The economic value of timetable changes

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th 45 European Transport Conference 2017, ETC 2017

The economic value of timetable changes Eric Kroesa,, Andrew Dalyb a aVU

VU University Amsterdam, the Netherlands Institute for Transport Studies, University of Leeds, United Kingdom

b b Institute

Abstract The paper addresses the consumer value of changes in service frequency for timetable-based transport systems such as bus, train, ferry and air transport. Instead of using “average waiting times” we propose a more appropriate model specification of the different impacts of timetable changes for individual travellers. This includes a decision whether to plan the journey or not, and (if planning) when to start the journey as a function of the services available. We also address the issue of how to aggregate the individual values across the travelling population to obtain estimates of total welfare. Because there is significant variation in preferences between individual travellers, logsum-type measures are used to derive expected utilities rather than mean values. We end our paper by illustrating the application of our method for a rail service. © 2018 The Authors. Published by Elsevier Ltd. © 2015 The Authors. by Elsevier B.V. This is an open accessPublished article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility for European Transport. Selection and peer-review under responsibility of of Association the Association for European Transport. Keywords: train timetabling; rail capacity allocation problem; rail infrastructure utilisation; mixed traffic operation

1. Introduction Values of travel time and of reliability are widely available for use in appraisal studies (Department for Transport 2015, Kouwenhoven et al. 2014). But less is known about the consumer value of changes in service frequency for timetable-based transport systems such as bus, train, ferry and air transport. In practical studies often “average waiting times” are used, computed as half the headway, and multiplied by an assumed value of waiting time (see e.g. Abrantes and Wardman 2011). But this is not a realistic reflection, and does not result in correct cost estimates. First, it has long been known that waiting times are not equal to half the headway. Surveys of actual waiting times (e.g. Daly and Zachary, 1977) show that passengers can wait much less than this (for reliable, long-headway services) or much more (for unreliable services). Secondly, the value of waiting time, i.e. at the actual bus stop or  Corresponding author: E-mail address: [email protected] 2214-241X© 2015 The Authors. Published by Elsevier B.V. Selection and peer-review under responsibility of Association for European Transport.

2352-1465  2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the Association for European Transport. 10.1016/j.trpro.2018.09.042

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railway station, is only part of the story, as passengers may have to wait at their origin, incurring ‘hidden’ waiting time1, for a suitable time to set out for their journey. Third, to obtain a value of waiting time, stated preference surveys are often used that collect travellers’ valuations of ‘waiting time’, interpreted as the travellers themselves see fit, quite likely without connection to the service being operated. To assess the value of timetable changes we need to consider the cost to passengers of headway, rather than that of waiting time. Fosgerau (2009) describes how the cost of headway can be determined, pointing out that public transport users typically incur scheduling cost due to the fact that the public transport vehicles do not necessarily depart exactly at the times the users want to travel. This is the scheduling cost for reliable services, operated according to the timetable. If the operation of the service is not exactly according to the timetable, i.e. if there is unreliability, the public transport users will incur additional cost. Fosgerau and Karlström (2010) describe how the pure unreliability cost can be determined. Others such as Tseng et al. (2012) and Lijesen (2014) show that unreliability may also increase the scheduling cost, but that travellers can eliminate part of the negative effect of unreliability on scheduling cost by adjusting their departure times. All these analyses have been based upon the assumption of equal intervals between subsequent transport services, perfect information among the travellers about the schedule, rational adjustment of the chosen departure times, and uniform preferences. In reality intervals may not be equal, passengers may not behave as perfectly informed strictly rational agents and may have heterogeneous preferences. The aim of this paper is to propose an alternative way to model the different impacts of timetable changes for individual travellers. We start with a review of some relevant literature, and then propose what we view as a suitable specification of the different impacts and costs of timetable changes for individual travellers. We also address the issue of how to aggregate these individual values across the travelling population to obtain estimates of total welfare. In some cases, when there is significant variation in preferences, logsum-type measures can best be used to derive expected utilities or costs. In other cases it may be more appropriate to use mean utilities or costs. We shall illustrate the application of the method for rail services, and demonstrate how the key model parameters influence the results. The paper concludes by providing some recommendations on the practical use of the methodology. While the example application is specific to the rail sector, the results are expected to be of interest for other transport services and possibly for other economic sectors as well. 2. Literature review In this chapter we first look at how various authors determine the expected cost of headway for timetabled transport services. Then we focus on a number of publications describing the so-called “Rooftops” model, a particular approach that is used in railways planning when assigning demand to different train services in time. We go on to look at some publications that broaden the Rooftops approach into a more general discrete choice-modelling framework. And finally we review a paper that describes ways to aggregate the results for individual travellers to those for entire populations. 2.1

The expected cost of headway

Fosgerau (2009) derives the user cost associated with the timetabled operation of public transport services, the cost of headway, separately for planning and non-planning users. He starts from a standard linear schedule delay cost function for a given preferred arrival time PAT: 𝐷𝐷(𝑃𝑃𝑃𝑃𝑃𝑃) = 𝛼𝛼(𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡) + 𝛽𝛽(𝑠𝑠𝑠𝑠ℎ𝑒𝑒𝑑𝑑𝑢𝑢𝑢𝑢𝑢𝑢 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒(𝑃𝑃𝑃𝑃𝑃𝑃)) + 𝛾𝛾(𝑠𝑠𝑠𝑠ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙(𝑃𝑃𝑃𝑃𝑃𝑃))

Fig. 1 indicates how the interval for which a given service is optimal is established around the PAT using D.

