. It includes the comparison of possible alternative strategies and the choice of one of them on the basis of expected characteristics, or attributes. en-route choice behaviour, which underlies user choices during the trip. This behaviour describes how users respond to unknown or unpredictable events. The type of en-route choice behaviour can be defined as: + indifferent: if users board the first arriving vehicle belonging to the set of alternatives; + intelligent: if users, when a vehicle belonging to the attractive set arrives, compare the disutility of the arriving vehicle with the disutility of the next arriving vehicles belonging to the set of alternatives.

Moreover, path choice model have to be specified according to user behavioural hypotheses that depend on specific user and service characteristics. The main service factors affecting user behaviour in path choice are frequency, regularity and information available to users. Service frequency can be related directly to the frequency of line / in the reference period, i.e. the number of runs belonging to line / in such a period or, for overlapping lines, to the cumulative frequency, i.e. the sum of the frequencies of all "attractive" lines connecting the O/D pair od. Usually service frequency is high if the average headway of vehicles is less than 12-15 minutes, while frequency is defined low if the average headway exceeds 15 minutes. Service regularity is a measure of to what extent the schedule is observed. If regularity is used to make assumptions on user behaviour in line-based systems, such as buses and trains, deviations from the schedule should be related to the average headway of runs belonging to the same line. Usually regular services are associated with low service frequencies, typical of extraurban systems (e.g. railways). On the other hand, irregular services generally refer to high frequencies as in urban or metropolitan areas, e.g. bus or underground lines. Pre-trip and/or en-route real-time information on services can be available to the user in different places (for example at stops or at home) and concerns at least waiting times of arriving vehicles at the chosen access stop, while more advanced information systems could give information on travel times and on-board occupancy, too. Static information on run schedule is traditionally available with timetable. Intelligent Transportation Systems (ITS) have significantly expanded the range of information available to the traveller through Advanced Traveller Information Systems (ATIS).

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ITS also improves the performance of transit services, in terms of regularity, through the use of Advanced Public Transportation Systems (APTS). As regards user characteristics, the main difference to consider concerns whether they are frequent or occasional users. Frequent users travel frequently and know routes and scheduled timetable as well as the real system functioning based on previous experience. Occasional users use sometimes transit services, so they only know some line routes (the most important) and their scheduled timetable. Of course different types of users that behave differently from those described above can be specified, but in the following we refer to these two user types that can be considered at the extremes of the possible range of user type to be taken into account. In addition, user behaviour is often related to trip purpose. Two main approaches can be used in transit path choice modelling: the frequencybased approach and the schedule-based one. The frequency-based approach considers services in terms of sets of runs (lines). In this case run scheduled times are not considered explicitly, but we refer to the line headways, or to their inverse (the service frequencies), from which the name of the approach derives. Therefore we are not able to calculate explicitly attributes that users consider in relation to single runs, but we can refer only to average values relative to lines. The schedule-based approach refers to services in terms of runs using the real vehicle arrival/departure time, and hence all the values of level of service attributes, evaluated at time in which users make their choices, can be explicitly taken into account. This approach allows us to take into account the evolution in time of both supply and demand, as well as run loads and level of service attributes. Given the modelling approach (frequency-based or schedule-based), as the main factor that greatly affects user behaviour is service frequency, the following sections will describe path choice models for both high and low frequency services, considering either the frequency-based and the schedule-based approach. User behaviour hypotheses and path choice models will be specified in the framework of (random) utility theory, in which users are assumed rational and behave with the aim of maximising their perceived utility.

Transit Path Choice and Assignment Model Approaches 3.1

103

Path Choice Models for High-Frequency Services

In relation to the high frequency of services we can assume that the origin departure time coincides with the desired origin departure time, so user arrival at the stop is not related to run departure scheduled times. Furthermore, it is assumed that users do not have full information before starting their trip and they follow a mixed pre-trip/enroute choice behaviour. The choice of boarding stops is considered to be made before starting the trip, since it is not influenced by unknown events, while en-route choices occur at stops and are relative to the decision to board a particular run or to wait for another run of the attractive set. 3.1.1 Frequency-Based Path Choice Models for High-Frequency Services In the sphere of the frequency-based approach, path choice for high-frequency services implies the choice of the boarding stop and that of the line (or sequence of lines) leading users to their destination. In the case of frequency-based models, the traditional approach in path choice modelling is that based on the concept of optimal strategy, in which users do not choose a precise path, but they rather identify with a pre-trip choice behaviour a set of attractive paths, within which they choose during the trip on the basis of an indifferent en-route choice behaviour at stops, minimising the perceived average trip cost, hi the following of this section we will refer to the line-based representation of the transit network reported in Figure 7, which is a convenient way to represent the line-based supply model reported in Figure 3.

Figure 7. Example of line-based transit networks

104


Under the assumptions of mixed pre-trip/en-route choice behaviour, a set of attractive paths can be represented on the transit network through a subgraph known as hyperpath. An hyperpath example for the transit network of Figure 7 is reported in Figure 8.

Figure 8. Example of hyperpath Given the en-route choice behaviour and the service functioning conditions, an average generalised cost C, can be associated with each hyperpath j. This cost is equal to the average generalised costs C* of paths belonging to hyperpath j, weighted by the probabilities q^j of choosing path k in the effective trip:

0) kej

kej

lek

where c\ represents the generalised cost of the generic link / belonging to the generic path k. To each diversion node i of the hyperpath j will correspond a diversion set L/j of attractive lines belonging to the considered hyperpath. To the boarding links l=(i,m) connecting the diversion node z with nodes m of the lines belonging to Lfj, it is possible to assign a diversion probability, rjij, that is the probability of using the line corresponding to link / connecting node z with node m of hyperpath j. Such a probability depends on the assumed user en-route choice behaviour and on service characteristics. In case of indifferent en-route choice, according to the hypothesis that users board the first arriving vehicle of a line belonging to the attractive set, and with the assumption that user and vehicle arrivals at stops can be modelled as Poisson random process, the


1 05

probability of boarding line / belonging to the set of attractive lines at stop z, L,j, can be expressed as:

where Us-) = prob(Vs + es> Vs- + es-)

s'*s, s'eSod

(10)

where Us is the perceived utility, sum of the systematic utility Vs and of a random residual es. The systematic stop utility is a function of the stop-specific attributes (e.g. access or egress times, presence of shops, etc.) and "inclusive utility" expressing the average utility associated with all runs available at stop s. As described for the run choice, deterministic or stochastic stop choice models can be specified according to the assumptions on random residuals ss. 3.2

Path Choice Models for Low-Frequency Services

From a practical point of view, low-frequency services, as regional bus and intercity railways, are usually characterised by regular service functioning and we hypothesise users have full information before starting their trip. In this case there is no difference in user behaviour for frequent and occasional users, and the presence of user information is unnecessary to support user choices, assuming that they at least know routes and timetable. Furthermore, as low-frequency services are typical of extraurban areas, it is common that only one access terminal as well as a single egress terminal are available, so stop choice can be easily simulated. The run choice is assumed fully pre-trip and, in addition to the other service attributes, we need to consider the disutilities that occur because of the difference (that can be considerable) between desired user departure time and vehicle scheduled departure time or between desired user arrival time at destination and run scheduled arrival. In the literature, this difference is called early schedule penalty or late schedule penalty. Figure 11 reports an example of early/late schedule penalty for a desired departure time (DDT) from origin at 1.30pm. Considering the connected arrival time at the stop at 1.50pm, the closest runs that allow us to reach the destination are train IR310, departing at 1.30pm, and train IR312, departing at 2.30pm. Hence users who will board train IRS 10 have to bring forward their origin departure time to 1.10pm, which implies an early departure penalty of 20 min to be taken into account in the utility function as an early schedule penalty (ESP) attribute. On the other hand, if the user boards train IR312 departing at 2.30pm, a late schedule penalty of 40min as a late schedule penalty (LSP) attribute has to be considered. The weights of early and late schedule penalties and hence the probabilities of choosing trains mainly depend on trip purpose, as will be shown in chapter 5.


111

Figure 11. Example of early/late schedule penalty 3.2.1 Frequency-Based Path Choice Models for Low-Frequency Services In the case of low-frequency services, the main problem of the frequency-based approach is that it does not allow to consider explicitly the above described early/late schedule penalties connected to departure/arrival in early/delay in relation to desired departure/arrival time. Moreover, the frequency-based approach in the case of low-frequency services leads to great approximations when attributes for path alternatives that include interchanges have to be assessed. In fact the use of a line-based representation of services does not allow a precise evaluation of level of service attributes at single run level, for which average values have to be considered. It implies relevant errors in path choice probability prediction. Furthermore, in the case of low-frequency services, the frequency-based approach is modelled by using the concept of optimal strategy or hyperpath (see section 3.1.1) that, as it is based on quite different assumptions about user and vehicle arrivals at stops with respect to those assumed in the low-frequency case, is not suitable to be used.

112


3.2.2 Schedule-Based Path Choice Models for Low-Frequency Services In order to take into account explicitly the early/late schedule penalties, the schedulebased approach is required. The probability p[r\ToJ of choosing run r, given the O/D pair od and desired departure time TDI (or desired arrival time r^/) can be written as

Attributes that can be considered in the systematic utility Vr are: access and egress times and costs; on-board times; transfer times; number of transfers; monetary cost; comfort; early/late schedule penalty. Most of schedule-based path choice models for low-frequency services take into account user target times (desired departure times at origins or destination arrival times at destinations), considering as path choice set alternatives the two "nearest" paths in terms of minimum early and late times with respect to the user target time (see Figure 12).

Figure 12. Example of path choice alternatives


113

Some models consider maximum earliness and lateness values to define a time slice around user target time, within which the path choice set is defined, while other models consider only the late alternative with respect to the origin target time or the early alternative in the case of desired arrival time at destination. If the path choice model is deterministic the minimum disutility alternative is considered, while if it is stochastic a certain probability to all alternatives of the choice set is associated with. Deterministic and stochastic path choice models can be specified according to the hypothesis of null er or otherwise. Assuming er different from zero, different random utility models can be specified. In particular, in the case of multi-class multi-service transit systems (e.g. railways), the existing correlation between alternatives have to be taken into account, and nested-logit or probit models should be used. Moreover, considering a learning process on attributes composing the systematic utility, day-to-day dynamic schedule-based path choice models for low-frequency services can be specified to take into account the day-to-day evolution of on-board loads and level of service attributes.

4.

ASSIGNMENT MODELS FOR TRANSIT SERVICES

Assignment models allow the on-board load of transit vehicles to be obtained. The level of output results depends on the path choice modelling approach (frequencybased or schedule-based). In particular, using a schedule-based approach, in which all runs of the transit services are explicitly considered, it is possible to obtain very detailed results in terms of on-board loads and level of service attributes for each vehicle. As this approach allows us to take into account the evolution in time of supply, demand and on-board loads, these models are intrinsically within-day dynamic. More aggregate results can be obtained by using a frequency-based approach that, considering services in terms of sets of runs (lines), allows to refer only to average values relative to lines and to compute only average line flows. The use of average line on-board flows, do not allow us to take into account peaks of loads on vehicles that can occur inside the reference period. Finally, the use of a constant average rate of user arrivals at stops, when this rate changes significantly during the period of analysis, could lead to relevant errors in vehicle load calculations. Whatever the used modelling approach is, the classification of assignment models usually refers to the classical one adopted for road networks, in which different

114


assignment models can be specified according to the type of behavioural path choice model (deterministic or stochastic), the type of link performance functions (flowdependent or otherwise, which lead to congested or uncongested networks), the assignment approach (network loading, user equilibrium, dynamic process, and so on), and the dynamic evolution (within-day and/or day-to-day) they take into account. Assignment models for transit networks were traditionally developed using the frequency-based approach on the basis of the optimal strategy or hyperpath concept to simulate path choice, while recently the innovative schedule-based approach, based on single space-time paths, have been considered. Hence, in the following description, when we talk about paths, we refer to an hyperpath in the frequency-based approach and to a single space-time path (which includes a run or a sequence of runs) in the schedule-based approach. Table 2 reports the classification parameters of assignment models for transit networks. Table 2. Classification of transit assignment models Path Choice Model Stochastic Transit network Assignment Deterministic approach Network uncongested SNL AON Loading [c=cost] congested

Equilibrium

DUE

SUE

[c=c(f)l

Dynamic

DDP or SDP

DDP or SDP

AON=A11 or Nothing; SNL=Stochastic Network Loading; DUE=Deterministic User Equilibrium; SUE=Stochastic User Equilibrium; DDP=Deterministic dynamic process; SDP=Stochastic dynamic process.

4.1

Frequency-Based Approach

In the framework of the frequency-based modelling approach, the AON (All or Nothing) assignment model allows us to load only the lines belonging to the least cost hyperpath, while using a probabilistic path choice model, the SNL (Stochastic Network Loading) assignment model allows to calculate the choice probability for all hyperpaths belonging to the hyperpath choice set, according to which lines belonging to hyperpaths are loaded.


115

For congested networks both equilibrium and dynamic models can be used. UE (User Equilibrium) assignment approach has been traditionally used to simulate congested transit networks using, at first, the variational inequality approach, and then the fixedpoint one, for which it is possible to demonstrate existence and uniqueness of equilibrium flows and costs. Recently, SUE (Stochastic User Equilibrium) assignment models have been specified on the basis of fixed point models. Stochastic models are more coherent with the problem, as the travel time associated with most links is better modelled as a random variable. For example, in addition to the user perception errors, link travel time dispersion depends on the traffic congestion on the transit mode and on the other interacting modes, as well as waiting time at bus stops depends on service irregularity. Day-to day dynamic process assignment models, both deterministic and stochastic, can be specified by considering a learning process on attributes and an updating mechanism of user choices. 4.2

Schedule-Based Approach

The type of assignment models presented in the framework of the frequency-based approach can be specified in the schedule-based approach, too. In particular, for uncongested transit networks, both AON (All or Nothing) and SNL (Stochastic Network Loading) assignment models can be considered to calculate on-board loads, as well as DUE (Deterministic User Equilibrium) and SUE assignment models can be specified for congested transit networks. For what concern the dynamic evolution, as in the schedule-based approach the timetable is explicitly considered, the within-day dynamic is a native characteristic of the schedule-based approach, in the sense that all schedule-based assignment models are at least within-day dynamic. Moreover, considering a learning process on attributes, the day-to-day dynamic can also be considered, leading to doubly dynamic (within-day and day-to-day dynamic) assignment models. The dynamic evolution can be simulated by dynamic process models, both stochastic and deterministic. A detailed description of the schedule-based assignment models, as well as a concise state of the art will be deepened in chapter 5.

