A decision support system for timetable adjustments

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A DECISION SUPPORT SYSTEM FOR TIMETABLE
ADJUSTMENTS
João Mendes-Moreira
Dept. of Informatics Engineering, Faculty of Engineering, University of Porto, Portugal
LIAAD-INESC Porto L.A., Portugal
E-mail: jmoreira@fe.up.pt

Eva Duarte
Department of Informatics, School of Engineering, University of Minho, Portugal
E-mail: evamcduarte@gmail.com

Orlando Belo
Department of Informatics, School of Engineering, University of Minho, Portugal
E-mail: obelo@di.uminho.pt


1 INTRODUCTION

For a public transport company, the process of defining trips offers is a central task because
trips are the main product they have to offer to their clients. As in other business areas the
offer should maximize clients’ satisfaction at a minimum cost. Traditionally, timetables were
defined assuming a deterministic travel time. However, with the investments done in the last
decade in Advanced Public Transportation Systems, a large amount of current data obtained
from Automatic Vehicle Location systems is now available. This data can be used to enhance
travel time modelling for timetable definition. This is the subject of this paper, which imposed
us the study of how to use current data in order to better define timetables aiding public
transport companies to accomplish their mission. We present a Decision Support System
(DSS) for the special case of timetable adjustments, assuming therefore that the schedule
under study is not new, i.e., there are actual trips for that schedule.
Firstly, in this paper, we present a brief state of the art review on defining travel times for
timetabling (Sect. 2) then we discuss the reasons for developing a DSS for timetable
adjustments (Sect. 3). In Sect. 4 we describe the DSS, and in Sect. 5 how to use it in typical
situations. We conclude with a discussion and guidelines for future research (Sect. 6).


2 RELATED WORKS ON TRAVEL TIMES FOR TIMETABLING

An important difference between the existing approaches on timetable creation concerns the
variables used. These variables depend on the purpose. If, for instance, a timetable for a new
line is needed, a variable such as the population density of the served area is important [3].
However, if the goal is to make small adjustments to the timetables, this variable is not

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relevant. We describe the variables for the latter case, i.e., for regular planning tasks (line
creation is a sparse event in a
scheduled travel time (STT), slack time (SlT), scheduled headway (
scheduled departure time. Let us assume that SCT is the scheduled cycle time, scheduled departure time. Let us assume that SCT is the scheduled cycle time,scheduled departure time. Let us assume that SCT is the scheduled cycle time,
??? ? ????? ????? ?????
where the indexes g and r represent the go and return trips (Fig.represent the go and return trips (Fig. 1).
Figura 1: Time
For urban areas, the departure times are usually defined by headway instead of
spaced. Irregularly spaced departure times are typically used for long distance trips or trips in
rural areas. We focus on the definition of the timetables' departure times by headway.rural areas. We focus on the definition of the timetables' departure times by headway.rural areas. We focus on the definition of the timetables' departure times by headway.
According to [10],
??? ? ? ? ??.
Fixing N, and assuming that travel times are exponentially distributed, slack times can be
optimized [10]. Using this approach, the shorter the slack time is, the shorter the scheduled
headway is. The objective function used is the passengers' expected waiting time function,headway is. The objective function used is the passengers' expected waiting time function,headway is. The objective function used is the passengers' expected waiting time function,
???? ?
??
?? ?1 ????????
???? ,
where ? represents the delay of the bus and
that passengers arrive uniformly at the bus stops, which is acceptable for short headways.
Headways equal or shorter than 10
literature [10].
However, Zhao et al. [10] argue that by using the function defined in
arrival at the bus stops, it is possible to adapt the solution t
this case,
relevant. We describe the variables for the latter case, i.e., for regular planning tasks (line
creation is a sparse event in a public transport company). Some typical variables are:
scheduled travel time (STT), slack time (SlT), scheduled headway (SH), fleet size (
? ????,

: Time-distance chart on different concepts of travel time.
For urban areas, the departure times are usually defined by headway instead of
spaced. Irregularly spaced departure times are typically used for long distance trips or trips in spaced. Irregularly spaced departure times are typically used for long distance trips or trips in
ng that travel times are exponentially distributed, slack times can be
. Using this approach, the shorter the slack time is, the shorter the scheduled . Using this approach, the shorter the slack time is, the shorter the scheduled
?
represents the delay of the bus and ?????? the variance of ?. This function assumes
that passengers arrive uniformly at the bus stops, which is acceptable for short headways.
Headways equal or shorter than 10 minutes are usually defined as short headways in the minutes are usually defined as short headways in the
argue that by using the function defined in [1]
arrival at the bus stops, it is possible to adapt the solution to problems with large headw
2
relevant. We describe the variables for the latter case, i.e., for regular planning tasks (line
company). Some typical variables are:
), fleet size (N) and
(1)
distance chart on different concepts of travel time.
For urban areas, the departure times are usually defined by headway instead of irregularly
(2)
ng that travel times are exponentially distributed, slack times can be
(3)
. This function assumes
that passengers arrive uniformly at the bus stops, which is acceptable for short headways.
for the passengers'
o problems with large headways. In