1

Alternatively or additionally, they may have to wait at their destination until the time they really needed to be there.

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Fig 1. Optimal interval around the PAT

For the planning user, in the case of a service with long headway ℎ, the expected cost is: 𝐶𝐶𝑝𝑝 (ℎ) =

ℎ𝛾𝛾𝛾𝛾 2(𝛾𝛾 + 𝛽𝛽)

where subscript 𝑝𝑝 refers to the planning user, and ℎ is headway. Note that the planning users do not have any waiting cost in Fosgerau’s model, which ignores unreliability. And note also that the expected cost is proportional to headway: the longer the headway, the higher the cost of headway. For the non-planning user, in the case of a service with short headway, the expected cost is for linear schedule delay cost D: 𝐶𝐶𝑢𝑢 (ℎ) = 𝐶𝐶𝑝𝑝 (ℎ) +

𝛼𝛼ℎ 2

where 𝛼𝛼 is the value of time2. This shows that the average schedule delay cost is the same for planning and nonplanning users, but that the non-planning users incur an additional expected waiting cost that is also proportional to headway. The planning users, on the other hand, may incur planning cost (which we have disregarded here). Note, by the way, that the scheduling cost has been derived assuming a uniform distribution of preferred departure times by the passengers, which is (within a limited time window) a reasonable assumption. And also it has been assumed that there are equal intervals between all subsequent services, so that headway is constant. The above results apply to a situation where the service is executed exactly according to the timetable, a reliable service. If there are deviations from the timetable, additional costs are incurred, both in terms of pure unreliability cost (the cost associated with the variability of travel time) and in terms of modified scheduling cost. Fosgerau and Karlström (2010) derive the value of reliability, and Fosgerau and Engelson (2011) consider the value of travel time variance. Tseng et al. (2012) investigate how unreliable trains also increase the scheduling cost for passengers, and how these should be accounted for in appraisal studies. They show that this effect is additional to the pure unreliability cost associated with the travel time variability, and can be quite substantial in the case of fixed departure times by the passengers. But they also show using a simple model how passengers can actually reduce these increased

2

This is actually the value of waiting time rather than travel time, which according to Fosgerau “is generally thought to be higher than the value of travel time”.

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scheduling cost in the case of unreliability by adjusting their departure times (anticipating departures). Lijesen (2014) uses observed punctuality statistics for Dutch trains to demonstrate that in practice passengers can reduce the increase in scheduling cost due to unreliability significantly by adjusting their departure times. That suggests that in fact the impact of unreliability on the scheduling cost can be limited, although the pure unreliability costs associated with varying travel times remain. The analyses of both Tseng et al. and Lijesen are based upon theoretical reasoning, assuming that all passengers know exactly the real distribution of travel times, and that all passengers behave strictly as cost minimisers when deciding when to travel. It is obvious that in reality these assumptions may not be entirely realistic. But we may still draw some conclusions from the above references:  users of public transport services incur scheduling cost because the public transport service departs only at specific moments in time (timetabled service);  these scheduling costs are proportional to headway, when headways are uniform and services are perfectly reliable;  the waiting cost for non-planning users is also proportional to headway;  unreliability of departure times of public transport services further adds to the scheduling cost, but the increase in scheduling costs can be reduced by anticipating departures. The intention of this paper is to focus on practical aspects of operation, in particular when different services provide alternatives for the same journey, possibly offering a non-uniform service (unequal headways, journey times, fares, etc.), and on how passengers value changes in timetables. We do not assume perfect reliability, but neither do we offer valuations of changes in reliability. 2.2

The Rooftops model

From a completely different perspective railway operators have developed models to predict how passengers distribute themselves among the different railway services available within a certain time window. These models are also used, and particularly, for situations where different departure times are offered by different railway operators, and where time intervals between subsequent departures may be uneven (not uniform headways). The classical approach that has been used for this by railway planners bears the name of the “Rooftops model”. The first publication we could identify that describes what was later called a Rooftops model is Tyler and Hassard (1973). They present a rail demand model for inter-urban rail, which includes a train-choice model based upon a preferred arrival time (PAT) and a measure of the dis-benefits arising from deviating from optimum arrival times. The name “Rooftops model” is taken from the shape of the graph that plots total cost consisting of travel time plus schedule delay cost (expressed in equivalent travel time) as a function of PAT, see Figure 2. The rail service in this example is regular but not uniform: train A departs 09.00 arrives 10.00 train B departs 10.00 arrives 11.20 train C departs 11.00 arrives 12.00 train D departs 12.00 arrives 13.20, and so on. A traveller wishing to arrive at, say, 10.32 has to accept 32 minutes of schedule delay cost which should be added to the 60 minutes travel time. For a traveller wishing to arrive at 10.00 there is no such addition. Note that in this case the value of 1 minute earlier or later arrival is taken as equivalent to 1 minute of travel time. According to Douglas et al. (2011) the Rooftops model was originally derived from spatial economics work by Hotelling (1929). Hotelling used a similar concept to describe where shops should locate to best serve their potential market, starting from the famous example of ice-cream sellers on a beach. The 1986 issue of the Passenger Demand Forecasting Handbook (PDFH), prepared for the British Railways Board, describes how the same Rooftops model was used to produce two important outputs: (1) which specific trains passengers chose in order to determine train loads, and (2) what scheduling costs were perceived by the passengers



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for a given timetable. The expected average travel time plus scheduling cost was obtained by evaluating the area under the “roofs”.