116 4.3

Advanced Modeling for Transit Operations and Service Planning Application Fields of the Different Approaches

In the last ten years the improvements in computer science and the increasing computation capacity have allowed the schedule-based approach to be developed and applied, sometimes replacing the commonly used frequency-based approach. The schedule-based approach was initially developed for low-frequency systems and has recently been proposed for urban transit systems in order to consider more coherent user behavioural hypotheses in relation to the characteristics of the analysed public transport system. Frequency-based models are suitable to be used when an high degree of detail is not necessary and results have to be obtained with few input data. For example in the case of the strategic planning of a new transit network, we need to specify only the line paths and stops in addition to service frequencies, and the average line loads, obtained as output, are sufficient for the aims of the project. In this case, the definition of the timetable and the output in terms of load for each run is not consistent with the level of detail of the other components of the simulation models. By contrast, in the sphere of operative planning (e.g. with the aim of supporting the definition of a new timetable or to assess the introduction of ITS systems) we need more precise and detailed results that can be obtained only through a schedule-based approach. Moreover, an example of problems that arise using the frequency-based approach, when bus arrivals are not uniform for both constant and time-varying user arrivals at stops, is reported in Figure 13. Let us consider a stop served by two lines (A and B) with the same service frequency (10 bus/h), but line B arrives just one minute after line A (see Figure 13). Even if assuming a constant user arrival rate at stop (e.g., 3 users/min), according the frequency-based approach, we have that users will board lines A and B according line frequency (that is, 50% on each line), while it is clear that all users will board line A, if congestion phenomenon does not arise, except for those arrived immediately after line A departure and before the departure of line B, leading to choice probability quite different from the 50%. This difference depends on user and vehicle arrivals at stops, requiring a modelling approach that takes into account both service timetable and time-dependent user arrivals.


117

Figure 13. Example of frequency-based approach disadvantages For what concern the operative application of such models, although all possible combinations of frequency, regularity and information to users are possible, three type of public transport systems (reference systems), which define three situations of greater operative interest, can be considered: - low-frequency regular services (extraurban systems); - high-frequency services without user information at stop (urban traditional systems); - high-frequency services with user information at stop (urban innovative systems). A summary of user behavioural hypotheses for the reference transit systems are reported in Table 3. Each of these three cases may require different supply network models and path choice models. They should be simulated using a scheduled-based approach, except for the case of high frequency services without user information that, under the hypothesis of constant user arrival rate and random vehicle arrivals as a Poisson process, can be simulated without distinction through a schedule-based or frequency-based approach, as it is possible to demonstrate that the hypothesis of indifferent en-route choice is fully acceptable. Table 3. Service characteristics and user behaviour transit system low- frequency high-frequency & information high-frequency & no information

stop choice pre-trip pre-trip pre-trip

line/run choice user arrival timetable-related pre-trip not en-route (intelligent) timetable related not en-route (intelligent or indifferent) timetable related

118

5.


REFERENCE NOTES

This section reports some references for public transport networks in the sphere of supply and path choice models for both the frequency-based approach and the schedule-based one. References to assignment models will be given only for the frequency-based approach, as chapter 5 will report in details that for the schedulebased approach. An effort has been made to give credit to the proper authors quoted from the literature. Nevertheless, for well-known results credits may be unintentionally omitted or misplaced. Apologies are submitted in advance for any error of this type. 5.1

Supply Models

In the sphere of the supply modelling, the diachronic network was proposed by Nuzzolo and Russo (1993) and was used in the schedule-based path choice and assignment models for low-frequency services presented by Cascetta et al. (1996), Crisalli (1999), Nuzzolo and Russo (1994, 1998b) and Nuzzolo et al. (2000). The diachronic network, with some changes in the demand subgraph and access/egress subgraph to take into account different demand and access/egress characteristics, was also used to represent high-frequency services (Nuzzolo and Russo, 1998a; Nuzzolo etal, 1999; 2001a; 2001b). The dual graph representation is described in Anez et al. (1996) and in de La Barra (1989). It was used in the integrated land use-transport model TRANUS (Modelistica, 1993) and in the initial phase of the set up of the transit network model of the largescale model system for the Copenhagen-Ringsted railway project (Nielsen and Jovicic, 1999). The mixed line-based/database supply representation have been widely used in literature (Carraresi et al., 1996; Florian, 1998; Hickman and Wilson, 1995; Hickman and Bemstain, 1997; Nguyen et al., 1997, 2001; Nielsen et al, 2000; Tong and Richardson, 1984; Wong and Tong, 1999), as it can be obtained by slight modifications of the traditional line-based approach to represent transit services, even if for large networks some computation problems could occur. This approach was initially used by Tong and Richardson (1984) and is described in Wong and Tong (1999). They use a line-based graph associating with line segments a data set which considers the number of runs of the transit line and their relative departure times at both upstream and downstream nodes. A similar approach is used in EMME/2 release


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9, described in Florian (1998), in which to reduce memory problems a mixed linebased/database supply model is used, but the clock-dependent congruence of run vehicles on the network is assured by expanding in real-time the strictly necessary part of the network to define the time-dependent congruence of single runs on the line-based representation. The mixed line-based/database approach is also used in Nielsen et al. (2000) to reduce computation times with respect to the use of the dual graph used at a first stage of their research (Nielsen and Jovicic, 1999). 5.2

Path Choice Models

The modelling approach (frequency-based or schedule-based) is the main criterion for the analysis and classification of existing path choice models. For this reason, section 5.2.1 reports path choice models developed by using the frequency-based approach, while section 5.2.2 presents path choice models specified through the schedule-based approach. Then, as user behaviour and choice dimensions depend on user and service characteristics (mainly the service frequency), classification considers models specified for high and low frequency transit systems 5.2.1 Frequency-Based Path Choice Models In the frequency-based approach, the simulation of path choice for high-frequency transit systems has been commonly performed on the basis of the concept of optimal strategies presented by Spiess and Florian (1989). Nguyen and Pallottino (1988) characterised and studied in detail the notion of the optimal strategies by using a graph theoretical approach, and developed the concept of hyperpath. Starting from these deterministic path choice models, Cantarella and Vitetta (200la) recently proposed a stochastic path choice model, which considers perception errors of path attributes. In the framework of low-frequency services, frequency-based path choice models are not present in literature and the same models specified for high-frequency services (optimal strategies) are often used by practitioners, even if they imply relevant approximations in path choice simulation due to different user behavioural hypotheses. 5.2.2 Schedule-Based Path Choice Models In the sphere of high-frequency transit services, Table 4 reports some of the existing path choice models, classified in relation to the possibility to take into account service

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regularity, user information, and different user type, as well as the used approach in utility and dynamic specifications. Table 4. Schedule-based path choice models for high-frequency transit systems user choice service user dynamic utility type approach approach mechanism regularity info Y D Hickman and Wilson (1995) FU SIM R-I WD Y SEQ S FU R-I WD Hickman and Bernstein (1997) FU S Y-N SIM R-I WD Nuzzolo and Russo (1998a) Y FU S SEQ R-I WD-DD Nuzzolo era/. (1999) S FU N SIM R WD Wong and Tong (1999,2001) FU-OU Y S SEQ R-I WD-DD Nuzzolo ef al. (200 la) FU S SEQ R-I Y-N WD-DD Nuzzolo et al. (200 Ib) R= regular services; 1= irregular services; N=without user information; Y=with user information; D=deterministic utility path choice model; S=random utility path choice model; SIM=simultaneous choice mechanism; SEQ=sequential choice mechanism; WD=within-day dynamic; DD=day-to-day dynamic; FU=frequent users; OU=occasional users. models

Although one of the first path choice models was the one introduced by Tong and Richardson in 1984 (described in Wong and Tong, 1999), the schedule-based approach in path choice and assignment was initially developed for low-frequency transit systems and only recently it has been proposed for urban transit systems. Hickman and Wilson (1995) specified deterministic path choice models, while stochastic ones were presented by Hickman and Bernstein (1997), Nuzzolo and Russo (1998a), Wong and Tong (1999, 2001), Nuzzolo et al. (2001a). Stochastic path choice models differ among them for the different random utility models (logit, nested-logit, probit) they use, and if the path choice set is considered explicitly or implicitly. Considering the dynamic feature of existing models, Hickman and Wilson (1995), Hickman and Bernstein (1997), Nuzzolo and Russo (1998a), Wong and Tong (1999, 2001) presented within-day dynamic path choice models that allow time variation in supplied services to be considered. The Nuzzolo and Russo (1998a) path choice models were extended in a doubly dynamic stochastic path choice model (Nuzzolo et al., 1999; 200la; 200Ib) that explicitly considers the within-day and day-to-day variations of services and user learning on attributes. For the simulation of low-frequency services, many schedule-based path choice models have been presented in literature (Nuzzolo and Russo, 1994; Carraresi et al., 1996; Cascetta et al., 1996; Nguyen et al., 1997; Florian, 1998; Nielsen and Jovicic, 1999; Nielsen et al., 2000; Nuzzolo et al., 2000); they differs among them for the way in which the path choice set and the path choice probabilities are defined.


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Most schedule-based path choice models for low-frequency services take into account DDT (desired departure times at origins) and DAT (desired arrival times at destinations) user target times, considering as path choice set alternatives the path of minimum disutility departing/arriving before and after user desired departure/arrival time (Nuzzolo and Russo, 1994; Cascetta et a/., 1996; Nielsen and Jovicic, 1999; Nuzzolo et al., 2000). Florian (1998) considers maximum earliness and lateness values to define a time slice around user target time, within which the path choice set is defined, while Carraresi et al. (1996) and Nguyen et al. (1997, 2001) consider only the late alternatives of minimum disutility with respect to the origin desired departure time or the early alternatives of minimum disutility in the case of desired arrival time at destination. Given the choice set, Florian (1998), Carraresi et al. (1996), and Nguyen et al. (1997, 2001) select the least generalised cost path in a deterministic way. Nuzzolo and Russo (1994), Cascetta et al. (1996), Crisalli (1999), Nguyen et al. (1997), Nielsen and Jovicic (1999), Nielsen (2000), Nielsen et al. (2000), Nuzzolo et al. (2000) use stochastic path choice models, both logit and nested-logit or probit, according to the hypothesis on the distribution of perceived utility random term. In particular, in the case of multi-class multi-service transit systems (e.g., railways), the existing correlations between alternatives have to be taken into account, and nestedlogit or probit models should be used (Nuzzolo and Russo, 1994; Nuzzolo et al, 2000; Nielsen, 2000). Table 5 reports a classification of some existing models for low-frequency transit services in relation to the random utility approach in path choice (deterministic or stochastic), and to the service type (monoservice or multiservice). Table 5. Schedule-based path choice models for low-frequency transit systems service utility type approach MONO Nuzzolo and Russo (1994) S MONO Carraresi et al. (1996) D MULTI Cascetta et al. (1996) S Nguyen et al. (1997,2001) MONO D MONO Florian (1998) D Nielsen and Jovicic (1999) MONO S Nuzzolo et al. (2000) MULTI S MONO Nielsen et al. (2000) S D=deterministic utility path choice model; S=random utility path choice model; MONO=monoservice; MULTI=multiservice. models

122 5.3

Advanced Modeling for Transit Operations and Service Planning Assignment Models

The above described path choice models are the core of the assignment models presented in literature for public transport networks. As schedule-based assignment models will be deepened in the next chapter, the following reference notes will concern the frequency-based approach only. In the framework of the frequency-based approach for high-frequency services, deterministic network loading models have been presented on the basis of the optimal strategy or hyperpath concept (Spiess and Florian, 1989; Nguyen and Pallottino, 1988), while deterministic and stochastic equilibrium models have been presented in order to take into account congestion and perception errors of path attributes (Nguyen and Pallottino, 1988; De Cea and Fernandez, 1993; Wu and Florian, 1993; Wu et al., 1994; Lam et al., 1999; Cantarella and Vitetta, 2001 a). Dynamic process assignment models have been formalised by Cantarella and Vitetta (200 Ib) in order to take into account the day-to-day variation of on-board flows. Further references on the frequency-based approach can be deepened in BouzaieneAyarietal. (1998).

REFERENCES Anez, J., T. de la Barra, and B. Perez (1996). Dual graph representation of transport networks. Transportation Research, 30B, 209-216. Bouzaiene-Ayari, B., M. Gendreau, and S. Nguyen (1998). Passenger assignment in congested transit networks: a historical perspective. In: Equilibrium and advanced transportation modelling (P. M. Marcotte and S. Nguyen, eds.), pp. 47-71. Kluwer Academic Publishers. Cantarella, G. E. and A. Vitetta (2001 a). Stochastic assignment to high frequency transit networks: models, algorithms, and applications with different perceived cost distributions. In: Mathematical methods on optimization in transportation systems (M. Pursula and J. Niittymaki, eds.), pp. 109-129. Kluwer Academic Publishers. Cantarella, G. E. and A. Vitetta (2001b). Frequency-based transit assignment models: stochastic equilibrium and day-to-day dynamic process, Teaching material of the Advanced course on transit networks, Dept Civil Engineering, "Tor Vergata" University of Rome, Italy.


