Practical modelling of trip re-scheduling under congested conditions

Transportation Research Part A 41 (2007) 788–801

www.elsevier.com/locate/tra

Practical modelling of trip re-scheduling undercongested conditions q

John Bates

Transport Planning Consultant, 1 Picklers Hill, Abingdon, OX14 2BA, UK

Received 13 January 2006; received in revised form 1 September 2006; accepted 9 November 2006

Abstract

There is plenty of evidence that drivers may make small changes in their time of travel to take advantage of lower levelsof congestion. However, progress in the practical modelling of such ‘‘micro’’ re-scheduling within peak period trafficremains slow. While there exist research papers describing theoretical solutions, techniques for practical use are not gen-erally available. Most commonly used assignment programs are temporally aggregate, while packages which do allowsome ‘‘dynamic assignment’’ typically assume a fixed demand profile.

The aim of the paper is to present a more heuristic method which could at least be used on an interim basis. Theassumption is that the demand profile can be segmented into a number of mutually exclusive ‘‘windows’’ in relation tothe ‘‘preferred arrival time’’, while on the assignment side, independently defined sequential ‘‘timeslices’’ are used in orderto respect some of the dynamic processes relating to the build-up of queues. The demand process, whereby some driversshift away from their preferred window, leads to an iterative procedure with the aim of achieving reasonable convergence.

Using the well-known scheduling theory developed by Vickrey, Small, and Arnott, de Palma & Lindsey, the basicapproach can be described, extending from the simple ‘‘bottleneck’’, to which the theory was originally applied, to a gen-eral network. So far, insufficient research funds have been made available to test the approach properly. It is hoped that bybringing the ideas into the public domain, further research into this area may be stimulated.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Re-scheduling; Congestion; Practical

1. Introduction

Highway congestion is increasingly common, and is set to increase substantially in the future unless appro-priate measures are taken. It is essentially a dynamic phenomenon, and has been extensively dealt with accord-ing to dynamic modelling principles in a number of academic contributions (see, for example, Ran and Boyce,1996; Han and Heydecker, 2006). However, given its importance for transport policy, the practical tools for

0965-8564/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.tra.2006.11.010

q Paper presented at 45th European Congress of the Regional Science Association, August 23–27, 2005, Amsterdam, special session:‘‘Choice analysis (N5)’’.

E-mail address: [email protected]

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J. Bates / Transportation Research Part A 41 (2007) 788–801 789

modelling congestion over reasonably sized networks remain rudimentary. While microsimulation models oftraffic are being increasingly used for highway assignment, they are not generally well integrated into overallmodels which can be used as planning tools. For practical purposes, such models need to reflect both demandand supply effects, and, while its validity may be disputed, a convenient working assumption is that the twomust be in equilibrium. In other words, the generalised cost that gives rise to a given pattern of demand mustbe compatible with the network’s ability (largely related to highway congestion) to accommodate the demandat that cost.

While much of practical transport modelling remains dominated by networks, often with assumptions offixed demand at any given time, there is now a general acceptance of the fact that the (generalised) costs of travelaffect the level of demand, whether these costs derive from deliberate policy (e.g., road pricing) or from capacityeffects. In the UK and many other countries, this has led to a revival of interest in the much criticised ‘‘fourstage’’ transportation model (see, for example, Bates, 2000). In this context, the need to recognise the differenttravel demands at different times of the day is now well understood. However, even for the most congested peri-ods, the standard approach is essentially ‘‘time-aggregate’’, in that the demand for highway travel for a substan-tial time period (usually for 2 or 3 h) is effectively factored to give the demand for a single hour (with practicevarying between average and ‘‘peak’’) and then ‘‘loaded’’ on to a network, making use of standard congestednetwork algorithms (equilibrium assignment), without any further temporal differentiation. The aim of thispaper is to assist in the practical specification of procedures which can make improvements in this respect.

Prompted initially by the findings of the SACTRA (1994) report, there has been ongoing effort in the UK todevelop reasonably standardised modelling approaches which allow for demand responses in a way that isaccessible to practical modellers. There has been general agreement about the need to reflect the responsesof frequency, distribution (or destination choice) and mode choice within the demand model, as well as routechoice (though this is handled routinely within the assignment module). It has proved more difficult to makeprogress on the choice of time of travel, but here too some general guidelines are being achieved.