Fig. 2 Total travel cost (in minutes) for different preferred arrival times

Several more recent publications describe comprehensive rail demand models that also include the Rooftops method for scheduling decisions, for instance Prior et al. (2011), Douglas et al. (2011) and Langdon and McPherson (2011). We have been informed informally that the Rooftops model is still in use today among train operating companies in the UK, and in Germany, Switzerland and the Netherlands the same Rooftops model approach is used to allocate passengers to specific trains. To conclude on the Rooftops model, this approach was established some 40 years ago to represent the scheduling decisions by train users: the choice of which train to use and the derivation of the average scheduling and travel cost to the passengers for a given timetable. The model assumes that:  all rail travellers are perfectly informed about the train time schedules;  all rail travellers choose the train that has the lowest total cost (cost associated with travel time plus scheduling) given their preferred arrival (or departure) time;  cost for travelling earlier and for travelling later are valued the same (and equal to travel time);  preferred arrival times are distributed uniformly over the period being considered. There is clearly a lot of similarity between Fosgerau’s approach and the Rooftops model. But both approaches have come from slightly different backgrounds (transport economics versus transport planning), and the models were originally developed for slightly different purposes (appraisal of user benefits versus determining train loads for railway operators). 2.3

More general choice-of-service models

The original Rooftops model is essentially a deterministic model of train choice: travellers are assumed to have perfect information about the timetables, and to choose exactly that train service that minimises their scheduling costs. A train that has only slightly lower cost takes all patronage, but a slightly dearer one captures none. In real life this is unlikely to happen. The passenger choice process is likely to be subject to differences in preferences and some

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error, due to imperfect information about the timetables and to random influences other than pure scheduling cost. And when one train is only marginally less costly to the average passenger than another, its market share should be only marginally higher. Douglas et al. (2011) therefore introduce an error term into the Rooftops model to make it probabilistic. They use the well-known logit model framework, adding a scaling parameter to reflect an amount of randomness in the train choices of the travellers. The size of this parameter has to be calibrated from actual train choice behaviour. Douglas et al. (2011) make some further extensions to the Rooftops model:  they replace the use of generalised time by utility, thus allowing the influence of fare differences also to be reflected in the attractiveness of the train services;  they vary the desired arrival time profile over different time bands, to reflect the well-known reality that passenger demand tends to be much higher at specific (peak) periods than at other (off-peak) periods of the day;  they calibrate the coefficients of their logit model (, ,  and fare and scale parameters) using data obtained through a Stated Preference experiment, where rail passengers have been asked to choose between different train services while trading-off earlier or later departure time against travel time and fare. With these extensions, a powerful tool is obtained to appraise the consumer value of changes in timetables, in first instance for individual travellers but potentially also for entire travelling populations after aggregation of the individual values. 2.4

Methods to aggregate individual values to population totals

Of course, for forecasting, appraisal and general planning, it is not of interest to know what any individual traveller will do, but to be able to forecast behaviour and appraise benefit for a population. For this we need to calculate overall averages, but there is an important issue here. The paper by Daly (1999) proposes a logit model of choice of competing rail services which takes account of scheduling, train speed and the possible need to change trains for some services. Fare is not included, but would not be difficult to add. Passengers are segmented into three groups:  those that do not know the timetable;  those that know the timetable and have a preferred departure time;  those that know the timetable and have a preferred arrival time. The distinction between the first and third groups parallels that made by Fosgerau (2009), but the second group, e.g. commuters leaving work at a fixed time, are not considered in his model. However, it is difficult to distinguish them from the first group. A central point that is brought out in the Appendix of Daly (1999) is that there are two separate aggregation functions by which we should calculate the average utility (negative cost) of travelling, taken over the set of available services:  when the process by which travellers come to use a particular service can be described as a choice, the utility needs to be calculated as the maximum for that particular traveller, given his or her preferences;  when the process has more the characteristic that the traveller has to accept what the system is offering, the utility needs to be calculated as an average in the usual sense. It is clear that the first aggregation gives a better (higher) utility than the second, because it takes account of differences in individual preferences between travellers. This expected maximum utility would be represented by a logsum formula for a logit model. Daly (1999) gives the difference between these two measures, which is related to the diversity of the choices available. Specifically, when the services available are seen as very different by the travellers, for example because they differ in cost, quality or speed, then the logsum formula allows for the possibility that different travellers will prefer different aspects of the service and the true benefit will therefore be



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greater than indicated by a simple average, which would effectively assume all the passengers had the same preferences. Daly (1999) gives the specific formula for the difference between the logsum and the average, which depends on the variance of utility differences across the alternative services and which can be called entropy. In the following section we describe how we recommend taking the insights from the literature described to propose a more complete model. 3. The proposed model approach 3.1 General Fosgerau’s approach, the Rooftops model and its extension by Douglas et al. (2011) provide us with a useful starting point, but they have some limitations:  Fosgerau distinguishes between planning and non-planning travellers, but he does not indicate who is planning and who is not;  Fosgerau assumes a uniform distribution of headways;  Fosgerau and the Rooftops model assume a deterministic choice between only two subsequent trains;  Fosgerau and Rooftops assume homogeneous preferences among the travellers (a single “representative passenger”);  Rooftops and its extension by Douglas et al. (2011) assume that all travellers are planning (and have perfect information). The model we propose addresses these limitations: it explicitly models who is planning and who is not, it accommodates any form of timetable, it simulates probabilistic choices among all available trains, and it allows for heterogeneous preferences between (groups of) different travellers. 3.2 Our basic approach We propose a modelling approach in two integrated steps. First, passengers are modelled as choosing whether to plan their journey or not. Second, passengers who plan will then choose when to start their journey as a function of the services available; passengers who do not plan or who cannot plan (because their departure times are constrained) will start their journey without reference to the services. This framework is illustrated in Figure 3.