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Carraresi, P., F. Maluccelli, and S. Pallottino (1996). Regional Mass Transit Assignment with resource constraints. Transportation Research, 30B, 81-98. Cascetta, E., L. Biggiero, A. Nuzzolo. and F. Russo (1996). A system of within-day dynamic demand and assignment models for scheduled intercity services. Proceedings of 24th European Transportation Forum Seminar D-E part 2, London. Crisalli, U. (1999). User's behaviour simulation of intercity rail service- choices. Simulation Practice & Theory, 17, 233-249. De Cea, J. and E. Fernandez (1993). Transit assignment for congested public transport system: An equilibrium model. Transportation Science, 27(2), 133-147. de La Barra, T. (1989). Integrated land use and transport modelling. Cambridge University Press. Florian, M. (1998). Deterministic time table transit assignment. Preprints of PTRC seminar on National models, Stockholm. Hickman, M. D. and D. H. Bernstein (1997). Transit service and path choice models in stochastic and time-dependent networks. Transportation Science, 31, 129146. Hickman, M. D. and N. H. M. Wilson (1995). Passenger travel time and path choice implications of real-time transit information. Transportation Research, 3C, 211-226. Lam, W. H. K., Z. Y. Gao, K. S. Chan, & H. Yang (1999). A stochastic user equilibrium model for congested transit networks. Transportation Research, 33B, 351-368. Modelistica (1993). TRANUS Technical description. Caracas, Venezuela. Nguyen, S. and S. Pallottino (1988). Equilibration traffic assignment for large scale transit networks. European Journal of Operational Research, 37, 176-186. Nguyen, S., S. Pallottino and M. Gendreau (1997). Implicit enumeration of hyperpaths in logit models for transit networks. Transportation Science, forthcoming. Nguyen, S., S. Pallottino and F. Malucelli (2001). A modeling framework for the passenger assignment on a transport network with time-tables. Transportation Science, 35, 238-249. Nielsen, O. A. (2000). A stochastic transit assignment model considering differences in passengers utility functions, Transportation Research, 34B, 377-402. Nielsen, O. A. and G. Jovicic (1999). A large scale stochastic timetable-based transit assignment model for route and sub-mode choices. Proceedings of 27th European Transportation Forum, Seminar F, 169-184, Cambridge, England.

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Nielsen, O. A., C. O. Hansen and A. Daly (2000). A large-scale model system for the Copenhagen-Ringsted railway project. Proceedings of 9th IATBR Conference, Gold Coast, Australia. Nuzzolo, A. and F. Russo (1993). Un modello di rete diacronica per 1'assegnazione dinamica al trasporto collettivo extraurbano, Ricerca Operativa, 67, 37-56. Nuzzolo, A. and F. Russo (1994). An equilibrium assignment model for intercity transit networks, Proceedings of TRISTAN II Conference, Capri, Italy, 1994. Nuzzolo, A. and F. Russo (1998a). A dynamic network loading model for transit services, Proceedings of TRISTAN III Conference, San Juan, Puerto Rico. Nuzzolo, A. and F. Russo (1998b). Departure time and path choice models for intercity transit assignment. In: Travel Behaviour Research: updating the state of play (J. Ortuzar, D. Hensher and S. Jara-Diaz, eds.), pp. 385-399. Pergamon Elsevier Science Ltd, Amsterdam, The Netherlands. Nuzzolo, A., U. Crisalli, and F. Gangemi, (2000). A behavioural choice model for the evaluation of railway supply and pricing policies. Transportation Research, 35A, 211-226. Nuzzolo A., F. Russo and U. Crisalli (1999). A doubly dynamic assignment model for congested urban transit networks. Proceedings of 27th European Transportation Forum, Seminar F, 185-196, Cambridge, England. Nuzzolo, A., F. Russo and U. Crisalli (2001a). Doubly dynamic path choice models for urban transit systems. In: Travel behaviour research. The leading edge (D. Hensher, ed.), pp. 797-812. Pergamon. Nuzzolo, A., F. Russo and U. Crisalli (2001b). A doubly dynamic schedule-based assignment model for transit networks. Transportation Science, 35, 268-285. Spiess, H. and M. Florian (1989). Optimal strategies: a new assignment model for transit networks. Transportation Research, 23B, 83-102. Tong, C. O. and A. J. Richardson (1984). Estimation of time-dependent origindestination matrices for transit networks. Journal of Advanced Transportation, 18, 145-161. Wong, S. C. and C. O. Tong (1999). A stochastic transit assignment model using a dynamic schedule-based network. Transportation Research, 33B, 107-121. Wong, S. C., and C. O. Tong (2001). Planning an urban rail transit system using a schedule-based network model. Proceedings of 9th WCTR Conference, Seoul, Korea. Wu, J. H. and M. Florian (1993). A simplicial decomposition method for the transit equilibrium assignment problem. Annals of Operations Research, 44, 245-260. Wu, J. H., M. Florian and P. Marcotte (1994). Transit equilibrium assignment: a model and solution algorithms. Transportation Science, 28, 193-203.

CHAPTER 5

SCHEDULE-BASED TRANSIT ASSIGNMENT MODELS0 Agostino Nuzzolo, "Tor Vergata" University of Rome, Italy, EU

1.

INTRODUCTION

In the last ten years improvements in computer science and increasing computation capacity have allowed the schedule-based approach to be developed and applied. The schedule-based approach was initially developed for low-frequency systems and has recently been proposed for urban transit systems. It considers the coherent user behavioural hypotheses in relation to user and service characteristics with respect to the well-known frequency-based approach which is based on the concept of optimal strategy or hyperpath (see chapter 4). As described in chapter 4, the schedule-based approach requires an explicit treatment for - the time segmentation of the origin/destination matrix, as user departure or arrival time distribution, both within-day and day-to-day, has to be considered; - supply modelling, as space-time characteristics of services during the considered periods have to be taken into account; Umber to Crisalli is co-author of sections 2, 3, 4, 5.1, 5.2, 5.3, 5.5; Pierluigi Coppola is coauthor of sections 5.4, 5.6

126

Advanced Modeling for Transit Operations and Service Planning - path choice model, with explicit within-day and day-to-day timedependencies.

This chapter describes some characteristics and the state of the art of the schedulebased approach in transit assignment modelling, and some application examples to realistically-sized networks.

2.

USER TARGET SEGMENTATION

TIME

AND

DEMAND

TEMPORAL

As previously mentioned, the times in which users desire to start or end their trips plays a key role in the schedule-based approach. These times are usually called user target times (TT) and can be classified into desired departure times (DDT), which represent the times in which users would depart from origin, and their desired arrival times (DAT), which represent the times which users would arrive at the destination. User target times TT can also be related to trip purpose: for exan p'e, a DAT can be considered for home-departure trips for commuter, business and st> idy trip purposes, while a DDT is generally used for home-destined trips. On account of the high frequency of urban transit services, only DDT, which are assumed to coincide with the user departure time TDI, are considered. For low frequency services, which are typical of extra-urban areas, either DDT or DAT, which are usually quite different from departure/arrival times of services, are used. This temporal characterisation of trips implies a temporal segmentation of the demand, in addition to the space features in terms of origin-destination zones. The reference period t is usually discretised in n elementary time intervals (e.g. of one minute each for high-frequency systems or ten minutes each for low-frequency ones), and given the generic time interval i. It is assumed that all user target times TT (DDT and/or DAT) in this time interval are aggregated into a single point TTTI (?Di and/or TAJ), e.g. in the middle point of the time interval /. An example of demand segmentation for both urban and extra-urban trips is reported in Figure 1. In the framework of the schedule-based approach, the O/D matrix should consider user target times in addition to the traditional space characterisation in origin and destination zones. For this reason, considering the reference period divided in time

Schedule-Based Transit Assignment Models

127

slices, an O/D matrix for each user target time is defined. For example, Figure 2 reports time-dependent O/D matrices in relation to origin departure times for each time slice of the reference period. In order to obtain time-dependent O/D matrices when they are not available, some procedures to estimate or update time-dependent transit O/D matrices from timevarying on-board counts can be used. The dynamic estimation problem can also be specified, thus extending the relationships between flows, counts and demand, which are formalised in the static approach in order to explicitly consider the time dependencies. In particular, it is necessary to describe the relationship between time-varying link counts and demand referring to the time interval /', in which demand relative to the O/D pair (od) leaves the origin and the time interval j, in which link counts^ are measured.

Figure 1. Example of origin departure times

128


Figure 2. Example of time-varying origin/destination matrix The flow on link / during the time slice j (fij) can be written as 0)

;=/ k

and considering all link counts in a matrix form, we have

f (d—T D1 ,.....,d—T

—

T

T

,d—T Dn) = Z—l—l.J Q T

/

Pd

=

(2) ^ '

i=J

where hj is the path flow vector, whose generic element hitk represents flow on path k relative to the O/D pair od and departing in the time interval /; Pj is the path choice matrix relative to time interval /', whose generic element p[k/od, ToJ represents the fraction of d°

>TD

' using path k; Qij is the crossing fraction matrix, whose generic

element q^1. represents the fraction of d° 'TD' using path k and contributing to flow on link / in time interval j; Mtj is the assignment matrix, whose generic element m° ' TD/ represents the fraction of demand d° 'TD' contributing to flow on link / during time interval/ The link flow vector^ can be calculated through a dynamic assignment model and is usually different from the vector of link counts relative to time interval j, f . , whose generic element // • represents the on-board count of vehicles represented by link / during time interval j. The estimation of the O/D trip demand vector d_ = (d_T

,d_T

, ...... ,d_T

) can be

carried out by "efficiently" combining traffic counts with all other available information. Estimators can be classified under classic estimators (like the Maximum


129

Likelihood or the Generalised Least Squares), if they combine experimental information (sample surveys) with traffic counts, and Bayesian estimators, if they combine non-experimental information ("a priori" information) with traffic counts. As formalised in the traditional static approach, the estimation of time-varying O/D matrices from time-varying on-board counts can be written as

x,

xn>0

(3)

~+z2[(f],...,fn);(f],...,fn)J} Equation (3) represents an optimisation problem made of the minimisation of the "distance" between the unknown time-varying demand x = (xj,...,xn) and the a priori time-varying demand dL = (d_T

,..-,d_r

), represented by the function z/f.J,

and between time-varying link flows f = (f,(x),...,f

(x)) , obtained using a

dynamic assignment model, and the observed link flows (time-varying on-board counts) f = (f.,..., f

), represented by the function z^/7- Functions zi[.J and z^/7

can be differently specified according to the chosen estimator. In the sphere of the dynamic O/D estimation, a further classification in simultaneous and sequential estimators can be done. Simultaneous estimators allow the joint estimation of O/D matrices, one for each time slice, to be obtained as reported in Equation (3). Sequential estimators provide a sequence of estimated O/D matrices, in which the estimation of the O/D matrix for the time slice j depends on the estimation of the O/D matrices relative to the previous time slices j-l,j-2,... ,j-n. Hence, the estimation of the demand vector — d.TDj , given the estimates V(d^ ,dr D2 , to _ T D I —T

,d ) —rTpj-1 '

relative to previous time intervals, can be written as

Xj;d

J +z2[f(Xj/drDl,...d

);f

1}

(4)

*^

Whatever the formulation and used estimators are, the core of the problem is the computation of the link flow vector fa of Equation (2) through the estimation of the assignment matrix MJJ, that, for transit networks, can be carried out using a schedulebased approach.

130


3. CLASSIFICATION OF SCHEDULE-BASED TRANSIT ASSIGNMENT MODELS The classification of schedule-based transit assignment models follows that introduced in chapter 4, in which different assignment models can be specified according to the type of behavioural path choice model (deterministic or stochastic), the type of link performance functions (flow-dependent or otherwise, which lead to congested or uncongested networks), the assignment approach (network loading, user equilibrium, dynamic process), and the dynamic evolution (within-day and/or day-today) they take into account. Note that, as in the schedule-based approach, the timetable is explicitly considered. The within-day dynamic is a native characteristic of the schedule-based approach in the sense that all schedule-based assignment models is at least under the within-day dynamic. Moreover, considering a learning process on attributes, the day-to-day dynamic can also be taken into account, leading to doubly dynamic (within-day and day-to-day dynamic) assignment models. In the case of transit, a further important element of classification concerns the system function, for which regular or irregular services can be taken into account (as a different run arrival sequence, with respect to the scheduled one, can produce quite different on-board loads). Service irregularity can be taken into account implicitly or explicitly. Implicit simulation of service irregularity is made by including the path choice model values of attributes calculated on the scheduled configuration of services. In addition, a further component to the random term is added in the perceived utility function of path alternatives in order to take into account service irregularity. Explicit simulation of service irregularity can be carried out by calculating values of attributes in path choice models on the basis of service configurations, generated starting from the scheduled timetable, and taking into account changes induced by service irregularity, in terms of arrival/departure times of run vehicles at stops. Table 1 reports the classification features of assignment models based on the abovedescribed criteria. The reader should consider the adjectives "deterministic" and stochastic reported in Table 1 as a peculiarity of the dynamic process features that, apart from the path choice model, can be either deterministic or stochastic.


131

Table 1. Classification of schedule-based transit assignment models Transit system functioning regular services

Transit Network uncongested [c=cost] Equilibrium approach congested Dynamic [c=c(f)] Process approach

irregular services implicit explicit simulation simulation

Dynamic Network Loading

Stochastic Supply Dynamic Network Loading User Equilibrium

Deterministic or Stochastic Dynamic Process (Deterministic Supply)

Deterministic or Stochastic Dynamic Process (Stochastic Supply)

In the case of regular services, or irregular services with an implicit treatment of irregularity, a DNL (Dynamic Network Loading) approach can be specified for uncongested transit networks; it can use a deterministic or a stochastic path choice model. For congested networks, if the day-to-day evolution is neglected, an UE (User Equilibrium) approach can be considered. DUE (Deterministic User Equilibrium) or SUE (Stochastic User Equilibrium) models can be specified according to the use of deterministic or stochastic path choice models. If a day-to-day dynamic evolution of the system has to be taken into account, a DP (Dynamic Process) assignment model, that can be Deterministic or Stochastic in the Demand component, is required. Different assignment models can be specified in the case of transit networks in which service irregularity is explicitly dealt with. For uncongested networks, a SSDNL (Supply Stochastic Dynamic Network Loading) can be used by considering average run loads calculated through successive averages of run loads carried out by different service configurations. In the case of congested networks, service irregularity can be explicitly treated using a SSDP (Stochastic Supply Dynamic Process) model.

4.