One of the key issues of time of travel choice is the level at which it may be considered. Following a clas-sification originally due to Bates (1996), it has become common to make a distinction between ‘‘macro’’ and‘‘micro’’ shifts, where ‘‘macro time-shifting’’ allows for the possibility of transferring between defined broadperiods (typically 2 or 3 h), as between peak and interpeak, and ‘‘micro time-shifting’’ may be defined as rel-atively small changes (of the order of minutes rather than hours) in arrival and/or departure time. Generally,micro-shifts are motivated by changes in the temporal profile of journey times – i.e., the variation in road jour-ney time between a fixed origin and destination, dependent on the exact departure time. If travellers have apreferred arrival (or departure) time, they will only shift from this if they receive some benefit in the formof reduced travel times. Typically, this will have implications for the arrival time, which, at least in the morn-ing peak, is likely to be critical.

Such variation in travel times is predominantly a manifestation of the build-up and dispersal of queues atvarious points in the network. Because this is essentially a dynamic process, it is not possible to represent itusing standard ‘‘time-aggregate’’ assignment methods – some account of the development and dispersal ofqueues is required. Ideally, this should be done using a fully dynamic assignment in continuous time.

There is plenty of evidence that drivers may make small changes in their time of travel to take advantage oflower levels of congestion. Of particular interest is the possible change in behaviour when additional capacityis introduced (see, for example, the work by Kroes et al. (1996) in connection with the completion of theAmsterdam Ring Road, and further examples cited by Small (1992a)), where the phenomenon of the ‘‘returnto the peak’’ (Antonisse et al., 1987) can be observed. However, progress in the practical modelling of such‘‘micro’’ re-scheduling within peak period traffic remains slow, despite theoretical research. As noted, mostcommonly used assignment programs are temporally aggregate, while packages which do allow some‘‘dynamic assignment’’ typically assume a fixed demand profile over the period being modelled.

At the same time, the main demand responses noted earlier (destination choice, etc.) are also typically mod-elled on a time-aggregate basis, even if broad distinctions (e.g. between peak and off-peak) are reflected. Thusit is important that the costs on which these choices are based are correctly deduced from the network con-ditions. One reasonable approach is to calculate a flow-weighted average of the costs over the profile of arrivaltimes, having achieved adequate convergence of the allocation of peak period demand. In this way it is alsopossible to reflect the scheduling costs, along the lines discussed by Small (1992).

790 J. Bates / Transportation Research Part A 41 (2007) 788–801

There are thus two essential questions. Firstly, how can a better understanding of the build-up of conges-tion within the peak be obtained, using reasonably accessible models? Secondly, how can we improve on thepotential errors imported into the overall demand-supply modelling process by not taking account of the pro-file within congested periods?

The aim of the paper is to present a more heuristic method which could at least be used on an interim basis.Currently this remains untested. It is hoped that by bringing the ideas into the public domain, further researchinto this area may be stimulated.

The key assumption is that the demand profile can be divided into a number of mutually exclusive ‘‘win-dows’’ in relation to the ‘‘preferred arrival time’’, while on the assignment side, independently defined sequen-tial ‘‘timeslices’’ are used in order to respect some of the dynamic processes relating to the build-up of queues.The demand process, whereby some drivers shift away from their preferred window, leads to an iterative pro-cedure with the aim of achieving reasonable convergence. In passing, we may note that there are some difficultdecisions to make as to what it means to say that demand is allocated to any particular time period – do wemean that it starts within that period?

2. Preliminaries

2.1. Theoretical background

The key starting point is the ‘‘schedule delay’’ formula, initially developed by Vickrey (1969), and furtherextended in the work of Small (1982, 1992b), and a series of papers by Arnott, de Palma & Lindsey (ADL: e.g.ADL (1994)). A major review is given in Bates (1996), available on the UK Department for Transport website(www.dft.gov.uk). However, the key aspects will be briefly presented here.

In the earliest expositions, all travellers wish to arrive at the same ‘‘preferred arrival time’’ (PAT), and thesystem is treated as having a single O–D pair. As long as the capacity of the network is sufficient, all travellerscan arrive at PAT. However, once capacity problems occur, this is no longer possible, and some travellers willbe early or late.