Fig. 3: Structure of our proposed model

To set up a comprehensive framework for these two steps in which we can integrate the various issues that have been raised in the literature and in our own work, we can set up a choice model for travellers choosing between the available services. In this model, each service would have a specified utility, defined by the following function,

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giving the utility to be obtained from choosing a service alternative: 𝑈𝑈(𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝, 𝑃𝑃𝑃𝑃𝑃𝑃) = 𝛼𝛼. 𝐼𝐼𝐼𝐼𝐼𝐼 + 𝛽𝛽. 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 + 𝛾𝛾. 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 + 𝛿𝛿. 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 + 𝜁𝜁. 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 + 𝛼𝛼.

𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 + 𝜅𝜅. 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 + 𝜀𝜀 𝜈𝜈

where 𝐼𝐼𝐼𝐼𝐼𝐼 is the travel time; this variable could be extended to represent different levels of comfort in different trains and/or to represent the disutility of crowding; 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 is the time by which the traveller has to leave before the preferred time; 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 is the time which the traveller arrives after the preferred time; 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 is a penalty incurred if the traveller arrives after a specified time (often, it is assumed that 𝛿𝛿 = 0); 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 indicates whether the traveller has incurred a planning cost; 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 is the fare to be paid, which is converted into units of travel time by dividing by the value of time 3; 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 is the penalty incurred at interchanges; typically this would involve a fixed penalty per interchange, perhaps depending on the circumstances (e.g. cross-platform, scheduled, etc.) and might include walking and waiting time at interchange points; 𝜀𝜀 is a random variable, which we shall take as independent and identically distributed Gumbel (extreme value type 1), which makes the model multinomial logit; 𝛼𝛼, 𝛽𝛽, 𝛾𝛾, 𝛿𝛿, 𝜁𝜁, 𝜈𝜈, 𝜅𝜅 are parameters which need to be supplied, e.g. by statistical estimation from data.

This function can be simplified for discussion purposes by defining a variable 𝐺𝐺𝐺𝐺𝐺𝐺 (generalised journey time) which includes travel time and its modifications by comfort and crowding as mentioned above, then adding in the fare and interchange components, so that we then obtain the simplified function 𝑈𝑈(𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝, 𝑃𝑃𝑃𝑃𝑃𝑃) = {𝛼𝛼. 𝐺𝐺𝐺𝐺𝐺𝐺 + 𝛽𝛽. 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 + 𝛾𝛾. 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 + 𝛿𝛿. 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖} + 𝜁𝜁. 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 + 𝜀𝜀

Note here that the part of the utility function enclosed by the {… } brackets is exactly what is considered by the Rooftops model. The 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 variable indicates the difference between passengers who plan their journey to fit with the schedule and those who do not or cannot. The implications of this difference are discussed in the following section. We note also that the use of this extended utility function implies that non-planning passengers may not take the next service in a literal sense but will choose the service that offers the highest utility (lowest 𝐺𝐺𝐺𝐺𝐺𝐺) to reach their destination, given the time they arrive at the departure station and assuming they then have full information about the next services. Most simply, the next train may be overtaken by a faster later train, but later trains may also be chosen because they are cheaper or involve fewer interchanges etc.. 3.3 The implications of planning The decision to plan will depend on:  the cost 𝜁𝜁 of planning, which itself depends on information provision by the operator and the experience of the passenger and for instance the memorability of the timetable 4;  the benefit that can be achieved by planning, which in turn depends on:

3

Of course, the fare may depend on the traveller type, and the value of time may vary within the population (like the other parameters), but we ignore these complications here. 4 An important reform in Switzerland has been to emphasise ‘clock-face’ timetables that are easy for passengers to remember. In the UK, large fare differences between tickets bought on-line for specific services and those bought on arrival at the station mean that passengers are almost forced to plan.



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the headway of the service, which determines how much can be gained by avoiding excess waiting time, and the reliability of the operation, which affects the ‘buffer time’ which a passenger has to allow waiting at the stop (rather than at the ultimate origin).