REFERENCE NOTES

In the sphere of schedule-based assignment models for high-frequency services for uncongested networks, Hickman and Wilson (1995) used a Deterministic Network Loading model to assess results of their path choice models, while Stochastic Network Loading models were used by Crisalli (1998), Nuzzolo and Russo (1998a), Nuzzolo et al. (1999) and by Hickman and Bernstein (1997). Congested networks have been studied using both the equilibrium and the dynamic process approach. Stochastic user equilibrium assignment models have been presented by Crisalli

132


(1999a), Nielsen (2000) as well as Wong and long (1999, 2001): they used an MSA (Method of Successive Averages) algorithm based on logit, hybrid logit and estimation by numerical simulation path choice models. Dynamic Process assignment models, Supply Deterministic and Stochastic, have been presented by Nuzzolo et al. (1999, 200la). In the case of low-frequency schedule-based assignment models, deterministic network loading models have been presented by Nuzzolo and Russo (1993), Carraresi et al. (1996), Florian (1998). Cascetta et al. (1996), Crisalli (1999b), Nguyen et al. (1997), Nuzzolo and Russo (1994, 1996, 1998b), Nuzzolo et al, (2000) proposed stochastic network loading models. Congested networks have been studied by Nuzzolo and Russo (1993) through a DUE assignment model, while Nielsen et al. (2000) extended to the stochastic user equilibrium assignment model proposed by Nielsen (2000) the extra-urban case. Moreover, considering a learning process on attributes, day-to-day dynamic schedulebased path choice models for low-frequency services could be specified to take into account the day-to-day evolution of level of service attributes, even if, to the author's knowledge, models of this type have not been presented in literature. The theoretical framework for the dynamic estimation of O/D matrices from traffic counts was defined in Cascetta et al. (1993) for road networks. In the framework of the schedule-based approach, only a few have been presented for transit networks: Wong and Tong (1998) presented an estimation method for urban transit networks on the basis of a maximum entropy estimator, while Nuzzolo and Crisalli (2001) used a generalized least squares estimator to solve the estimation problem for uncongested low-frequency transit services.

5.

APPLICATION EXAMPLES

As reported previously, the schedule-based dynamic approach requires a more detailed description of services and demand; it is more time-consuming with respect to the frequency-based approach, but obtains more accurate results in terms of vehicle loads and level of service attributes. Applications to real cases show the advantages of this class of path choice and assignment models and highlight the importance of software tools, like the Decision Support Systems, for the operations planning of transit networks.


133

In the following some application examples of schedule-based assignment models will be presented. In particular, section 5.1 reports an application to regional transit services; section 5.2 describes the application to a national railway network; section 5.3 shows some applications to urban transit systems; section 5.4 reports an application in the field of short-term forecasting of on-board loads and run arrival times at stops for urban transit services; sections 5.5 and 5.6 describe a possible use of schedule-based assignment models to estimate transit O/D matrices and to support the timetable setting. 5.1

Regional transit services

In the following, some application examples of a system of schedule-based assignment models (at the regional district level) will be described. These models were implemented in DY-RT, a software tool that simulates regional transit services by using a schedule-based dynamic approach on the basis of the system of models proposed by Nuzzolo et al. (2000). The DY-RT mode-run choice model considers car, park & ride (car + train), bus and train mode alternatives. For transit services, it takes into account two run alternatives: the ones of minimum generalised cost (including early and late schedule penalties) departing before and after the user target time. The run choice is fully assumed pretrip, and the structure of the mode-run choice model of DY-RT is reported in Figure 3.

Early Train (bus+train)

Early Park-Ride

Late Bus

Figure 3. The structure of the mode-run choice model In the framework of the random utility theory, the probability pff/TrrJ of choosing mode-run alternative j, given the od pair and target time TTT, can be written as = prob(Vj/~TrrJ+£j>

Vj

for each alternative y '^j belonging to the mode-run choice set.

(5)

134


Attributes composing the systematic utility F/ for each alternative are those reported in Table 2. Table 2. Path choice model - alternatives and attributes alternatives

attributes

description

CAR

EB

ET

EPR

LB

LT

LPR

car

early bus

early train

early park&ride

late bus

late train

late park&ride

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X X

X

X

X

X

X

TB

transit travel time

TC

car travel time

X

C

travel cost

X

EP

early schedule penalty

LP

late schedule penalty

A/E

access/egress time

HI

high income user

PR

park&ride dummy

CAR

car dummy

EBUS

early bus dummy

X

X

X

X X

X

X X

Figure 4 describes the functional architecture of DY-RT. On-board loads for each run of transit services, and relative revenues, are computed as the result of a sequence of phases consisting of: -

definition of operating characteristics and timetable of transit services; construction of the supply model, both road and transit; calculation of path choice sets and level of service attributes; estimation of time-dependent O/D matrices for each mode (car, bus, train); calculation of path choice probabilities for each od pair and target time TTT', assignment of O/D matrices for each target time TTT to the diachronic

-

network /2 (within-day dynamic network loading procedure) according to path choice probabilities computed in the previous step; determination of on-board run loads and revenues.


135

Figure 4. DY-RT functional architecture Given a design scenario, which includes a supply configuration (routes and timetable of transit services) and a space-time O/D demand, DY-RT can compute the on-board loads for each run of each transit mode (bus and train) as the result of two types of within-day dynamic assignment procedures: - run choice simulation, in which the demand of each mode (a priori estimated) is assigned to the network to calculate on-board loads and revenues for each run of bus and rail services; - mode-run choice simulation, in which the total demand of the scenario is given. At first, the mode choice is simulated to estimate the O/D matrices for each mode in relation to level of service attributes of the design scenario. Subsequently, on-board loads and revenues for each run are calculated in relation to the estimated demand. DY-RT has been widely used to support operations planning of many regional transit networks in Italy (e.g., Lazio and Veneto districts). In particular, this software allows us to consider effects of rail-bus integration for regional transit trips and to assess the effects of network rationalisations, in order to reduce costs and improve service quality. In the following, two application cases will be briefly described. The definition of minimum regional bus services in Lazio district Lazio study area was divided into 395 traffic zones at municipality level, except for the metropolitan area of Rome, which was divided into 9 different zones. The regional bus services were made of 8892 runs per day, 2769 of which were relative to the morning peak reference period (6.00-10.00am). The railway services were made of 1764 train runs (405 of which were scheduled from 6.00am to 10.00am).

136


Considering the relevant number of bus and railway runs, diachronic supply networks have great dimensions: for example the bus diachronic graph is made of about 230000 nodes and 560000 links. The demand relative to the reference period (6.00-10.00am) was divided into 8 time slices of 30 minutes to take into account different user target times. Given a possible configuration of the regional transit network (see Figure 5), the within-day dynamic assignment allows us to obtain on-board loads for each run of each scheduled service over each section of the network. An example is reported in Figure 6 for the six runs of line Rieti-Rome, passing over section Poggio NativoNerola.

Figure 5. Example of regional transit services (Lazio district)


137

Figure 6. Example of run on-board loads Figure 6 also shows the advantages of the schedule-based approach, through which the variation in the reference period of on-board loads for different runs of the same line over the same section can be taken into account. On-board run loads are quite different from the average on-board line value, which can be obtained using the frequency-based approach. Moreover, the calculation of level of service attributes at the single run level allows us to support the route design. This allows the definition of the minimum set of main and feeder lines to cover the Lazio district, as well as to assess the timetable set up that minimises the interchange times between runs of main and feeder lines as well as between bus lines and trains. The integration of regional rail-way-bus services in Veneto district Aiming to design regional bus and rail services in a strictly integrated network, Veneto district was divided into 195 zones, served by 237 trains and 1776 bus. The diachronic networks, which represent these services, were made of about 25000 nodes and 55800 links for regional bus services, while railway services were represented by 3300 nodes and 10600 links, approximately. With respect to the reference scenario, the design was built after considering the railbus integration through 112 new railway runs, 10 new terminals and 28 park & ride areas. In particular, bus and rail timetables were changed to minimize the impacts of interchanges through the synchronization of bus and train arrivals and departures. The

138


train timetable was designed using the "clock-headway" concept, which is characterized by a timetable of easy-to-memorize repetitions of departure time, usually in multiples of an hour or less (e.g. at 5, 10, 20, 30, 40, 45, 50 and 60 minutes). In addition to on-board loads for each scheduled run of transit services described in the previous example, DY-RT allows us to calculate the level of service attributes and the performance indexes at the single run level, as well as to assess integration policies implemented in the design scenario (e.g., see Table 3 for railway services). This result can be obtained using the schedule-based approach only. Table 3. Example of comparison between level of service attributes railway comparison (reference period 6. 30am- 10.30 am)

attributes access time (min) egress time (min) on-board time (min) transfer time (min) number of runs for O/D

5.2

design

reference

variation

10.0 8.2 27.2 10.1 8.0

11.8 10.6 30.2 13.5 4.5

-15.3% -22.6% -9.9% -25.2% 77.8%

Long-distance railway services

This section describes a system of within-day dynamic railway service choice and assignment models, which are used as the modelling basis for a large Decision Support System (DSS) for the operational planning of rail services, named SASM, that was recently developed by the Italian Railways Company. The decisional variables considered by the DSS concerns the operating characteristics (paths, stops, timetables and train composition) and price of the service. The support system was designed to help by both analysing the current situation and defining marketing strategies. SASM computes the performance of railway and competitive mode services, passenger loads and related returns of each single train for a design scenario through a sequence of phases consisting of (Figure 7): - formulation of the design scenario, with the definition of operating characteristics and prices of railway services; - construction of supply models of scenario rail/services; - calculation of alternative runs and level of service attributes for demand models;

139


calculation of probabilities of each alternative mode-service run configuration for each demand segment and for each O/D pair od, by means of the system of demand models; application of such probabilities to "current" rail O/D matrices for each demand segment and computation of new O/D matrices for each railway service (high speed, slow speed); assignment of O/D matrices to networks of the relative railway services (within-day dynamic network loading procedure); determination of the rail service passenger flows; determination of railway revenues.

Rail service passenger flows

Figure 7. Functional architecture of the Decision Support System SASM The core of the system is the path choice model, which was calibrated on the most important railway corridor in the northern part of Italy that runs from Turin to Venice through Milan. The calibration phase of the path choice model is briefly described. In order to apply the service, run and class choice model, current railway demand was subdivided into market segments given by the combination of: - 9 trip purposes: commuter home-work and vice-versa, home-business and v.v., home-school/university and v.v., home-other purposes and v.v., non home-based trips;

140

Advanced Modeling for Transit Operations and Service Planning - 15 desired arrival times at destination (DAT) for home-departure trips and/or non home-based trips and 15 origin desired departure times (DDT) for homebound trips; - 3 user categories: with reimbursed travel cost, with medium-high family gross income (> 20000 €), with a low family gross income (< 20000 €).

To obtain the distribution of users for the various categories starting from current demand data, on the basis of analyses conducted on the user sample of the on-board train survey, the following assumptions were made: - distribution by purpose depends on the service and class currently used and on the type of O/D relation; six relation types were considered combining three zone types (Metropolitan Area, Provincial Town and Other) with two O/D distance band (less than or greater than 100 km); - distribution by target time depends on the purpose and O/D distance band (200 km); - distribution by user category depends on service, class and trip purpose. An on-board train survey was conducted in order to estimate the rail O/D matrix, to calibrate the path choice model and to obtain the market segment distribution. The survey consisted of: - counts of passengers (approx. 50000) boarding high speed (intercity) or slow speed (regional) trains on a workday, with interviews on origin and destination; - survey of a sample of about 10000 users, with questions about used terminals, access and egress mode, trip purpose, target time and socioeconomic characteristics. In order to compute the level of service attributes, a supply model was implemented. Italy was divided into 484 traffic zones (with about 100000 inhabitants per zone). The road infrastructure network model, which also includes the zone centroids and passenger service terminals, has 4969 nodes and 16414 links. The railway infrastructure network model consists of 448 nodes, 441 of which are railway terminals, and 1086 links. The best specification obtained for the service/run/class choice model is that reported in Figure 8, and the probability p[cl/ser,r,m] of choosing the elementary path alternative identified by railway service type ser, run (or sequence of runs) r, class cl, given the O/D pair od and the market segment m (which defines the user target time TTT) can be written as:


141

where:

in which symbols assume the meaning reported in Table 4. The same table reports parameter estimations and test statistics for commuter work purpose. All estimated parameters have correct signs and most of them are statistically significant. The scale parameters Sser and 8r assumes values quite different from one, proving the hypothesised correlation among alternatives.

Figure 8. SASM choice model structure

VOTs (Values Of Time) have the expected values in relation to trip purpose; they also respect the difference between the two income classes. Indeed, VOTs relative to the medium-high income class are greater than those for the low-income class. The significant difference between the two classes could be explained by the fact that the average income of low class railway users, especially for commuter work purpose, is less than the average income of other mode users. As we can see in Table 4, the early (late) schedule delay parameters have considerable weight. The expected symmetry between early and late penalties for outgoing and returning trips is clear: we can observe a greater disutility, for outgoing trips, associated with a late arrival schedule delay at destination rather than with an early one. It is related to trip purpose (commuter work) and these results are also confirmed in other studies. The opposite phenomenon can be observed for returning trips, where

142


a late schedule delay, with respect to the desired departure time, is perceived less than an early one. The scheduled delays are considered in relation to trip distance, in order to take into account their greater influence for shorter trips than for longer ones. Table 4. Coefficients and VOTs Attribute

symbol

units

P

t-statistics

TV

hours

-3.752

-3.92

VOT (€)

h. 7.20 Total travel time

1. 2.82 Total travel cost for mediumhigh (h) income class

CH

€

-0.521

-8.49

Total travel cost for low (1) income class

CL

€

-2.552

-24.2

-5.816

-7.78

Early arrival schedule delay with respect to DAT, divided by the distance (outgoing trips)

ESPG

Late arrival schedule delay with respect to DAT, divided by the distance (outgoing trips)

LSPG

Early departure schedule delay with respect to DDT, divided by the distance (returning trips)

ESPR

Late departure schedule delay with respect to DDT, divided by the distance (returning trips)

LSPR

hours/ 100km

hours/ 100km

hours/ 100km hours/ 100km

h. 11.17 1. 4.38 h. 16.32

-8.5

-8.15 1. 6.39 h. 20.20

-10.52

-11.53 1.7.91

h. 13.75 -7.161

-10.52 1.5.39

Access (egress) logsum for outgoing (returning) trips

AC

1.481

3.11

Egress (access) logsum for outgoing (returning) trips

EG

0.22

0.93

High-speed train dummy

1C

0/1

-0.225

-0.74

Second class of high-speed train dummy

IC2

0/1

0.341

1.83

Run scale parameter

5r

0.79

2.65

Service scale parameter

5ser

0.81

2.19

2

Rho squared

P

0.575

Rho squared correct

p corr

0.569

Number of observations

N. obs.

1275

2

The dummy variables are relative to the efficient combination (in terms of model response) in order to avoid correlations in the calibration phase. A negative value for high-speed train dummy (1C) and a positive value for second class of high-speed train


143

dummy (/C2) showed user preference for the cheaper service (due to the daily trip feature for commuter work purpose). Moreover, a limited significance of the egress variable coefficient can be observed. This is probably due to the calibration process which was based only on railway choices, where in many cases, there are no real alternatives for the egress terminal, thus reducing the explanatory power of this variable. Obviously, logsum values of access/egress mode choice could play a different role in a model that also considers other modes (car, bus and aeroplane). The model was validated using the goodness of fit p statistic showed in Table 4 and reproducing the sample distribution among services, runs and classes, through the computation of RMSE% values. These results show a good capacity to describe the sample choices, as reported in Table 5. Table 5. RMSE% values Choice

RMSE%

service

4.8%

service/class

8.7%

service/run/class

16.6%

A more sophisticated model structure which includes other choice dimensions, implicitly considered in the structure of Figure 8, could be taken into account. For example, the access/egress mode choice could be explicitly considered, but the introduction of a further level in the nested-logit structure might induce a very large number of theoretical possibilities. An example of access-egress choice models for railway services can be found in Crisalli and Gangemi (1997). Some test applications of the DSS were carried out in order to verify its capability to reproduce the actual state of the railway services. The current distribution of train loads was reproduced (an example is reported in Figure 9 that shows the load of train IC640 from Turin to Venice), comparing the total number of users boarding each train by the model and the number of users counted in the corridor surveys. The comparison between counted and simulated passenger is shown in Figure 10 and appears quite satisfactory for operational uses.