The schedule delay formula is a functional form for the utility of arriving at times other than the PAT, tak-ing account of the possible time advantages of so doing. If we denote s as the actual arrival time, and n(s) asthe travel time for those who arrive at s, then by far the most popular proposal for this utility is that due toSmall (1982), a development of Vickrey (1969), whereby

UðsÞ ¼ �anðsÞ � b SDE� c SDL� ddL ð1Þ

where all four terms a, b, c, d are positive, and the terms SDE (‘‘Schedule Delay Early’’), SDL (‘‘ScheduleDelay Late’’), and dL (‘‘dummy (0,1) for late arrival’’) are defined as

SDE ¼MaxðPAT� s; 0Þ ð2aÞSDL ¼Maxðs� PAT; 0Þ ð2bÞdL ¼ 1 if s > PAT; 0 otherwise ð2cÞ

The negative signs in Eq. (1) reflect the fact that most authors have worked in terms of cost rather than utility,and have been inserted to maintain consistency with the sign of the parameters. Note that the terms in theutility function [b SDE, c SDL, d dL] give the variations in utility associated with each possible arrival timeper se: the sum of these terms constitute the schedule utility. Clearly this is at a maximum (of 0) when s = PAT.

It may be noted that d, which represents a penalty for being late per se, is in fact omitted from many of thestudies using this general formulation: implicitly, it is set to zero or subsumed within the c parameter. For rea-sons of simplicity we henceforth ignore the d parameter.

If now we maximise the utility with respect to the arrival time s, we obtain the well-known key demand-sidefirst order conditions on the gradient of travel time n 0(s):

for early shift ðs < PATÞ n0ðsÞ ¼ b=a ð3aÞfor late shift ðs > PATÞ n0ðsÞ ¼ �c=a ð3bÞ

http://www.dft.gov.uk


The interpretation of these conditions is that if shifting is to take place, the network will need to deliver theappropriate travel time gradients.

It may be noted that in most of the theoretical work (by the authors previously cited), a particularly simpleform of network is used – the so-called ‘‘bottleneck’’ model. This has the property whereby free-flow times aremaintained as long as capacity is not exceeded: thereafter a deterministic queuing process begins, in which the(additional) travel time is directly proportional to the length of the queue. In this simple case, it is possible toderive the conditions for the network to satisfy the gradient requirement directly, thus allowing the departuretime profile to be derived analytically. With a ‘‘real’’ network, however, this is no longer possible, and an iter-ative procedure is required to find the equilibrium.

As noted, the theoretical work tends to assume a homogeneous population in respect of PAT and the utilityparameters, though work has been done by Small and ADL to relax these restrictions. Ideally the demandshould be expressed in continuous time, but, as noted earlier, for practical purposes we will assume that itcan be expressed in terms of discrete ‘‘windows’’ of PAT.

For convenience of exposition, we assume that total demand over the whole of the peak period is fixed, andthat outside this period (in what we may refer to as the pre-peak and the post-peak), the level of congestion isnot affected by variations in demand, though we allow for the fact that it could be different between the pre-and post-peaks.

2.2. Notation

Based on the recommendations of Bates (1996), it is crucial to avoid any confusion between variables whichare indexed by departure time and those which are indexed by arrival time, and this approach is thereforefollowed here. In essence, the key notation is set out in the following paragraphs:

As a general convention, we use ‘‘t’’ to indicate departure time, and ‘‘s’’ to indicate arrival time. Thedifference between these, s � t, represents the journey time, or, perhaps better, journey duration.

Viewed from the arrival point of view, we write the journey duration, given arrival at time s, as n(s), andcorrespondingly, viewed from the departure point of view, we write the journey duration, given departure attime t, as H(t). The fundamental linking identities can then be written as:

s � t þHðtÞ ð4aÞt � s� nðsÞ ð4bÞnðsÞ � Hðs� nðsÞÞ ð4cÞHðtÞ � nðt þHðtÞÞ ð4dÞ

We assume a base demand matrix for the peak period T, in the sense that if free-flow times prevailed, this is thelevel of demand which we would assign using an time-aggregate approach. It is convenient to act as if thisrepresents all demand wishing to arrive within the peak period (although, as noted earlier, the exact specifi-cation of what it means to say, for example, that ‘‘the number of travellers wishing to travel in the peak isN’’ remains somewhat elusive). In Section 3.1.1 below we make allowance for the fact that travellers will actu-ally wish to arrive at different times within the period.

Since all calculations are on a matrix level, we will suppress any ‘‘ij’’ notation, but it is implicit throughout.