Passengers who choose to plan will be those who have high values of the delay parameters (𝛽𝛽, 𝛾𝛾) and/or low values of the cost (𝜁𝜁) of planning, for example because they use the service regularly or have convenient access to online information while they are travelling. Passengers who choose not to plan will have low values of the delay parameters and high values of the cost of planning, for example because they have no easy access to public transport planning tools or are uncomfortable or slow in using these. The utility components (𝐺𝐺𝐺𝐺𝐺𝐺 + 𝛽𝛽. 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 + 𝛾𝛾. 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙) are the basis for the Rooftops approach. The most general model would represent variation of 𝛽𝛽 and 𝛾𝛾 within the population, along with variation of 𝜁𝜁 (and possibly of the components of 𝐺𝐺𝐺𝐺𝐺𝐺, particularly of the value of time). This could for instance be done using an error components logit or mixed-logit model. However, in a simplified model all these variations over the population could be represented in 𝜀𝜀. The implications of planning are therefore, for the two groups of travellers:  non-planning users will simply arrive at the station and will take the next train; assuming their arrivals at the station are uniformly distributed over the period being studied, their average cost will be that of the ‘next’ (i.e. best) train at the moment they arrive at the departure station, plus the waiting time; this is not a choice process, so that a simple average is the appropriate calculation for aggregation;  users who know the schedule and who have a preferred arrival time or a constrained departure time will be able to choose their preferred service; in this case instead of averaging we must calculate the expected minimum user cost, i.e. for a logit model we calculate a logsum rather than an average for aggregation. The consequences of these conclusions for forecasting and appraisal are discussed in the following section. 3.4 Aggregate forecasting To make aggregate forecasts of the choice of service and the user benefit it is necessary to calculate the average for the whole population of the probability that the utility of each service is the best and then the overall average utility, conditional on the choice. Because of the different behaviour with respect to planning of the two population segments described in the previous section, the issue of planning must be dealt with first. The benefit of planning can be seen from the difference between the average cost incurred by the first group (non-planners) and the expected minimum cost or logsum experienced by the second group (planners). The cost of planning, which we have labelled as 𝜁𝜁, will vary in the population as we have indicated. Thus a sub-model of the decision to plan can be developed from suitable data and with input from the sub-model of choice of service. The model of the choice whether to plan thus has three components:  the cost of planning, which could depend on the nature of the system and the provision of timetable information, as well as perhaps on personal characteristics of each passenger;  the cost of travelling including waiting when not planning; and  the cost of travelling (without waiting but potentially with a buffer time) when planning. Thus the costs of travelling for planning and non-planning passengers must be extracted from the models of choice of service for those two conditions. A detailed specification of the model is given in Appendix 1. One way of applying these models would be to specify a distribution over the population of PAT and of the behavioural parameters (𝛼𝛼, 𝛽𝛽, 𝛾𝛾, 𝛿𝛿, 𝜁𝜁, 𝜈𝜈, 𝜅𝜅). Then, drawing a sample from this distribution to represent an individual, we would be able to determine whether the individual would plan and, whether planning or not, which service he or she would choose. This approach, suggested for example by Daly (1999), would offer a very complete treatment of

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the issues, but might prove difficult in practice. A slightly simplified approach would be to specify segments within the population for which representative values of (𝛼𝛼, 𝛽𝛽, 𝛾𝛾, 𝛿𝛿, 𝜁𝜁, 𝜈𝜈, 𝜅𝜅) could be specified and use these for each PAT to determine the choices that would be made. An approach with further simplification is to assume that all of the variation in the population can be summarised in the random component 𝜀𝜀 in the utility function, which (again for simplicity) is assumed to follow an extreme value distribution. This would imply that non-planning passengers are assigned to the next service, with the corresponding (non-random) utility, but that planning passengers can choose among all of the services available to them, obtaining a logsum utility (which is better than obtained by the non-planners). These two utilities are then input to the choice of whether to plan. In this model the parameters (𝛼𝛼, 𝛽𝛽, 𝛾𝛾, 𝛿𝛿, 𝜁𝜁, 𝜈𝜈, 𝜅𝜅) are assumed to be fixed. Values for the parameters need to be provided. Some information is available from published literature but ideally local estimates should be made from revealed preference or stated preference data. In the example that follows we have used illustrative ‘consensus’ values that we have taken from existing literature and own experience. 4. An example application Having specified our proposed model approach, we now show results of an example application. We do this for three different types of passengers, and two different timetables. The different passenger types serve to illustrate the need to account for heterogeneity within the travelling population, and the two timetables will allow us to evaluate the costs of a timetable change, which was our ultimate objective. But before we can show the results we need to specify the parameters of our model, for the different passenger types, and the different timetables:  Passenger type 1 is arrival time constrained and has no difficulty with planning: 𝛽𝛽 = −0.1, 𝛾𝛾 = −0.4, 𝜁𝜁 = 5 minutes.  Passenger type 2 is departure time constrained and has no difficulty with planning: 𝛽𝛽 = −0.4, 𝛾𝛾 = −0.1, 𝜁𝜁 = 5 minutes.  Passenger type 3 is neither arrival nor departure time constrained but does have serious difficulty with planning: 𝛽𝛽 = −0.2, 𝛾𝛾 = −0.2, 𝜁𝜁 = 20 minutes. The specification of the other model parameters that we have assumed is given in Appendix 2. The behaviour of passengers with these parameter values is simulated for two timetables:  Timetable “a” is a service with 4 departures per hour, at 0, 10, 30 and 40 minutes past the hour, and we have assumed an average generalised journey time (GJT) of 100 minutes for each service.  Timetable “b” is a service with 2 departures per hour, at 0 and 30 minutes past the hour, and we have assumed the same average generalised journey time (GJT) of 100 minutes for each service.