144


Figure 9. Passenger flow on train 1C 640 from Venice to Turin 800

700600

500 --

1400 S

300200 100

0

100

200

300

400

500

600

700

800

counts

Figure 10. Comparison between train counts and model results 5.3

Urban transit services

This section presents assignment results carried out considering the system evolution both during a single day period (within-day variation) and in following periods (day-to-day variation) on the basis of the doubly dynamic schedule-based assignment models described in Nuzzolo et al (200Ib). These models were implemented in a software tool (DY-BUS) that can be used to simulate urban transit services Given a design scenario, which includes a supply configuration (paths and timetable of transit services) and a transit space-time demand, DY-BUS can compute the on-board loads for each run of the transit services as the result of different types of assignment procedure:

Schedule-Based Transit Assignment Models •

•

•

1 45

Within-day dynamic assignment, which simulates the evolution of on-board loads within a single-day period on the basis of the supply configuration of a given day t; the assignment model is a Stochastic Network Loading (SNL) model, since it uses a stochastic path choice model. Stochastic user equilibrium assignment, which allows the calculation of onboard loads for each run in the case of congested networks with the regular service functioning. Dynamic process assignment that simulates the evolution of on-board loads on different days in which user choice is influenced by the learning process on congestion-related attributes and by the different supply at their disposal due to service irregularity, leading to different level of service attributes from one day to another. Service irregularity is explicitly considered through a sequence of supply configurations (one per day), obtained by using Monte Carlo simulation techniques, starting from the scheduled timetable, as described in Nuzzolo et al. (200 Ib).

In order to simulate the day-to-day dynamic evolution, a learning process on attributes experimented and forecast on previous days is considered. It is specified using an exponential filter in which the attribute value X{° forecast on day t is expressed by a convex combination of the attribute forecast on day t-1, X{°} , and of the value realised on the same day t-1, X**j , that is:

(8) where ye [0,1] is the weight given by users to attributes realised on day t-1. In order to verify the treatment of realistically-sized urban transit networks to support operations planning, results of applications of the above described tool are here reported. The test network was developed on the topological basis of the transit network of the town of Salerno, situated in southern Italy. The service supply was changed to simulate different functional conditions. The study area was divided into 62 traffic zones characterised by a transit demand of about 32000 users, which was assumed constant during the simulation period (morning rush hour: 7:30-8:30), which was, in turn, subdivided into 60 time slices of one minute each. On the basis of the 58 lines of the transit system a service timetable with 523 runs was defined. Such a supply allowed us to hypothesise a high frequency service for all O/D pairs and random

146


arrival of users at stops. The diachronic network used to represent the service supply consists of 44846 nodes and 81654 links. At the moment DY-BUS uses a logit specification, even if some problems related to the IIA assumption could occur. Given an od pair and an origin departure time TDI, path choice probabilities were calculated considering a sequential choice mechanism described in chapter 4, in which the specification of the systematic utilities of the arriving run r and of the next ones r' are: (9)

(10)

where TWr- is the waiting time (equal to the difference between the arrival time of run r' and the arrival time of run r, supplied by the information system), TBr and TBr- are on-board times, TCr and TCr- are transfer times, NTr and NTr' are the number of transfers; CFBr and CFBr' are the "route" on-board comfort (function of on-board crowding degree in the following links, up to egress stop), CFWr and CFWr' are the on-board comfort at stops when users have not yet boarded (function of run crowding degree at stop), TPr is the time already spent at stop (equal to the difference between arrival time of run r and the user arrival time Tots at stop). The choice of stop s is calculated considering a logit model in which the systematic utility is given by: V,=PTATAS+PHHS

(11)

where TAS is the access time of stop s from the considered origin o, Hs is the inclusive utility of runs at stop s. A value 7=0.05 for the day-to-day learning process of Equation (8) is considered, while attribute parameters of Equations (9, 10, 11) are those reported in Nuzzolo et al. (200 Ib). In the following the results of different assignment models will be presented. Within-day dynamic assignment The use of a schedule-based stochastic path choice model in a within-day dynamic network loading procedure allows us to perform an SNL (Stochastic Network Loading) assignment, which computes on-board loads for each run in the reference period.


147

Figure 11. Example of within-day dynamic assignment results As an example, Figure 11 reports on-board loads for runs of line 20 over section 115127, where several bus lines are in competition for the same od pairs. Single run loads differ from one another and from the average line value for many reasons, such as different bus headways, presence of competitive runs of other lines and so on. Stochastic user equilibrium assignment In order to take into account congestion, it is possible to explore assignment results using both equilibrium and dynamic process approaches. In the framework of regular service functioning, the day-to-day utility variations are exclusively due to comfort attribute dependence on on-board loads, and an equilibrium approach can be used. Using a SUE (stochastic user equilibrium) assignment model, Figure 12 reports onboard loads for all runs of line 20 passing over section 169-173. The results are based on the application of an MSA (Method of Successive Averages) algorithm on costs. The capacity, defined the same for all vehicles, is equal to 100 users.

runs Figure 12. Example of equilibrium assignment results

148


Figure 12 shows the capability of the model to smooth (when it is possible) congestion effects. The values of on-board loads exceeding capacity, obtained with the MSA algorithm, are due to severe high saturation conditions. It is because the model cannot balance both for the lack of alternative runs in relation to user departure time, and for the impossibility to use capacity-explicit bounds in the equilibrium assignment approach without damaging the existence and uniqueness of the problem solution. Such values, even if higher than capacity are, in any case, still lower than the correspondent values obtained through the SNL model. Dynamic process assignment For the day-to-day dynamic, the results of applications to transit services with regular and irregular functioning, respectively, are reported in the following. In the case of regular service functioning, as an example of results of a deterministic dynamic process assignment model, Figure 13 reports the on-board load of line 20 run 22 and of line 1 run 18 over the same section 115-127 obtained for a 90-day simulation. Such values show system convergence towards points that seem to be fixed point attractors. In fact, the load value of line 20 run 22, which was initially near capacity (100 users/vehicle), decreases because of the learning process on the comfort attribute, that tend to disadvantage this run use when compared with alternative runs. This can be observed in the increase of line 1 run 18 over the same section in Figure 13.

Figure 13. Example of day-to-day assignment for regular services In the case of irregular service functioning, the on-board load changes from day to day mainly due to service irregularity (i.e. if a run is delayed, alternative runs are much more heavily loaded compared to days when they arrived earlier) and congestion (i.e. high on-board load are considered through learning and updating mechanisms that lead to changes in user choices on different days, even if runs are on


149

time). An example is reported in Figure 14 where, for the same run described in Figure 13, the on-board load in the case of service irregular functioning is pictured.

Figure 14. Example of day-to-day assignment for irregular services 5.4

Short-term forecasting of on-board loads and arrival times at stops for urban transit systems

An interesting application of schedule-based assignment models is to support the short-term forecasting of on-board loads and run arrival times at stops by the use of Advanced Traveller Information System (ATIS) to provide to users en-route descriptive information on the network state in the attempt to facilitate their travel choices. Conceived essentially for private transport networks, in recent years ATIS has increasingly been applied to transit networks, where users may be interested in: (1) real-time waiting times at stops or equivalently the arrival time of runs at the stop; (2) degree of occupancy of upcoming runs at the stops. The latter information can affect to great extent en-route traveler's choices, especially in congested transit systems, where travellers may choose to skip overloaded runs and wait for less crowded ones by trading off between longer waiting-time and higher on-board comfort. In order to provide information of the type 1 (i.e. runs arrival time), it is necessary to develop an algorithm able to predict real-time link travel time on the network that is based on current vehicle location and historical data needs. On the other hand, to provide information of type 2 (e.g. loads or runs occupancy) a modelling procedure of the overall system needs to be set up. In fact, the real-time prediction of loads on runs of transit network requires the simulation of travellers' path choice and the way in which they propagate on the network. The prediction of the O/D travel demand

150


pattern is also required. To this aim, a modelling procedure which explicitly simulates within-day dynamics in transit networks from both the demand and supply side, has been built up. This is based on a schedule-based approach and is described in the following of this chapter. The conceptual scheme of the overall system is depicted in Figure 15, where the main components of the 775 (Transit Information System) are clearly outlined.

Figure 15. Schematic representation of the considered Transit Information System The surveillance system, by means of monitoring and communication devices (e.g. DGPS, radio modem, etc.), detect and transmit to the Operations Control Centre (OCC) the current location of the vehicles and the number of passenger that boarded or alighted at stops. The OCC elaborates the real-time data gathered to predict arrival times and occupancy of runs at stops, and transmits them to user interfaces through communication devices (e.g. long-range radio communication systems, and so on). Arrival times and runs occupancy predictions are based on the modelling framework schematically depicted in Figure 16. In principle, arrival time prediction is simpler than run occupancy prediction since it does not require the simulation of traveller behaviour with respect to the current network condition and to the information provided. Examples of algorithms to predict link travel times, and hence arrival times at stops, are widely reported in the literature. On the other hand, to predict run occupancy, a comprehensive modelling framework based on the schedule-based approach has been built up. It consists of (see Figure 16): - a time-varying O/D matrices estimation procedure based on real-time observation of the number of passengers boarding and alighting from vehicles at stops; - a supply model to represent a time-dependent transit network, whose temporal coordinates are updated real-time in relation to the information on vehicle location; - a sequential path choice model based on random utility theory, simulating public transport traveller behaviour;

Schedule-Based Transit Assignment Models -

151

a within-day dynamic assignment procedure following a schedule-based approach, estimating the loads on each run of the transit system at any time of the reference period.

The system of schedule-based supply, path choice and assignment models are those presented by (Nuzzolo et al., 200la). The algorithm adopted to implement the modelling framework follows an event-based simulation approach. An event is defined by the arrival of a signal from the surveillance system to the OCC. This happens when new information on vehicle location and/or passengers who boarded/alighted on/from a given run at a given stop are available. When an event occurs (say at time T), the input variables of the schedule-based transit assignment models previously described are updated, that is: - the service configuration (diachronic supply model) is updated, based on the current vehicles location and on the link travel times estimates; -

the O/D matrix is estimated for each time slice TDJ, based on the passenger boarded/alighted counts and current network performances. Then, traveller path choice is simulated and the loads on the runs for the remaining time of the whole reference period (i.e. the period of time from T to the end of the reference period) are estimated.

Figure 16. Modelling components of the Operations Control Centre (OCC)

152


In the following, a preliminary application example is described that is used to test the approach and its potential use to support operations planning. The considered test network is schematically depicted in Figure 17. Four centroids have been considered; these are connected to nodes 1,2,8 and 9; the transit services consist of 3 lines and 9 runs; their diachronic representation consists of about 65 nodes and 200 links. Given a uniform demand pattern (i.e. constant arrival rates at stops) within a reference period of 1 hour (from 7:00 to 8:00), the experiment was aimed at investigating travelers' path choice variations for two cases of information provision: case 1: travellers are provided only with waiting times of upcoming runs at the stops; case 2: travellers are provided with waiting times and loads (i.e. run occupancy) of upcoming runs at stops.

Figure 17. Test network Let us consider travel demand from centroid 2 to centroid 9: path choice alternatives are represented by line 1 (on-board travel time equal to 30 min) and line 2 (on-board travel time equal to 20 min). In fact, the alternative of taking line 1 and then transferring to line 3, in this case, is dominated by the alternative of taking line 1 straightforward to destination. Moreover, let us suppose that, according to the transit schedule, runs of line 1 are expected to arrive at stop 2 at 7:10, 7:30 and 7:40, while runs of line 2 at 7:14 and 7:44.


153

Case 1 Let us consider travellers arriving at stop 2 at 7:15 (i.e. right after the departure of first run of line 2). These travellers are provided, through the Transit Information System (e.g. by means of a VMS or other User-Interface device), with the following information: they have to wait 15 minutes for line 1 and 29 minutes for line 2. The effects of the information can be seen using the graph depicted in Figure 18, which was derived from the path choice model considered here (adapted to the present case study from Nuzzolo et al., 200Ib). From the graph, it can be seen that assuming the arrival time of the runs remain on schedule (i.e. estimated waiting times at stop linearly decrease with absolute time), 85% of travellers would choose line 1 and 15% would choose line 2. This represents the trade-off between more waiting time with less on-board time. At time 7:20, waiting times have been perturbed in the attempt to simulate irregularity of service. In particular, the waiting time of line 1 has been increased until it is greater than the waiting time of line 2. Accordingly, path choice probabilities vary: at time 7:44, when the run of line 2 arrives at stop 2 (i.e. when the travellers waiting at the stop actually make their choice), then 95% of the travellers would choose to board line 2, as shown in Figure 18. Case 2 Let us suppose now that travellers at stop 2 also have information on passenger loads in the upcoming runs, these being 25 passengers for line 1 and 75 passengers for line 2. The path choice probabilities are slightly different with respect to case 1. In fact, according to the scheduled services (e.g. at time 7:15), choice probability of line 1 would be equal to 91% (vs. 85% of case 1) due to the greater attractiveness of boarding a less loaded and, hence, more comfortable vehicle (see Figure 18). Similarly, when the travellers actually make their choice (i.e. at time 7:44) choice probability of line 1 is equal to 8% (vs. 5% of case 1). Note that due to service irregularity, the estimates of loads on line 1 vary with time. In fact, as a result of the delay in line 1, there is an overall increase of loads on line 1 (from 25 to 43 passengers). This results from the fact that traveller arrival at stops have been assumed to be uniformly distributed over time: This means that the longer a run is delayed, the number of travellers arriving at the stop (and boarding the vehicle) will also increase.