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????? ? exp??????/?
exp????????
??
?
, (4)
where ???? ? ????????? is the utility function, ??????? is the expected waiting time of an
arrival at time ? and ? and ? are constants that must be defined from empirical data. The
derivation of the passengers' expected waiting time for large headways has not yet been done.
Using an economic perspective, Carey defines as objective function, the cost expressed in
terms of STT, lateness and earliness unit costs [2]. Using this approach it is possible to define
the optimal STT and SlT for given ratios between the STT unit cost and both the lateness and
earliness unit costs. Another contribution of this work is the inclusion in the model of the
effect of relaxation when the SlT is larger, i.e., it is known that when the schedule is tight, the
actual travel time is shorter than when it is large. Carey calls it the behavioural response.
What Carey shows is that the timetable definition should be neither too tight, to avoid delays
in departures, nor too large, to avoid behavioural inefficiency.
In all the studies on the definition of travel times, it is not explicit how global cost is defined,
i.e., the cost for the passengers and for the company. In Carey's approach, these costs are
implicit in the unit costs, but the author does not explore how to estimate them (it is not the
goal of the paper). The work by Zhao et al. uses just the passengers' cost, i.e., the expected
time the passengers must wait at the bus stop. The operational costs are not considered.
The above mentioned approaches assume that the purpose is to adjust STT and SlT, i.e., the
timetable is already defined (even if roughly) and the goal is just to tune it. However, there are
several studies on methods for the creation of bus timetables, with different purposes. For
instance, Ceder [3] addresses the definition of the frequency related to the problem of the
efficient assignment of trips to running boards (i.e., bus duties). Input variables such as the
population density of the area served by the line, bus capacity and single mean round-trip
time, including slack time, are used [3]. This work was extended in order to address the
synchronization of certain arrivals [4]. In [6] the goal is to minimize total schedule delay costs
for the users. In [7], the goal is to define the bus departure rate as a function of passengers'
arrival rate. In all these works [3,4,6,7], it is assumed that the travel time is deterministic.
The approaches used by Carey and by Zhao et al. benefit from the existence of abundant
archived data from AVL systems, in particular the one by Zhao et al.. This work has the
appeal of being an analytical approach. However, for the schedulers, rather than a method that
solves the (partial) problem in a deterministic way, they need a tool to give them insights into
the best solution, at least while there are no answers to questions such as “what are the
optimal ratios between STT and lateness unit costs and between STT and earliness unit costs
(in Carey's approach)?”, or, “when should the scheduler put on an additional bus, i.e., how
does passengers' waiting time compare with the operational cost of an additional bus?”, or
even, “what is the impact of reducing the SCT on operational costs?”.

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4
3 A DSS FOR TIMETABLE ADJUSTMENTS

In the previous section we presented the limitations of the existing methods for the creation of
bus timetables with regard to the value of travel time. We pointed out that one of the
difficulties is the inherent multi-objective nature of the problem of finding the optimal value,
namely the minimization of both the expected passengers' waiting time at the bus stop and the
operational costs. In this paper we propose a DSS, which allows the person in charge of
timetable planning to assess the impact of different scenarios in both objectives. He or she can
test different values for the scheduled travel time, slack time and headway, and obtain a set of
descriptive statistics that allow this person to evaluate the impact of this scenario using data
from a past period similar to the one that the timetable is going to cover.
The reason for using this approach is that the existing ones can give optimized solutions when
some of the variables are fixed but do not allow the planner to easily evaluate the sensitivity
of the solution to each decision variable. Furthermore, the objective functions used by these
approaches do not simultaneously cover the two objectives above mentioned. This DSS can
be seen as an integrated environment for analysis and is compatible with the use of optimized
solutions like the one described in [10]. In fact, such solutions can always be developed and
integrated in this DSS as a default solution for fixed given values.


4 SUPPORTING DECISION IN TIMETABLE ADJUSTMENTS TASKS

The person in charge of planning (the planner) starts by selecting the data to be used for the
analysis of travel times. It is expected that the planner will choose past data that might be
representative of the period (in the future) that is going to use the new version of the
timetable. The planner is able to choose the characteristics of the analysis, such as period of
time (days), the time of the day and the line/route. There are three types of analysis that
depend on the characteristics of the line and the objectives of the analysis the planner wants to
perform: single direction, double direction and circular route analysis.

4.1 Single direction analysis
In a single direction analysis (Fig. 2) the information provided is:
• A time plot of travel times: it provides visual information about travel times. It allows
the user to observe the dispersion of travel times during the period of analysis. This
can show, for instance, the difference in travel times along the seasons of the year or
days of the week. It is also possible to identify outliers and to obtain information on
the trips by clicking the mouse.

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Figure 2: Single direction analysis.
• A chart of the accumulated relative frequency of travel times: it allows the user to
have an idea of the route performance, just by looking at it. The steeper is the slope of
the plot, the less is the variability of travel times. We can observe that the duration of
the majority of the selected past trips for line 602 is between 50 and 62.5 minutes.
• Information about the sample (‘amostra’ in Portuguese): it characterizes the sample
past data being used in the analysis.
• A table with statistical information on travel times: it provides the user with statistical
information for the 25th, 50th and 75th percentile. For these three travel times, say travel
time ?, one presents the percentage of travel times in the intervals: ? − ∞,? − 5 ????,
?? − 5 ???,? ? 5 ????, ?? ? 5 ???,? ? 10 min ?, ?? ? 10 ???,?∞?. The value of ?
that maximizes the percentage of travel times in the interval ?? − 5???,? ? 5????
is also calculated and presented in the table. This is the initial information provided in
the table. The percentiles and the intervals being used were chosen by the planners we
have been working with. This information complements the plot of the accumulated
relative frequency of travel times. However, the user may add lines to the table by
choosing the values of percentile or duration he wants to analyze. The information
provided in this table is very useful for the transit planners because it helps them to
estimate the effects on delays and early arrivals, when choosing the duration for a trip.