Thus, comparing passenger costs of timetable b with those for timetable a allows us to evaluate the impact of a frequency reduction. The results obtained with the different methods are given in Table 1. For each combination of timetable and passenger type we give the following information:  The percentages of passengers planning and not planning, according to our model;  The total cost of travel including the scheduling cost, according to our model (in equivalent minutes IVT). Note that this is the cost for the entire population, taking planning and non-planning passengers together;  The total cost of travel including the scheduling cost, according to Fosgerau (in equivalent minutes IVT). Note that this approach does not give the percentage of passengers who are planning. Note also that we have slightly modified his scheduling cost calculation to account for the uneven intervals in Timetable a 5; 5

We have replaced h by h2/h



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

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The total cost of travel including the scheduling cost, according to the Rooftops model (in equivalent minutes IVT). Note that the Rooftops model assumes that all passengers are planning. Table 1: Results of our model compared with those of Fosgerau and the Rooftops model. % Planning Our model

% Non-plan Our model

Passenger type 1 arr. Constrained

82%

18%

Passenger type 2 dep. Constrained

80%

Passenger type 3 slow plan

Segment

Cost (min) Our model

Cost (min) Fosgerau6

Cost (min) Rooftops7

-Plan -Non-Pl -All

106.18 115.67 104.91

103.33 119.00 NA

103.33 NA 103.33

20%

-Plan -Non-Pl -All

106.71 115.67 105.32

103.33 119.00 NA

103.33 NA 103.33

26%

74%

-Plan -Non-Pl -All

122.55 115.67 113.80

104.17 119.83 NA

104.17 NA 104.17

Passenger type 1 arr. Constrained

95%

5%

-Plan -Non-Pl -All

110.26 129.00 109.92

106.00 135.00 NA

106.00 NA 106.00

Passenger type 2 dep. Constrained

95%

5%

-Plan -Non-Pl -All

110.36 129.00 110.02

106.00 135.00 NA

106.00 NA 106.00

Passenger type 3 slow plan

58%

42%

-Plan -Non-Pl -All

126.80 129.00 123.23

107.50 136.50 NA

107.50 NA 107.50

Timetable a:

Timetable b:

We can observe that 80% to 95% of passengers of types 1 and 2 are planning, while 42% to 74% of passengers of type 3 are not planning; this is in line with what could be expected. For timetable b, with two services per hour, we find higher percentages passengers planning than for timetable a, with four services per hour, again as expected. Some further observations:  Our model is the only one that gives results for the entire population while taking account of both planning and non-planning passengers;  The costs according to Fosgerau and Rooftops are identical for planning passengers; this illustrates that both approaches are essentially the same for these passengers and these timetables;  In our model the cost for planning passengers for passenger type 3 is much higher than for Fosgerau and Rooftops; that is because Fosgerau and Rooftops do not include planning cost here;  In our model the non-planning passenger cost is lower than for Fosgerau; that is because our model does not include scheduling cost for non-planning passengers;  In our model the cost for All passengers is lower than the cost for both planning and non-Planning; this is due to the fact that we use the logsum of planning choice as the measure of expected maximum utility, thus allowing each passenger to choose or not choose planning as they prefer. 6

Separately for Planning (Plan) and Non-planning (N-Pl) passengers, but not for entire population (All) because shares are not specified in Fosgerau. 7 The Rooftops approach assumes that all passengers are planning.

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We now look at how the different methods lead to different conclusions with respect to the impact of the changed timetable, which is what matters in a cost benefit analysis. Table 2 gives the results for the three methods. Table 2: Change in cost due to timetable change according to different approaches (assuming a Value of Time of 12 euro/hour). Change in cost Our model (euro/trip)

Change in cost Fosgerau (euro/trip)

Change in cost Rooftops (euro/trip)

Passenger type 1 arr. Constrained

+1.00

NA

+0.53

Passenger type 2 dep. Constrained

+0.94

NA

+0.53

Passenger type 3 slow plan

+1.89

NA

+0.67

It is clear that our approach produces substantially higher estimated increases in costs as a consequence of the frequency reduction than Rooftops for passenger types 1 (arrival time constrained) and 2 (departure time constrained), and for passenger type 3 (people who have difficulty with planning) in particular. This is due to the assumption in the Rooftops approach that all passengers are planning at zero planning cost; the frequency change impacts non-planners much more strongly than planners. The great difference in estimated costs can have a very important effect on the conclusions of an appraisal study! No results have been given for Fosgerau, as it is unclear what the weights of planning and non-planning passengers should be in determining the population total cost. Conclusions In this paper we have reviewed the possibilities to model the different impacts of timetable changes for individual travellers, and proposed an alternative approach. We have shown that the “half headway” model is not realistic, as it ignores the important scheduling issues. Fosgerau (2009) does not ignore these, and provides a thorough analysis of the cost of headway, while distinguishing between planning and non-planning travellers. But he does not say who is planning and who is not, and his analysis assumes equal intervals between subsequent trains and a deterministic model of train choice. Further, apart from the distinction between planners and non-planners, he does not allow for heterogeneity in the travellers’ preferences. The Rooftops approach, developed many years ago in the railway sector, provides a pragmatic solution for modelling travellers’ scheduling decisions for real timetables, with uneven intervals and different travel times for different trains allowed. But it assumes that everybody plans, and like Fosgerau it uses a deterministic choice model and ignores heterogeneity in the travellers’ preferences. Douglas et al. (2011) make extensions to the Rooftops model, allowing for randomness and therefore (some) diversity of choice, as well as incorporating fare differences. However, they preserve the assumption that every passenger plans their journey, and they do not allow for variation in the preference parameters of the model. We have proposed our own alternative approach, which takes components from the previous approaches but also adds various elements:  it allows any type of timetable to be analysed, with uneven intervals between different trains, different travel times, different fares, different numbers of interchanges required and so on;