154


P"

U-

10 5

5

7

c

70 60

a40 10

5

7

Figure 18.

5.5

Information provided by TIS (i.e. waiting time and loads) and path choice probability for travellers departing at time 7:15 from centroid 2 to centroid 9

Estimation of time-varying transit O/D matrices from time-varying traffic counts

This section reports a method and an application of a Generalised Least-Squares (GLS) estimator, which can be used to carry out transit O/D matrices segmented for user target time, on the basis of the methodology described in section 2. In the sphere of uncongested transit networks and considering user origin departure time fixed, the use of a simultaneous estimator is fully acceptable. Using a GLS estimator (Cascetta, 1984), Equation (3) can be written as

(12)

+ If IM, / •*/ -/JJ0TYIM,- / •*,- -/J j; j=i 1=1

-

i=i

~

where W_ is the variance-covariance matrix relative to vector e_ = (e_1>

,£,•,-•••,£„)

of assignment model and counting errors, characterised by E(s)=0_ and var[£j=W;


155

d_j is an available estimation of the unknown demand vector x, obtained through a random sample such as dj=xj+ilj

(13)

in which JJ/ is a random vector that considers demand sample errors, characterised by =0_ and var[rjj]=Zj. In order to solve problem (12), it is necessary to calculate the assignment matrix MJJ defined as Mij-Qij-Pi

(14)

The use of the schedule-based dynamic transit modelling approach allows a more precise and direct way of calculating the crossing fraction matrix Qtj and the path choice matrix Pj. In fact, as each link of the diachronic graph represents a precise location in both space and time, it is possible to calculate, in a simpler way, the crossing fraction matrix QJJ, which is made up of elements qf'1. , because each path k is individuated on the graph by a sequence of links that allowed the fraction of path load /?,-,k and the crossing link a in time interval j to be defined directly. Elements of path choice matrix Pj, which represents the probability of choosing path k (individuated by run r) for users travelling on the O/D pair od and departing in time interval /., can be calculated using the schedule-based path choice models described in chapter 4. This approach allows problem (12) to be easily solved by using a traditional project gradient algorithm. This method is applied to estimate the Italian railway O/D matrices at a national scale, which is made of 13420 O/D pairs for each service (high speed and low speed trains) and travel class (first and second). The estimation is carried out on the basis of about 868 time- varying counts on 257 of the 334 trains of the railway services over several sections. The reference period (one-day) was divided into 15 time intervals of one hour each in the rush hours, while wider intervals for other periods were defined. An origin departure time for each time interval was considered. In order to calculate path choice probabilities, the path choice model structure, attributes and parameters are those reported in section 5.2.

156


This methodology is actually used in the Decision Support System of the Italian Railways to estimate the railway O/D matrices at national scale. The starting O/D matrices are the railway O/D matrices of sold tickets, which are available from automatic station reports. These O/D matrices differ from the real ones as they do not consider tickets sold on-board, railway passes and free tickets. In addition, it also does not fully account for tickets users who purchase tickets that could be used over the next 60 days. For this reason, in order to simulate in a more precise way railway services, the updating of railway O/D matrices of sold tickets from traffic counts is crucial. The accuracy of a solution is evaluated by the difference between the generic O/D vector d and the real one d_. Different statistics like the root mean square error (rmse), the relative mean error (rme) and the mean square error (mse) are used. Performance of the used estimator have been considered by computing average values of statistics on the basis of different starting matrices d_ obtained through random perturbations of the real O/D vector d_ as df = df

+ (p(u-0.5) df

(15)

where u is a value extracted by an uniform (0,1) random variable and ^ is a parameter that is 0.7, 1.4, 2.0 to which corresponds to variation coefficients of 0.35, 0.70 and 1, respectively. Estimated O/D matrices have been carried out using a project gradient algorithm, in which the optimal solution is reached when the difference between two successive iterations does not exceed 1 % (£=0.01). Table 6 reports statistics for the three different estimations ((f>= 0.7, 1.4, 2.0). It can be seen that the method has succeeded in improving the starting vector, which showed absolute reductions are larger when the perturbation is higher. Different applications with different number of time intervals and perturbations have been considered, showing that the percentage of the reduction is approximately constant and improvements are slightly larger for higher number of intervals (i.e. smaller interval widths).


157

Table 6. Comparison between starting and estimated demand matrices

* 0.7 1.4 2.0

5.6

rme 0.03 0.05 0.09

mse 794.0 1216.7 2300.4

Rmse

0.26 0.32 0.44

Timetable setting

Assessing an optimal timetable can improve, to a great extent, effectiveness and efficiency of the transit network, both on the demand-side and on the operator perspective. In fact, an optimal timetable is the result, on the one hand, of a good match between bus supply and passenger demand and, on the other hand, of the minimization of the number of the operating bus fleet. In the light of these objectives it is worthwhile to distinguish between high-frequency and low-frequency systems. In high-frequency systems, the high frequency of transit lines and the random arrival of travellers at stops means that the marginal benefit of improvements (in terms of waiting and transfer time) is minimal. Thus, typical timetable schedules are derived from the optimisation of crew and vehicle schedules which may induce a higher marginal benefit in terms of operating cost. For low-frequency systems, the problem of constructing timetables can be addressed as a problem of defining the scheduled departure-times of the runs of the transit lines. Once the frequency of each line is determined, i.e. the total number of runs departing in the reference period (e.g. the morning peak-hour), the number of runs of each line can be determined at prior planning levels. The method of determination can be grouped into those that simulate passenger flows, those that deal with ideal networks and those based on mathematical programming. A review of these methods can be found in chapter 2. Given the frequency of each line, the problem of optimal timetable setting is that of determining the departure-time of each run r of the line /, (Ort /. This can be achieved through different criteria such as maximizing synchronization of rides at transfer nodes, tuning ride departure-times versus user demand bases (from traffic counts) at stops, or simulating user choice. Among the latter methods, a method for optimal transit setting (Coppola, 2002) which is based on the schedule-based dynamic transit assignment modelling (Nuzzolo et al, 200la) is described. Typically the following assumption are made, i.e.:

158 -

Advanced Modeling for Transit Operations and Service Planning routes and stops of all transit lines are fixed, as well as the roundtrip travel time of each line;

-

capacity of the bus operating on each line is a priori determined;

-

the origin-destination demand profile in the reference period is known and fixed with respect to the transit timetable.

The timetable setting problem can then be defined as the minimization of an objective function R(co) specified according to different perspectives: the users' and the operator perspective. From the operator perspective, the goal is to minimize operative costs which in turn means minimizing the operating fleet size on one hand and maximizing the load factors on the other. Assuming that all the buses are operating within the system, the operative cost does not depend on run departure times and the objective function results in a function only of the perceived users' costs. The perceived users' cost, which is assumed to be the total travel time the users spent on the network, can be written as:

(16) where: -

d° 'TD' is the demand flow on the O/D pair od leaving the origin during time interval /';

~ Pk/odTDi (^) *s me probability of choosing path k connecting origin-destination -

pair od departing in time slice /. WTk/odTD.(G}) is the waiting time on path k connecting O/D pair od departing in time slice /';

~

TTk/odToi(Q) *s me transfer time on path k connecting O/D pair od departing in

-

time slice /; BTj(/ociTD.(a>) is the on-board time on path k connecting O/D pair od departing

-

in time slice /. the sum is extended to all paths of all O/D pairs and time slices of the reference period.

In order to assure the consistency between link and path flows, the following constraining expression of the flow on each link of the diachronic network is the sum of all the flows of paths including that link, and can be considered as:


159

where - fj the flow on the generic link 7' of the diachronic network; - Fk/0(jTD.(&) is the flow on the generic path k of the diachronic network, -

connecting the O/D pair od of users departing in time slice /; 8jtk the element of the link-path incidence matrix, equal to 1 if linky belongs to path k, 0 otherwise.

Furthermore, technical constraints can be added to the optimization problem. To guarantee a minimal space distance between the buses of two subsequent runs, a minimum value of the headway is introduced (MinHdw): MinHdw 0, a e A since ^ va = Vt, i e TV. aeA +

Then, by introducing new variables w(., which denote the total waiting time of all trips at node /, w =

V. ' , i € TV, one obtains the equivalent problem Z-/X at A*

subject to

The objective function (8) is now linear and the 0-1 variables are only used in constraints (9), which are the only nonlinear constraints. These constraints may be relaxed by replacing (9) with

which yields the linear programming problem (8), (12), (10), (11). It may be shown by using the extreme point properties of the solutions of a linear programming model, that this problem is equivalent to (8)-(12). The dual problem of this last linear program is g.H,.

subject to

where ut, Uj are the dual variables corresponding to (10) and jua are the dual variables corresponding to (9).

Frequency Based Transit Route Choice Models

169

Let (v*,W) and (u*,ju') denote the optimal solutions of the primal and dual problems. The weak complementary slackness conditions are and

In both the primal and dual formulations, the transit route choice model has a close resemblance to the shortest path route choice model. The latter corresponds to the shortest path problem when none of the links of the network involves waiting; thus fa —> oo and vt>. -> 0. The solution algorithm which solves the transit route choice model bears a strong resemblance to the label setting algorithm for computing shortest paths. The solution algorithm is composed of two parts. In a first pass, from the destination nodes to all origins, the arcs which carry flow, A', and the expected travel times u* from each node i, ieN, to the destination nodes are computed. In a second pass, from all origins to the destination, the demand is assigned to the arcs a, a e A *. The algorithm is stated below (the convention 0 oo = 1 is used):

Transit Route Choice Algorithm

Part l:Find A* Step 1. 1 (Initialization) : ut 0 when va —» va ,' where v"a is the saturation flow of line a. In order to formulate the corresponding equilibrium model some additional notation is required. Qy is the space of feasible link flows, Qv =[0, oo)1 and Q^ is the space of destination feasible link flows, Q;f=[0,oo) f

respectively.

Mid

f-\fa\,a&A

f(v) = {fa(v)}aeA

. t(a\ h(a)

are the tail and head nodes of an arc,

\

is the vector of nominal line frequencies and

is the vector of effective frequencies. c(v) = {ca(v)}aeA is the vector

of arc travel costs, which are assumed to be continuous. In the following development it is convenient to refer to a node strategy which is a subset of the arcs A*. Each node /, i&N is an origin when a flow traverses it towards destination q. A strategy can be decomposed into a set of node strategies one for each node in TV*, the nodes of the network which carry flow for strategy k. ( ( N k , A k j is the subgraph associated with strategy k, or the hyperpath k, such named by Nguyen and Pallotino (1982)). Qs is the space associated with node strategies, Q^ -R^ms where ms = ^\Kf\, where Kt is the set of non-empty subsets of ,4*. & e A\ is a node ieN

strategy for node / and hqk is the flow of the node strategy k towards destination q. Sqk(v) is the expected time for travel from i to q by using node strategy k and r J(v) is the expected cost of the shortest hyperpath from i to q with frequencies fa(y)

and

time ca(v). Last but not least, A^v) (respectively A'iq{v), A^g(v), A*q(v)) is the set of arcs a e / 4 ^ such that f 0 (v) + r* ( f l ) 0 the variables rf are computed recursively for each destination by:

(see the algorithm for the linear cost model). It was shown by Cominetti and Correa (2001) that T is an implicit continuous function of / and c. But /(v) and c(v) are explicit functions and hence T is an implicit function of v. hi the following all values of T will be denoted rf(v). For each v e Q ^ such that /(v)>0 the cost of a node strategy k&A* may be computed recursively as

The equilibrium model is formulated by stating Wardrop's (1952) user optimal principle as:

Finally, let V(K) = {v e Q^ : (h, v) e Q5 x Q^}. There are several properties which allow the characterization of minimal cost node strategies. They demonstrate that in a general network, the common lines problem

Frequency Based Transit Route Choice Models

175

studied by Cominetti and Correa (2001) is solved implicitly for each node destination pair. These are stated without proofs, which may be referenced in Cepeda (2002). If k,k} and k2 are subsets of A] such that k = kluk2,qeQ and v e Q ^ then the strategy costs are relatively

If k is a node strategy of minimal cost for the pair (19) and ve Q^, then k n A^(v) . In addition, k n A^(v) - . Finally, the node strategy k = A 0, aeA

(7)

Capacity Constrained Transit Assignment Models and Reliability Analysis

191

Constraint (3) represents flow conservation at each node while (4) constrains the wait at each stop to be inversely proportional to the sum of the relevant service frequencies. Constraint (5) divides the flow leaving a node between relevant links according to the relative frequencies of service on those links. Finally, (6) and (7) are binary variable and non-negativity constraints respectively. Following Spiess and Florian (1989), PI may be transformed into the following compact linear program: ?2: Minimise Za&4 cava + Z/e/ W( with respect to va, a eA, and Wj, iel, subject to

where

is the total wait (as opposed to the wait per passenger) at node i measured in units of passenger-hours per hour. The logic of (8) is that for passengers leaving node i the probability of choosing link a=(i,j) eA is either proportional to the frequency of service on that link (if the constraint is active) or 0 (if the constraint is inactive). In the former case, the link is on the hyperpath, while in the latter case, the link is not on the hyperpath. Cases where the constraint is not active but va > 0 do not arise (see Spiess and Florian, 1989). At the solution to ?2 (denoted by *), we can determine the sets IN*(i) and OUT*(i) as follows: a=(i,j)eIN*(j) and a=(ij) eOUT*(i) if and only if v*a > 0. This is so because every link in the hyperpath will be used, since/, > 0. Having determined IN*(i) and OUT*(i), (1) and (2) allow the determination of transition probabilities and nodal delays. In order to solve ?2, it helps to formulate the dual problem: PS: Maximize Z/e/ 0,-w,- with respect to U{, iel, and wa, a eA, subject to

PS may be solved by a Dijkstra-like procedure (see Spiess and Florian, 1989).

192

8.