3. Outline of the approach

3.1. Temporal disaggregation

3.1.1. Segmenting by PAT

We assume that the base demand T is segmented by PAT bands or ‘‘windows’’. The notation allows forany number of such bands within the peak period, and we index the bands as ‘‘k’’. There is an implicationthat the bands need to be relatively narrow, since, although we will concentrate on the differences in traveltime between the bands, we cannot ignore the possibilities that there will be variation within the bands. By


keeping them narrow, we limit this effect, and provide some justification for the use of the term ‘‘gradient’’ todescribe the variation between bands. There is also probably some value in keeping the intervals the samewidth.

Then band k is defined on the interval Jk = [PATk1, PATk2], and the demand that falls within this band willbe written as Tk. It is implied that:

X

k

T k ¼ T ð5Þ

and that the bands cover the whole peak period.The split into matrices Tk is independent of any network considerations, and is thus fixed. We are ignoring

here how it would be done in practice, and from now on, we assume that the split has been achieved. For con-venience, we will assume that the distribution within each PAT window is uniform, though this might be ques-tioned if the width of the bands becomes larger.

3.1.2. Assignment

In order to introduce some ‘‘dynamics’’ into the procedure, a sequential assignment procedure is carried outfor different timeslices within the peak period. The actual way in which this is done will depend on the assign-ment program. Here we try to give a general description.

It is convenient to assume that the separate assignment time-slices relate to departure periods, though it isrecognised that there are some questions of interpretation here. We will treat this as a technical issue, and notdiscuss it further at this stage.

There is no requirement, in principle, for the assignment time-slices, which we index by ‘‘r’’, to bear anyrelationship to the PAT windows, and we treat them quite independently. Once again, they need to be definedas abutting intervals which we write as Ir = [tr1, tr2]. For each assignment time-slice, we require a trip matrix Ar

and we assume that the internal dynamics (e.g. queue-passing) are effectively handled, so that the assignmentsof successive time slices are not independent. Given an equilibrium assignment for time-slice r, we can ‘‘skim’’the network to find the implied matrix of minimum times (i.e. journey durations) along the optimum routes forthat time-slice: we write this as Hr. In addition, we have the fixed matrices H1 and H2 from the pre-peak andpost-peak assignments.

3.1.3. Time period choice

A crucial role for the demand model is to reflect the way in which travellers wishing to arrive in PAT win-dow h allocate themselves to a particular assignment time-slice. This is the departure time model componentand it is useful to view it as essentially carrying out the following task:

for each PAT segment k, calculate the proportion pk,h of total demand Tk allocated to each Arrival time

window hNote that while, following the conventions of this paper, the ‘‘ij’’ arguments are suppressed, this proportion

pkh is also implicitly a matrix with a separate value for each ij element.Although it will need to be checked in actual circumstances, we expect the profile of continuous journey

times n(s) gradually to rise to a peak value, and thereafter decline. We will denote the arrival time associatedwith the highest n(s) as s*. Moreover, we may expect that the gradient on the ‘‘early’’ side (s < s*) will typ-ically be shallower than that on the late side – this is related to the general expectation that b/a < c/a, reflectingthat most travellers would rather be early than late.

Consider the behaviour of travellers in PAT segment k, with average travel time nk (note that the actualvalue will need to be obtained by interpolation, in a way which we discuss later). These travellers have threepossible choices for their acceptable arrival times h:

within PAT window for segment h ¼ k

earlier window h < k

later window h > k


Which option they choose will depend on the gradient of n (and, of course, their values of the schedulingparameters – we are here assuming homogeneity in this respect).

Assuming a uniform interval size J for each PAT window, there will be no early shifting

if bðk � hÞJ þ anh > ank for h < k;

and no late shifting if cðh� kÞJ þ anh > ank for h > k;

(remembering that a, b and c are positive).In other words, for early shifting (h < k), we require:

ðnk � nhÞP ðb=aÞðk � hÞJ ð6aÞ
while for late shifting (h > k), we require:
ðnh � nkÞ 6 �ðc=aÞðh� kÞJ ð6bÞ
This, of course, rules out, on the early side, any shifting if the gradient n 0 in PAT segment k is negative, and onthe late side, any shifting if the gradient n 0 is positive.
While these conditions are straightforward to derive, their implications are less straightforward. Suppose,for example, that we are on the early side, considering the choices for PAT segment k, with the difference(nk–nk�1) P b/aJ, and the difference (nk–nk�2)P2b/aJ. This would appear to suggest that the traveller wouldbe willing to shift early to arrival window (k � 2).