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  

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it explicitly models who is planning and who is not planning, based primarily upon the trade-off between the passengers’ scheduling preferences and the cost of planning; for the planners it models a probabilistic choice among the different departing trains, and determines the expected minimum cost of that choice; that cost is lower than the cost of the single best alternative; it acknowledges that even within the group of planning travellers there may be different preferences for the different scheduling and other choice dimensions, leading to different choices and different costs.

We have illustrated our approach with some examples. We have shown that our approach can produce a substantially higher cost impact of a change in timetable, where the frequency was reduced from 4 trains per hour to 2 trains per hour, than the traditional Rooftops approach. This is a consequence of the use of a probabilistic choice model, with train choice among 5 or 9 alternative services rather than 2, and the explicit modelling of planning choice (planning versus non-planning). The fact that the Rooftops approach ignores non-planning travellers is clearly an omission, particularly for more frequent services. Our approach is much more flexible than Fosgerau and Rooftops in handling different types of timetables and incorporating the behaviour of different types of passengers, but also it is more sensitive in picking up the (dis)benefits of changes in timetables for travellers choosing among a range of possible services. References Abrantes, P.A.L. and Wardman, M.R. (2011): Meta-analysis of UK values of travel time: An update, - Transportation Research Part A: Policy and Practice, Volume 45, Issue 1, January 2011, Pages 1-17. British Railways Board (1986): Passenger Demand Forecasting Handbook, Technical Report, Section B5. Daly, A.J. and Zachary, S. (1977) Bus Passenger Waiting Times in Huddersfield, Local Government OR Unit Report T67. Daly, A.J. (1999) The use of schedule-based assignments in public transport modelling, European Transport Conference, Association for European Transport. Department for Transport (2015): Values of travel time savings and reliability: final reports, part of Transport appraisal and strategic modelling (TASM) research reports, Road network and traffic, and Rail network. Douglas, N.J., Henn, L. and Sloan, K. (2011): Modelling the ability of fare to spread AM peak passenger loads using rooftops, Australian Transport Research Forum, 2011 Proceedings. Fosgerau, M. (2009): The marginal social cost of headway for a scheduled service, Transportation Research Part B, Vol. 43, pp. 813-820. Fosgerau, M. and Karlström, A. (2010): The value of reliability, Transportation Research Part B, Vol. 44, pp. 38-49. Fosgerau, M. and Engelson, L. (2011): The value of travel time variance, Transportation Research Part B, Vol. 45, pp. 1-8. Kouwenhoven, M., G.C. de Jong, P. Koster, V.A.C. van den Berg, E.T. Verhoef, J. Bates, P.M.J. Warffemius (2014): New values of time and reliability in passenger transport in The Netherlands, in: Research in Transportation Economics, 47, pp. 37-49. Langdon, N. and McPherson, C. (2011): CLICSIM: Simulation of passenger crowding on trains and at stations, European Transport Conference, Association for European Transport,. Lijesen, M.G. (2014): Optimal Traveller Responses to Stochastic Delays in Public Transport, Transportation Science, Vol. 48, pp. 256-264. Prior, M., Vickers, J. Segal, J. and Quill, J. (2011): Modelling open access train services, European Transport Conference, Association for European Transport,. Tseng, Y-Y., Rietveld, P. and Verhoef, E.T. (2012): Unreliable trains and induced rescheduling: implications for cost-benefit analysis, Transportation, Vol. 39, pp. 387-407. Tyler, J. and Hassard, R. (1973): Gravity/elasticity models for the planning of the inter-urban rail passenger business, Proceedings of Seminar G, PTRC Summer Annual Meeting, University of Sussex.

APPENDIX 1: Mathematical Specification of the Model The probability of planning 𝑝𝑝𝑝𝑝,𝑎𝑎 , given a preferred arrival time 𝑎𝑎, is 𝑝𝑝𝑝𝑝,𝑎𝑎 =

where

exp 𝜃𝜃(𝑉𝑉𝑝𝑝,𝑎𝑎 + 𝑈𝑈𝑝𝑝 )

exp 𝜃𝜃(𝑉𝑉𝑝𝑝,𝑎𝑎 + 𝑈𝑈𝑝𝑝 ) + exp 𝜃𝜃𝑉𝑉𝑛𝑛,𝑎𝑎

0 < 𝜃𝜃 ≤ 1 is the structural parameter relating the choice of whether to plan to service choice;

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𝑉𝑉𝑝𝑝,𝑎𝑎 is the utility of the train journey for time 𝑎𝑎, given there is planning; 𝑈𝑈𝑝𝑝 is the utility of making the effort of planning, i.e. minus the cost of planning; 𝑉𝑉𝑛𝑛,𝑎𝑎 is the utility of the train journey for time 𝑎𝑎, given there is no planning.