LINE LOADING BY MARKOV CHAIN

A Markov chain is characterized by a transition matrix defining the probability of an entity (in this case a passenger) moving from one state to another state. Conservation requires the rows of this matrix to sum to one. In this context, the states represent the transfer nodes of the transit network, the origins, a destination and a notional "bin" where trips that "fail" collect. In this context, a trip "fails" when there is insufficient capacity on the line on which the passenger seeks to board. The transition probabilities are both line- and destination-specific. The destination and the bin constitute absorbing states, in the sense that this state once entered is not left. In the event of multiple destinations, these have to be considered one at a time. In order to establish the available capacity at each stop it is necessary to load the lines in the chronologically correct sequence. Destination-specific transition probabilities may be used to preserve the boarding sequence. Let: tfji =

Probability of transition from stop i to stop j by line / on hyperpaths to destination 5

then

Further define

and

From the definition

or in matrix notation

Capacity Constrained Transit Assignment Models and Reliability Analysis

193

where y = [yj, o = [oj, T = [tij+], Tt = [tyj and / is an identity matrix of appropriate dimension. This series is convergent provided T" —> 0 as n —> oo. If the row sums of T are bounded by 1 and at least one row sum is strictly less than 1, and provided each destination is reachable from any node (in which case it is an ergodic system), it follows from the PerronFrobenius Theorem (see Cox and Miller, 1965) that all the eigen values of T are strictly less than 1 and so T" —> 0 as n —> oo.

9.

ENFORCING CAPACITY CONSTRAINTS

So far, a capacity constraint has not been enforced. Current thinking (see Spiess and Florian, 1989; De Cea and Fernandez, 1993; Cominetti and Correa, 2001) seems to be that, in order to enforce this constraint, the "effective frequency" of the line should be reduced at the point where the overloading first arises. The justification for this is that some passengers will not be able to board the first vehicle that arrives at that stop due to the overloading. De Cea and Fernandez (1993), the first offer a solution, suggest using BPR-type waiting time functions at the stop, where the capacity available at a stop on each line depends on the line capacity and the passenger "through traffic" on that line at that point. The effective frequency is proportional to the inverse of the waiting time. Cominetti and Correa (2001) then look at replacing the BPR-type functions by queuing models. As a result of the capacity constraint, we require that the boarders at each transfer node / should not exceed the available capacity across all lines serving the stop that are in the attractive set.

To prevent capacity being infringed, some passengers will not be able to board. Hence

Note that (12) and (13) are simultaneous equations for a given set of hyperpaths. However, one would expect the probability of not boarding to influence the set of hyperpaths in some way.

1 0.

CAPACITY CONSTRAINED TRANSIT ASSIGNMENT

The risk of failure to board can be included in the determination of hyperpaths by defining the following generalised cost of travel:

194


where ft is a weight representing the risk averseness of the traveller. The following problem finds a risk-averse transit assignment for given failure-to-board probabilities: P4: Minimise Za=(«,/;>&4 gijVa with respect to va, a

where 9 is the coefficient of perceptional variation. n\ is the utility associated with route k in the associated physical sub-network, expressed as: < = n , + 7 V ' f + ?Vpf,

ViJsU^

(4)

where y^ is a mode-specific constant; t\ is the travel time on route k; and pf is the monetary cost associated with route k, which can be specific to the particular mode in Class-3 as taxi charge or gasoline cost etc. 2.4

Multi-modal Utility Functions

A route in this SAM network generally consists of a sequence of node-to-node direct links ad on a transport mode plus transfer links a, between modes. In-vehicle travel times, transit fares, and modal preferences constitute the utility of the node-to-node direct links ad; whereas the transfer time and transfer penalties are considered in the transfer links at. To simplify the analysis, we further consider that link utilities i//a,

242


a e A = A r u Arf , in the SAM network are additive. Let A" be the utility of route p between origin-destination pair rs in the SAM network: )-¥a,

(5)

where S(p,a] = 1 if link a in the SAM network is on route p ; zero otherwise. Each link in the SAM network is either a transfer or direct link, whose utility is determined from (1), (2), or (3). Within this SAM network, we can further divide travelers into different classes with different utility functions to capture their specific travel choice characteristics. For simplicity, in the example shown in Section 4, we only consider the case of different coefficients $,, $ , yb, 7, and y2 in (1), (2) and (4) for different traveler classes, as will be discussed more in Section 4.

Figure 5. The original 3-modal network The above discussions lead to the formation of the SAM network. For illustration purposes, consider the multi-modal network shown in Figure 5. Passengers departing from node 1 to node 4 have three transport modes and transfer possibilities at nodes 2 and 3. Adopting the transfer rules in Figure 1, the associated SAM network is formulated as shown in Figure 6. 2.5

Multi-modal Network Assignment

After a multi-modal network is transformed to a SAM network, it can be considered as a simple network without the need to attend to transfer feasibility, nonlinear fares, and related issues. A route in the SAM network automatically combines the modetransfer choices, which can be decoded for the specific modes used and transfer locations selected. In the following example, we formulate the multi-modal assignment problem as a SUE route choice problem.

Modeling Competitive Multi-modal Services

Figure 6.

243

The resultant SAM network of the original 3-modal network

The perceived utility (A") of route p between origin-destination (OD) pair rs in the SAM network is modeled as the sum of a systematic term (A") and an error term (e") that is independently and identically distributed: A" = A1 + srps

VrsellS,p€ Prs

(6)

where RS is the set of OD pairs in the SAM network; Pn is the set of routes linking OD pair rs. Assuming that the random term is Gumbel distributed, the resultant SUE pattern follows this logit relationship: expf/l-A") „" _

* \

"I

Wr, ,- D« V

RS

(7)

teP"

where w" is the proportion of passengers between rs using route p; /I is a parameter on travel utility perception variation. Moreover, the flow on route p (h") satisfies the demand constraint: h^-w^-q"=Q, V~rse'RS,VpeP7s

(8)

where q" is the total demand on OD pair rs. By multiplying h™ to (8), we obtain the following complementary form:

244

Advanced Modeling for Transit Operations and Service Planning RS,VpePs

(9)

If h" > 0 , then (8) must be satisfied or h™ is apportioned according to (7). Theoretically, the nature of the SUE assignment would assign a positive flow to each route, rendering the case of h™ = 0 nonexistent. Thus, this SUE problem can be written as a nonlinear complementarity problem (NCP) in the form of:

h>0 where h = (/*;, \/7s e~RS,\/p&P7s)

and F(h) = (/£ -wj ^

This NCP can be solved with a mathematical programming technique (Lo and Chen, 2000; Lo and Szeto, 2002) or an approach based on the Method of Successive Averages (MSA). For brevity, we do not describe the solution procedure here.

3.

FARE COMPETITION AMONG TRANSIT SERVICES

The SAM network provides a structure to model travelers' combined route-modal choices in a network of multi-modal transit services. Using the SAM network as a base, it enables various kinds of sensitivity analysis of transit service modifications and competitions. One example is studying the impact of fare competition on the services' revenues, the resultant fares, and network congestion. This analysis has important implications on government regulations. Our previous study (Lo and Yip, 2002) examined this problem in a simplified fashion by adjusting the fare of each service until the fare elasticity of each service becomes negative one. This simplified approach cannot be extended for an intertwined multi-modal network with many transfers between services. In this study, we propose a formulation via the Variational Inequality Problem (VIP) that seeks a Nash equilibrium among the transit services, in which no operator can increase its revenue further by unilaterally changing its fare. This equilibrium fare structure represents the result of an unregulated fare competition among the transit services. Let p = (/?,,••-,p ( 0 ,-~PK) be the fares of K transit services engage in a fare competition, and y/ = (^],...,^(B,...,^) be the services' revenues. According to (2), changing the fare of a service will affect the utilities of its associated direct links, and


245

hence its patronage and revenue, therefore one can express y/ = \y (p, h (p)) = y (p) . +

/*

*

K

* \

A Nash equilibrium is achieved when a fare vector p = \p\ ,...,/9 (U ,.../? Ar Je9I + is found such that:

where p*^ = (p\,...,p*ia_^p'(a^,...,p'K \ . In other words, for each individual service CD , charging any fare that is different from p*u will result in a lower revenue. When this occurs, none of the services will have an incentive to deviate from this equilibrium fare structure. To find p" , we assume that the revenue for each service co , y/^ , is a continuously differentiate function and concave with respect to pa ("n" shape). This assumption is reasonable as it implies that the optimal fare for service a> is neither zero (which will result in zero revenue) nor infinitely high due to the presence of competition, but somewhere in between. Mathematically, for each service co, this concave property with respect to p^ can be written as:

.-/O^.(p)

02)

where p ', p e 9?f are any two arbitrary fare vectors. Setting p ' = p* , we have:

M*>n^(A.V.)*r.(p).

(13)

As long as one can find p* such that: or -

then (13) becomes: ^ (/?*)> y/m (p) , which is the same as (1 1) - a fare that service co does not want to deviate from. Repeating this argument for each service and summing its corresponding condition (14), we have the variational inequality problem VTP(F,9lf): finding a vector //e9^ such that: -V¥(p}T(p-p)>V,

Vpe^f,

(15)

246


, v where -Vy/{p ]=

OI//AP i oy/^p i o\i/K\p i -—'-,..., -—'-,..., -—'- . More discussions on

this type of oligopolistic market equilibrium can be found in Nagumey (1993). We reiterate the relationship that y/(p) = y/"(p,h(/?)) or any change in the fare structure will affect the route flows in the multi-modal assignment, which in turn affect the revenue distribution. To solve this VIP(F,9tf), a number of projection methods can be employed. In this study, we adopt the projection method developed in Han and Lo (2002). hi the interest of space, we omit the details but only summarize the main algorithmic steps as below:

(16) M P

=pk-tke(pk,pk)

(17)

where P^H is the projection onto SRf; pk refers to the fare vector at the kth iteration; flk, tkare parameters to be set, which can be set to vary from iteration to iteration; and e(pk,j3k\ is a measure of the error of convergence at the kth iteration. Through applying (16)-(17) repeatedly, the algorithm gradually reduces the error measure e(pk,j3k]

to zero, ensuring that the solution obtained at the last iteration,

pk, fulfills the VIP (15).

4.

CASE STUDY

To illustrate the applicability of the SAM network, we study the impact of fare competition among services connecting the Hong Kong International Airport (HKIA) to the downtown area in Kowloon West. After establishing and calibrating the SAM network of the study area with field data, we determine the market equilibrium under fare competition using the VIP formulation discussed in Section 3. 4.1

HKIA - Kowloon West SAM Network

A sample of seven hundred travelers at the arrival hall of the HKIA was surveyed on 18 and 19 January 2000. The survey revealed the variety of transit modes and combinations selected by travelers. In addition to auto, travelers chose among the rail


247

services, bus services (A-Bus, E-Bus & S-Bus), shuttle services (H-Bus), and taxis. The survey also recorded passengers' transfer locations. The model considers three pairs of OD demands from the airport, to: Tsim-Sha-Tsui (TST), Mongkok (MK), and Kowloon West (KW). Travelers are stratified into two classes: "arriving passengers": who have just finished a trip overseas and arrived at the airport and "domestic passengers": who went to the airport to pick up friends or work at the airport. The hourly demands of both traveler classes are shown in Table 1. The multi-modal network is shown in Figure 7(a). Each roadway link is labeled (which will be referred to in result discussions). Links 1 to 5 are highways that have spare capacities; whereas links 6, 7,9, 10 are congested urban roadways. Shown along each node (or location) are the alighting or boarding modes available. We define the probable transfer states as those that were actually used by travelers according to the survey, as shown in Figure 7(b). Note that: (i) the Airport Express Line (AEL) cannot reach nodes 4, 6, and 7 directly, so transfer is necessary for trips heading for those destinations; (ii) some AEL passengers chose to walk from the AEL station at node 5 to the destination node 7, the walk link 8 between nodes 5 and 7 is thus introduced; (iii) States 8 and 9 both correspond to the Mass Transit Railway (MTR) line. They are designated different states due to different prior mode usages. State 8 refers to trips transferred from AEL; State 9 refers to those transferred from SBus. For trips transferred from the AEL service, the MTR trip is free. This demonstrates how states in the SAM network can capture collaborative fare discounts between transit operators. The disutility functions include these attributes: fare, in-vehicle travel time, and waiting time. Their associated coefficients are calibrated from the survey data, as shown in Table 2. To simplify the calibration procedure, we assume the same attribute coefficients but different mode specific constants for the two traveler classes with different destinations. Also, as auto and taxi costs vary as a function of travel distance and time, we convert their associated monetary costs into equivalent in-vehicle times, combine with the actual in-vehicle time, and calibrate the in-vehicle time attribute accordingly. This is why the coefficients of in-vehicle time of these two modes are different from the others. A generic transfer penalty term is added to the utility for each transit transfer. Using the maximum likelihood estimation procedure developed in Lo and Yip (2002), we calibrate the disutility functions. By combining the calibrated disutility functions with the SAM network analysis, we obtained the hourly passenger flows by class on each network link, as shown in

248


Figure 8. The loads on each roadway link, each bus line, and each rail service are depicted. Table 1.

Average hourly demands

O-D pair Origin

HKIA (node 1)

Destination

Arriving passengers

Domestic passengers

TST (node 7)

1167

652

MK (node 6)

288

621

KW (node 4)

121

152

Table 2.

Utility

Demand

Attribute coefficients of the disutility functions

Class f Class 1 Class 1 Class 2** Class 2 Class 2 (for (for (for (for (for (for MK) KW) TST) TST) MK) KW)

AEL

-2.3

-3.6

-9.2

-1.5

-1.3

-21.5

MTR

-5.4

1.5

-4.5

-2.0

-4.6

A-Bus

-7.3

-0.8

11.5

-1.9

-3.4

Fare

In-Veh Waiting time time 0.0470

0.1787

1.8

0.0387 0.0387

0.0470

0.1787

5.0

0.0387

0.0470

0.1787

0.0470

0.1787

E-Bus

36.2

0.6

-11.4

36.7

17.3

-18.2

0.0387

Auto

-7.5

-1.0

-9.9

-2.0

-4.5

11.5

0

0.1507

0.1787

H-Bus

-6.7

NA

NA

-3.2

NA

NA

0.0387

0.0470

0.1787

Taxi

-6.6

-0.3

14.2

-2.3

-5.0

-0.4

0

0.1905

0

S-Bus

23.0

-1.5

13.5

28.8

22.3

-21.7

0.0387

0.0470

0.1787

Walking

-2.6

NA

NA

50.57

NA

NA

0

0.0470

0

Transfer Penalty: Note:

2.62

Class 1 refers to arriving passengers; Class 2 refers to domestic passengers. NA is denoted if no passengers of the specific class took this mode.


