A: DTU Management Engineering B: Rapidis APS
*Corresponding author: Morten Eltved, Stud. MSc. Transport & Logistics, DTU Management Engineering, Ph: 30702705, E: MortenEltved@hotmail.com
Joint modeling of schedule- and frequency-based services in public transport assignment models
Danish: Integreret modellering af frekvens- og køreplansbaserede rutevalg i kollektiv transport Morten EltvedA*, Otto Anker NielsenA, Thomas Kjær RasmussenA, Rasmus Dyhr FrederiksenB
Background
Public transport assignment models are traditionally either following a schedule-based or frequency- based approach. This paper suggests a new framework where both frequency-based and schedule- based services are included. It has not been possible to find any literature describing such a unified model and the authors therefore consider this work as the first of its kind in public transport assignment models.
The background for considering this problem is twofold. The public transport system in the capital region of Denmark is a combination of both frequency-based lines (i.e. A-busses and Metro) and schedule-based lines (i.e. trains and bus lines with low frequencies). It would therefore be more behaviorally realistic to include the frequency-based lines as frequency-based services in the model (i.e. that the passengers only know the frequency, but not the exact time of departure).
The second problem arises when the analyst makes analyses of future scenarios with changes in the network. If the entire timetable for intercity trains in Denmark is changed, then a large effort need to be made to ensure that the corresponding bus-services are still coordinated well. If the corresponding busses were frequency-based this problem would not arise, as the model could (approximately) deal with low-frequency lines without need for coding new time-tables for them.
Framework and used methods
The general framework of the model is based on a choice set generation phase and a subsequent application of a discrete choice model. For each origin, destination and desired departure time, different alternative paths are identified, and given the utility of these alternatives, a path size correction logit model is applied to find the distribution of traffic over the different alternatives.
To create a choice set for each desired departure time, the event dominance principle in Florian (2004) is used to dynamically unfold the graph. This gives more flexibility to decide which events to include, but the idea in our work is also to allow “non-optimal” events to be included within a certain threshold. This means, that there is not a strict event dominance implemented, but the event dominance is relaxed, so that “dominated” events are still kept in the heap. In this way, not only the shortest path is found, but also paths with a slightly higher utility are included in the choice set. The choice set then consists of space-time paths through different routes in the space dimension. Different heuristics can be applied to ensure that for example no routes with loops are included.
One of the main challenges in a joint frequency- and schedule-based model is to identify the transfer times between frequency-based and schedule-based services and vice versa. There are four possible
transfer types, which are shown in Table 1. The modeling is closely related to finding a consistent method to solve especially the case where a transfer is made from a frequency-based line to a schedule-based line.
Transfer from/to A) Frequency-based line
C) Timetable-based line
A) Frequency-based
line Transfer time is
following a distribution
There is a probability to catch the line, and for this a distribution of transfer times C) Timetable-based
line
Transfer time is following a distribution
Transfer time is deterministic
Table 1 - Possible transfer types between frequency- and schedule-based services
For the paths in the choice set for a specific departure time choice, the path size correction (PSC) factor as described in Prato (2009), is used to find the overlap correction factor based on the length of the overlap on different line segments. The choice model is a standard logit model (with path size correction) as shown below, where 𝑃𝑘 defines the probability of taking path k among the set of paths in the choice set:
𝑃𝑆𝐶𝑘 = − ∑ (
𝑎∈𝑇𝑘
𝐿𝑎
𝐿𝑘ln∑ 𝛿𝑎𝑙
𝑙∈𝐶
)
𝑃𝑘 = exp(𝑉𝑘+ βPSC∗ 𝑃𝑆𝐶𝑘)
∑𝑙∈𝐶exp(𝑉𝑙+ βPSC∗ 𝑃𝑆𝐶𝑙)
Preliminary results
As the work is still ongoing, only a limited number of tests on small example networks have been performed. One of the examples is described below including some initial results.
The first test is performed in a network with four main alternatives using the relation DTU to Copenhagen airport as the example. In terms of route choice, there are two main decisions in this small example:
1. From DTU to Nørreport
a. Taking bus 150S or 15E directly from DTU to Nørreport (respectively frequency- and schedule-based)
b. Or taking a bus to Lyngby st. and then take the S-train to Nørreport st. (all schedule- based)
2. From Nørreport to CPH airport
a. Taking the Metro (frequency-based)
b. Or taking the regional train (schedule-based)
The overall probabilities of taking each alternative are based on having a launch each minute of an hour, and the average of the probabilities for each launch give the overall probabilities of taking either of the alternatives. For the frequency-based services a waiting time of ½ headway is assumed. Three different ways of generating the choice set has been tried in relation to the choice between 150S and 15E, where the last two ways take the common line problem into consideration:
1. The passenger chooses between 150S and 15E and these are therefore separate alternatives.
2. The passenger is assumed to know the arrival time for both 150S and 15E to Nørreport, and therefore takes the service arriving first to Nørreport.
3. The passenger is not aware of the timetable for 150S and therefore takes the first departing service. There is therefore a probability of taking 150S given the waiting time for 15E and for this a set waiting time. The rest of the probability is assigned to taking 15E.
It is assumed that the passenger has decided whether to take the Metro or the regional train prior to starting the trip. The results of the different choice set generation techniques are shown in the table below. The alternatives including the Metro are, as expected, strongly preferred compared to taking the regional train. The split between taking 150S/15E or travel via Lyngby st. is highly dependent on how the alternatives are generated, and the modeling should therefore closely consider these challenges.
Tech- nique
150S ->
Metro
15E ->
Metro
150S ->
Reg. train
15E ->
Reg. train
300S ->
S-train ->
Metro
300S ->
S-train ->
Reg. train
1 30% 36% 4% 5% 22% 3%
2 46% 7% 41% 6%
3 44% 10% 40% 6%
Table 2 - Distribution of passengers between the alternatives with different choice sets
Proposed category
This paper regards both transport modeling and public transport, but as the work primarily concerns public transport assignment, it is suggested that the paper is put in the category “Transport models and their applications”.
References
Florian, M., 2004. Finding Shortest Time-dependent Paths in Schedule-based Transit Networks: A Label Setting Algorithm, in: Wilson, N., Nuzzolo, A. (Eds.), Schedule-Based Dynamic Transit Modeling: Theory and Applications. Kluwer Academic Publishers, pp. 43–52.
Prato, C.G., 2009. Route choice modeling: past, present and future research directions. J. Choice Model. 2, 65–100. doi:10.1016/S1755-5345(13)70005-8