The train journey utilities are given in turn by 𝑉𝑉𝑛𝑛,𝑎𝑎 = 𝜆𝜆(𝐺𝐺𝐺𝐺𝐺𝐺(𝑛𝑛𝑎𝑎 ) + 𝑤𝑤(𝑛𝑛𝑎𝑎 )) 𝑉𝑉𝑝𝑝,𝑎𝑎 = log ∑

𝑡𝑡∈𝐶𝐶(𝑎𝑎)

exp(𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆(𝑡𝑡) + 𝛽𝛽. 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒(𝑡𝑡) + 𝛾𝛾. 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙(𝑡𝑡))

where 𝜆𝜆 < 0 is the scale of the service choice model; 𝐺𝐺𝐺𝐺𝐺𝐺(𝑡𝑡) is the generalised journey time for the service at time 𝑡𝑡; 𝑤𝑤(𝑡𝑡) is the waiting time for service 𝑡𝑡; 𝑛𝑛𝑎𝑎 is the ‘next’ service (in the sense explained in the text) for time 𝑎𝑎; 𝐶𝐶(𝑎𝑎) is the set of services that are relevant for time 𝑎𝑎; 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒(𝑡𝑡) and 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙(𝑡𝑡) are the schedule delays for service 𝑡𝑡.

The utility of planning 𝑈𝑈𝑝𝑝 can be expressed as 𝜆𝜆𝜆𝜆, where 𝜁𝜁 is the planning cost expressed in minutes of travel time. The probability of a planner choosing service 𝑡𝑡 ∈ 𝐶𝐶(𝑎𝑎) is then 𝑝𝑝𝑡𝑡,𝑎𝑎 =

exp(𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆(𝑡𝑡) + 𝛽𝛽. 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒(𝑡𝑡) + 𝛾𝛾. 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙(𝑡𝑡)) exp 𝑉𝑉𝑝𝑝,𝑎𝑎

The overall cost of the journey is then, at time 𝑎𝑎,

for planners: 𝑊𝑊𝑝𝑝,𝑎𝑎 =

for non-planners:

𝑉𝑉𝑝𝑝,𝑎𝑎 ⁄ , to which the planning cost 𝜁𝜁 needs to be added 𝜆𝜆

for the population:

𝑊𝑊𝑛𝑛,𝑎𝑎 = 𝑊𝑊𝑎𝑎 =

𝑉𝑉𝑛𝑛,𝑎𝑎 ⁄ 𝜆𝜆

log(exp 𝜃𝜃(𝑉𝑉𝑝𝑝,𝑎𝑎 + 𝜆𝜆𝜆𝜆) + exp 𝜃𝜃𝜃𝜃𝑛𝑛,𝑎𝑎 )⁄ 𝜆𝜆𝜆𝜆

The average cost over the period being modelled can then be calculated from these last formulae by averaging over 𝑎𝑎: 𝑊𝑊𝑝𝑝 = 𝜁𝜁 + ∑𝑎𝑎∈𝐴𝐴 𝑞𝑞𝑎𝑎 𝑊𝑊𝑝𝑝,𝑎𝑎 ,

𝑊𝑊𝑛𝑛 = ∑𝑎𝑎∈𝐴𝐴 𝑞𝑞𝑎𝑎 𝑊𝑊𝑛𝑛,𝑎𝑎 ,

where 𝑞𝑞𝑎𝑎 is the fraction of flow occurring at time 𝑎𝑎 𝐴𝐴 is the period being considered.

𝑊𝑊 = ∑𝑎𝑎∈𝐴𝐴 𝑞𝑞𝑎𝑎 𝑊𝑊𝑎𝑎

A reasonable assumption in many cases is that the flow is evenly distributed over the period and that we can model by making the calculations for each short interval (e.g. minute) 𝑎𝑎. In this case we get 𝑞𝑞𝑎𝑎 = 1⁄𝐴𝐴, where 𝐴𝐴 is now the length of the period, e.g. 60 minutes, and the equations above become:



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𝑊𝑊𝑝𝑝 = 𝜁𝜁 + ∑𝑎𝑎∈𝐴𝐴 𝑊𝑊𝑝𝑝,𝑎𝑎 ⁄𝐴𝐴,

𝑊𝑊𝑛𝑛 = ∑𝑎𝑎∈𝐴𝐴 𝑊𝑊𝑛𝑛,𝑎𝑎 ⁄𝐴𝐴 ,

𝑊𝑊 = ∑𝑎𝑎∈𝐴𝐴 𝑊𝑊𝑎𝑎 ⁄𝐴𝐴

APPENDIX 2: Parameter values used in the example In the example application the following parameters have been used:           

8

scale of planning train model 𝜆𝜆 = -0.2 𝛼𝛼 travel time (min) = -0.2 (after scaling) 8 𝛽𝛽 schedule delay early (min) varies by passenger type: -0.1/-0.2/-0.4 𝛾𝛾 schedule delay late (min) varies by passenger type: -0.1/-0.2/-0.4 𝛿𝛿 penalty for being late = 0 𝜁𝜁 cost of planning varies by passenger type: 5 (types 1 and 2) or 20 minutes (type 3) value of waiting time (min) = -0.4, i.e. twice the value of travel time scale of planning choice model 𝜃𝜃 = 0.8 (MNL) total generalised journey time including vehicle time, fare, interchange = 100 minutes modelled period: 180 minutes results averaged over 60 minutes

Note that 𝛼𝛼 and 𝜆𝜆 cannot be determined separately and we have set them to be equal.

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