However, it could be the case that the ‘‘incremental difference’’ (nk�1–nk�2) < b/aJ. This makes it clearthat the conditions just stated are not in fact complete. In these circumstances, travellers will not shiftall the way to window (k � 2), since having got as far as (k � 1), the further shift cannot be justified. Essen-tially, this is a consequence of dealing with discrete intervals, rather than requiring a continuous conditionon the gradient. What is actually happening is that, somewhere between the mid-points of arrival segments(k � 1) and (k � 2), the gradient falls below the critical value. Without doing further (non-linear) interpo-lation, it is not obvious how much, if any, of the total demand for PAT segment k should be allocated toarrival segment (k � 2). Note that this uncertainty implies that we should limit our expectations of the levelof convergence that may be attained: we are dealing with approximations to a continuous dynamicrepresentation.

Suppose that for travellers with PAT window k we define the maximum value of the utility function (Eq.(1)) over all possible arrival time windows as V k�. Then the essential equilibrium conditions on the demand-side can be stated as:

for each PAT segment k

V kh 6 V k� if pk;h ¼ 0

V kh ¼ V k� if pk;h P 0

where V kh ¼ �bðk � hÞJ � anh if h < k

¼ �ank if h ¼ k

¼ �cðh� kÞJ � anh if h > k

ð7Þ

It will be seen that these correspond with the standard Kuhn-Tucker conditions for a constrained optimisationproblem.

Now for an equilibrium, these conditions also have to be compatible with the travel times delivered by thenetwork, given the pattern of departure times. What we therefore need is a procedure which will (a) deliver thequantities nh and (b) allow the quantities pk,h to be calculated.

For a given set of {nh}, we can calculate pk,h along the lines just described. The assignment process is notconcerned with people’s preferred arrival windows k, only the windows h which they have chosen as optimal.By summing over PAT windows k, we can obtain the total implied actual arrival demand for each arrival timewindow h. For the purpose of the assignment program, this can then be translated into the departure timeslicesr, by allowing for the travel time nh. Some interpolation will be required, because of the differences in defini-tion between the assignment timeslices and the arrival windows.


3.2. General algorithmic approach

3.2.1. The equilibrium principle

An analogy can be constructed with equilibrium assignment in route choice. Suppose we enumerate all pos-sible routes, applying some ‘‘sensible’’ criterion to avoid ‘‘cycles’’. The equilibrium conditions tell us that allroutes actually used must have the same cost. But we do not deduce from this that all the enumerated routeshave the same cost! Rather, we have to compute how many of the enumerated routes need to be used.

Another useful way of envisaging both problems (i.e., route choice and departure time choice) is along thefollowing lines. Assume any demand matrix, and then consider what happens as we allow it to increase byapplying a single uniform growth factor to all the cells (so that the matrix retains a ‘‘fixed shape’’ but withvariations in the total demand). When the total demand is low, relative to the network capacity, all vehicleswill select the single minimum cost route for each O–D pair, and concomitantly, all vehicles will choose thedeparture time which allows them to arrive at their PAT. As total demand increases, the performance ofthe (free-flow) minimum cost route will gradually deteriorate, until it is equal to the 2nd best route, and atthis stage both routes will be brought into use. These two routes will then ‘‘deteriorate’’ at the same rate untilthey reach the cost of the third best route etc, so that the size of the set of used routes depends on the volumeof demand. Precisely similar developments relate to departure time choice – the window of acceptable arrivaltimes expands with the volume of demand.

In practice, of course, the interdependence of links in a network makes this more complex, so that, forexample, what was the third best route under free-flow conditions need not be the same as the third routewhich is actually brought into use as the volume increases. Further, we have appealed to the network ‘‘capac-ity’’ which is difficult to define in practice. Nonetheless, none of this subtracts from the essential validity of theprinciple described.

We have described the ‘‘raw material’’ of the procedure, and what is now required is to describe the inter-faces which permit the algorithm to proceed. For ease of illustration, we will assume a single homogeneousdemand ‘‘segment’’ with respect to the schedule parameters a b c. However, the approach can in principlebe extended to multiple populations with different schedule parameters.