249

o.

(a) The P h y s i c a l M ulti-m o d a l N e t w o r k of the case s t u d y s c e n a r i o

Figure 7.

(b) State T r a n s i t i o n D i a g r a m of the s a m e n e t w o r k w i t h Identified Probable Transfers Physical network and probable transfer states for the case study

250


Figure 8. 4.2

Passenger volumes on the exiting multi-modal, network

Transit Fare Competition

We study the scenario wherein these privately operated services engage in a fare competition, including AEL, MTR, A-Bus, E-Bus and S-Bus. Other modes are excluded from this competition because they are either private services (e.g. auto and H-Bus) or tightly regulated (e.g. taxi). The results in Tables 3 - 4 and Figure 9 show the impact of this competition on fare, revenue, and travel pattern. Table 3 depicts the fare and aggregate revenue of each service as a result of the competition. All modes raise their fares, ranging from 10% for AEL, 27% for S-Bus, and doubling or more for A-Bus, E-Bus and MTR. Note that the existing AEL fare is higher than the other modes5. The competition, if unregulated, would allow A-Bus, E-

5

The rail services must recover the right-of-way and infrastructure costs, whereas the bus services use the roadway infrastructure that is provided and maintained for free by the government. Therefore, justifiably the Hong Kong government allows the rail services with a greater flexibility to define their fares.


251

Bus and AEL to all raise their fares so that they become about the same. This of course defeats the purpose of intending the bus services as affordable modes. This result indicates the possible occurrence of collusion if allowed, wherein all transit operators simultaneously raise their fares to exploit travelers. In the end, most of the operators are able to increase their revenues from the current levels, especially for the rail service MTR and to a great extent for the other rail services AEL and E-Bus. On the other hand, S-Bus will lose out in this competition despite a substantial fare increase. Table 3. Impacts on Fare and Revenue initiated by Transit Fare Competition S-Bus Mode AEL MTR A-Bus E-Bus Fare Change (%) +26.9 +10.3 +147.3 +91.3 +222.0 -25.3 Revenue Change (%) +59.3 +4.1 +14.3 +19.9 Figure 9 shows the passenger volumes as a result of the fare competition. In crossreference with Figure 8, it shows a major shift of travelers (especially domestic passengers) from E-Bus to AEL, A-Bus, and MTR, after E-Bus more than doubles its fare. E-Bus, originally designed as a service with low fare but a long detour route, has a relatively high mode-specific disutility (Table 2). This large fare increase leads to a significant drop in E-Bus's patronage. This of course defeats its purpose as an affordable mode. Due to substantial fare increases of all transit modes, a considerate amount of passengers now prefer to take taxis or autos, resulting in an increase in network congestion. Table 4 shows the travel time of each roadway link of the network. It shows an increase in the travel time for all the links. The percentage changes in travel times are particularly substantial for links 7 and 8, which are already congested main roads leading to the downtown area. This shift to taxi and auto trips further exacerbates the congestion problem. All in all, the fare competition leads to these outcomes: • • •

Travelers generally pay higher fares for all of the transit services The network gets more congested, especially on the already congested roadways Most of the transit service operators enjoy high revenues.

This case study indicates that if the system were allowed to engage in a totally unregulated fare competition, both travelers and the network would suffer. It challenges the notion that more, freer competition would always benefit consumers.

252


Table 4. Travel Time of each Roadway Link of the Network PART 1 - Auto/Taxi Travel Time (min) 1

1

Road

Current auto travel time

10

7.0

15.0

15.0

10.1

12.2

13.1

12.9

Travel time after competition

7.0

15.1

15.0

10.4

12.7

13.1

Percentage Increase (%)

0.4

0.4

0.0

2.4

4.4

0.2

5.2

9.0

15.4

8.2

9.1

9.5

19.9

56.0

1.8

7.1

PART 2 - Bus (A-Bus, E-Bus & S-Bus) Travel Time (min) Road

1

1

10 2

Current bus travel time

15.0

20.0

20.0

15.2 N.A.

N.A.

12.9 N.A.

11.2 N.A.

Travel time after competition

15.1 20.1

20.0

15.5 N.A. N.A.

15.4 N.A.

11.4 N.A.

Percentage Increase (%)

0.4

0.0

2.4

N.A. N.A.

19.9 N.A.

1.8 N.A.

Notes: 2

0.4

See Figure 7(a) for reference. No bus routes on links 5, 6, 8 or 10.

Figure 9.

Passenger volumes as a result of fare competition


5.

253

CONCLUDING REMARKS

The paper presented a framework to model multi-modal networks. Two key considerations addressed by the SAM network approach are the incorporation of probable transfer states and the introduction of nonlinear fare and utility structures. We have coupled this SAM network with a logit assignment approach. For combinedmode trips with transfers, mode-segment overlaps between transfers are likely to happen, which may violate the assumption of independent and irrelevant alternatives (IIA) of the standard logit approach. To avoid this problem, one can extend the consideration to a nested logit approach. On example of such an extension is shown in Lo et al. (2002b). In general, how to couple the SAM network structure with the nested logit approach so as to better describe travelers' route-modal choice behavior, while maintaining the model to be tractable and conveniently calibrated is a future research direction. As demonstrated in this study, combining this SAM network framework with game theoretic approaches forms a very rich platform for analyzing the competitions between operators as well as for studying the case of regulations. Such analysis would shed light on creating and maintaining the thin balance between corporate profitability, traveler disutility, and overall network congestion. In broader terms, how should the government create an environment such that the private sector can be relied upon to provide for the services while maintaining the public welfare? Is there a role for government regulations on fare and service quality so as to create "win-win" situations, in which private investors are enticed into quality services provision and travelers continue to enjoy affordable services? Analysis leading to answering this question are of significant interest and potential application to many transit-oriented cities.

ACKNOWLEDGEMENT This study is sponsored by the Competitive Earmarked Research Grants, HKUST6083/OOE and HKUST6161/02E, of the Hong Kong Research Grant Council.

254


REFERENCES Bell, M. and Y. lida (1997). Transportation network analysis. John Wiley & Sons, Inc., Bertsekas, D. (1995). Dynamic Programming and Optimal Control. Belmont, Mass. Athena Scientific. De Cea, J. and E. Fernandez (1993). Transit assignment for congested public transport systems: An equilibrium model. Transportation Science, 27, 133-147. Fernandez, E., J. De Cea, M. Florian, and E. Cabrera (1994). Network equilibrium models with combined modes. Transportation Science, 28, 182-192. Gabriel, S. and D. Bernstein (1997). The traffic equilibrium problem with nonadditive path costs. Transportation Science, 31, 337-348. Han, D. and H. Lo (2002). Solving nonadditive traffic assigment problems: A decent method for co-coercive variational inequalities. European Journal of Operational Research. Submitted. Lam, W. H. K., Z. Y. Gao, K. S. Chan, and H. Yang (1999). A stochastic user equilibrium assignment model for congested transit networks. Transportation Research, 33E, 351-368. Lo, H. and A. Chen (2000). Traffic equilibrium problem with route-specific costs: Formulation and algorithms. Transportation Research, 34B, 493-513. Lo, H. and W. Y. Szeto (2002). A methodology for sustainable traveler information services. Transportation Research, 36B, 113-130. Lo, H. and C. Yip (2002). Deregulating transit services: Winners and losers in a competitive market. Journal of Advanced Transportation, 35, 215-235. Lo, H., C. Yip, and K. Wan. (2002a). Modeling transfers and nonlinear fare structure in multi-modal transit network. Transportation Research B. In press. Lo, H., C. Yip, and K. Wan. (2002b). Modeling competitive multi-modal transit services: A nested logit approach. Transportation Research C. In press. Nagumey, A. (1993). Network Economics: A Variational Inequality Approach. Kluwer Academic Publishers. Nielsen, O. A. (2000). A stochastic transit assignment model considering differences in passengers utility functions. Transportation Research, 34B, 377-402. Nguyen, S. and S. Pallottino (1988). Equilibrium traffic assignment for large scale transit networks. European Journal of Operational Research, 37, 176-186.


255

Spiess, H. and M. Florian (1989). Optimal strategies: A new assignment model for transit network. Transportation Research, 23B, 83-102. Wu, J., M. Florian and P. Marcotte (1994). Transit equilibrium assignment: A model and solution algorithms. Transportation Science, 28, 193-203.


CHAPTER 10

MODELING URBAN TAXI SERVICES: A LITERATURE SURVEY AND AN ANALYTICAL EXAMPLE Hai Yang, Min Ye, Wilson H. Tang, Department of Civil Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, P.R. China, and S. C. Wong, Department of Civil Engineering, The University of Hong Kong, Hong Kong, P.R. China

1.

INTRODUCTION

In most large cities, taxis are an important transportation mode that offers a speedy, comfortable and direct transportation service. Unlike regular transit such as trains, trams, and buses, taxis are usually operated by a large number of private firms, frequently owned and operated by individuals. With such a market setting, individual taxi drivers or firms can freely choose their working schedule in response to market profitability and operating cost as well as opportunity cost of being in service in different time of the day. A remarkable characteristic of taxi service in most

258


metropolitans is the hourly variation of customer demand and taxi service intensity throughout the whole day. hi addition, cruising rather than dispatch taxi service is the norm; this is particularly true in metropolitans in developing countries. The taxi market is highly regulated. Nearly all over the world, local governments limit the number of taxi licenses issued, set the fares to be charged, and impose and enforce standards of quality and rules of conduct. In the urban area of Hong Kong, taxis currently form about 25% of the traffic stream, hi some critical locations, taxis form as much as 50% to 60% of the traffic stream (Transport Department, 1986-2000). Taxis make considerable demands on limited road space and contribute significantly to traffic congestion even when empty (cruising for customers). In general the taxi industry is subject to various types of regulation such as entry restriction and price control, hi Hong Kong, taxi operations are subject to service area demarcation as well. The urban taxis operate throughout the territory, while the others are fundamentally confined to the rural areas of the New Territories and Lantau Island. Currently, annual taxi service surveys (surveys at sampled taxi stands and roadside observation points) have been conducted since 1986 to gather the information on customer/taxi waiting time, taxi utilization and taxi availability for the city of Hong Kong (Transport Department, 1986-2000). These types of information have been used for the evaluation of taxi services and government decision-making with respect to the increase in the number of taxis and/or adjustment of taxi fares. Furthermore, a simple modal split model of taxis has been developed in Hong Kong for the prediction of taxi person and vehicle origin-destination matrices, which has been incorporated as an important component in comprehensive transport studies (Transport Department, 1993). Traditionally, many economists have examined the models and economics of urban taxi services under various types of regulation, such as entry restriction and price control in an aggregate way. It was only recently that urban taxi services was modelled in a network context. A realistic method has been proposed to describe vacant and occupied taxi movements in a road network as well as taxi drivers' search behavior for customers. A few extensions have been made to deal with demand elasticity, multiclass taxi services with service area regulation, and congestion effects together with development of efficient solution algorithms. The models have been extended to the multi-period dynamic demand-supply equilibrium of taxi services. Calibration and

Modeling Urban Taxi Services: A Literature Survey and An Analytical Example 259 validation of the network taxi service models have been conducted towards their practical applications. This chapter presents a comprehensive survey of the static and dynamic modeling methods for urban taxi services. Both the conventional aggregative demand-supply equilibrium model and the recent network equilibrium models are covered, with emphasis placed on the interplay among taxi supply, customer demand and externality of service consumption on customer waiting time and taxi utilization. The models and analysis presented here are useful to understand the manner in which the demand and supply are equilibrated in the presence of regulations in terms of entry restriction and price control, thereby providing information for government decision-making in terms of such regulations.

2.

AGGREGATE MODELING

2.1

Overview

The taxi market is heavily regulated mainly by local government in most large cities. Regulation makes the taxi market interesting, because price does not play the usual role of clearing the market, and taxi supply and customer demand are brought in balance through two intermediate variables: taxi availability and taxi utilization. The interest of economists for the analytical aspects of the taxi market can be traced back to the early sixties. In an appendix of his first provisional edition of "Price Theory" (Friedman, 1962), Friedman included the issue of "licensing taxicabs": a problem of delightful subtlety. This problem soon attracted interest by professional economists (Lipsey and Steiner, 1966; Orr, 1969). A subsequent stream of papers followed the topic continually kept up to date (Douglas, 1972; Beesley, 1973; De vany, 1975; Shrieber, 1975, 1977; Abe and Brush, 1976; Manski and Wright, 1976; Foerster and Gilbert, 1979; Beesley and Glaister, 1983; Schroeter, 1983; Frankena and Pautler, 1986; Gallick and Sisk, 1987; Rometsch and Wolfsteter, 1993; Hackner and Nyberg, 1995). Early works centered on the general recognition of pervasive market failures in this sector and the analysis of the effect of regulation of fares and also investigated entry under alternative assumptions regarding the market structure and the organization of the service. Recent notable studies in the topic have made significant improvements in our understanding of the market mechanism (Arnott, 1996, Cairns and Liston-Heyes, 1996, Bergantino and Longobardi, 2001; Fernandez et a/., 2001). In addition, there have been many contributions towards the study of empirical

260


aspects of taxi regulation/deregulation around the world (Teal and Berglund, 1987; Garling et al, 1995; Dempsey, 1996; Gaunt, 1996; Gaunt and Black, 1996; Morrison, 1997; Radbone, 1998; Schaller, 1999; Flath, 2002). The economics of taxi service has been overwhelmingly examined in an aggregative manner and the general analytical framework can be described below:

,

dF

-

,

dW

(1)

W=W(V\ — 0) is a positive parameter whose value depends on the size of the service area and the distribution of taxi stands over the service area. Substitute Equation (7) into the demand function (4), we have

Modeling Urban Taxi Services: A Literature Survey and An Analytical Example 263 0

(8)

Note that O (Q) is a continuous function of Q e (0, N/T) . Furthermore, .0 > 0

(N-TQY and _ f f TP ^1 / \ N Lim < 0 and Z-/w O(2) = — > 0

We thus conclude that for a given fleet size N and fare F per taxi ride, there exists one and only one equilibrium value of Q from the above equation within the feasible domain (0, N/T).

We can thus write the equilibrium demand function as

Q = Q(F,N,

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