3.2.2. Interfaces

The equilibrium conditions were set out in 3.1.2 above, in terms of the travel times {nh}, that is, the traveltime viewed from the standpoint of the arrival time window. However, we do not have direct access to this –we only have the values from the assignment, which relates to different period definitions. Since we haveassumed that these yield Hr rather than nr, these need to be converted (essentially using Identity 4d above).The details of this conversion will depend on the assignment package. In some packages (for example,CONTRAM – ‘‘CONtinuous TRaffic Assignment Model’’: see http://www.contram.com) the actual arrivaltimes s are calculated, so n can be obtained directly. However, we treat this as a technical issue which is inprinciple soluble. Here we are effectively assuming that a package such as SATURN (‘‘Simulation and Assign-ment of Traffic to Urban Road Networks’’: see http://www.saturnsoftware.co.uk) will be run for successivetime-slices, with ‘‘queue-passing’’.

Assuming therefore that we have nr relating to an arrival time sr for each assignment time-slice, we now uselinear interpolation to translate these into the required nk values defined at the mid-point of each PAT seg-ment. For example, supposing that sr and sr+1 are the arrival times associated with the mid-points of two suc-cessive assignment timeslices, and that the mid-point of arrival time window k is sk, such that sr < sk < sr+1.We can then calculate the implied value of nk as: nk = nr + (nr+1 � nr).(sk � sr)/(sr+1 � sr). We illustrate theprocedure in Section 3.4.

We can now proceed to calculate the utilities for each arrival timeslice, and hence the allocation of demand,separately for each PAT segment.

Because of the assignment time-slice procedure, a further calculation is required to map the contributions ofeach PAT segment to the assignment timeslices. First we sum over each PAT segment k, to calculate howmuch of the total demand T is allocated to each arrival time window h: from this, we can arrive a cumulativeprofile Xh of arrivals at time s. Then, we translate back, by means of the travel duration nh, to obtain the cor-responding cumulative profile Qt in terms of departure at time t. From this we can calculate what part of the

http://www.contram.com

http://www.saturnsoftware.co.uk


total demand T falls in each assignment time-slice r, thus giving us the required assignment matrices Ar. Again,some interpolation is required.

We may illustrate the essential progress of the iterations as follows:

1 ‘‘M

Ar !assign=skim

Hr !convert

nr !interpolate

nh !arrival time choice

Xh ! Trconvert=interpolate

There are three essential elements in the procedure.Firstly, the skimmed quantities from the assignment need to be converted to arrival time windows: this

involves the use of identity (4a) and interpolation.Secondly, separately for each of the PAT windows, a method is required to allocate the total demand

among the possible arrival time windows.Thirdly, the implied cumulative distribution of arrival times needs to be converted back to a departure time

window basis: this again involves interpolation, together with identity (4b).The least straightforward process is element 2. With the analogy of equilibrium assignment in mind, it will

be essential to have an effective iterative procedure. Given the inherent approximation in the proposedmethod, a straightforward procedure such as MSA1 would seem to be appropriate, as opposed to more‘‘mathematical’’ programming approaches such as Frank-Wolfe. To implement MSA, within each iterationn the arrival time with maximum utility must be chosen, and the demand then averaged with the estimate fromthe preceding iteration according to the proportion 1:n � 1. After adding across all PAT windows, this leadsto an estimate of the total allocation to each arrival window. We describe this in more detail in the followingsection.

3.3. Proposed Algorithm (using MSA)

MSA is a member of the family of ‘‘convex combination’’ approaches to optimisation (see, for example,Sheffi, 1985), in which, given a current estimate in iteration n of X(n), an ‘‘auxiliary’’ or ‘‘target’’ estimateY(n) is obtained, and the updated estimate for the next iteration is defined by the combination:X(n+1) = (1 � k) Æ X(n) + k Æ Y(n). While the Frank-Wolfe algorithm seeks to find the optimum value of k at eachiteration, MSA adopts a straightforward rule whereby k is set to 1/n (i.e. the reciprocal of the iterationnumber).

In any iteration n, assume that we have travel time estimates nk for each PAT segment k: these have beenobtained by (a) conversion of the assignment journey duration matrices Hr to nr by associating the durationswith the appropriate arrival times, and (b) interpolating across the entire peak period.

We now wish to estimate the number of travellers in PAT segment k who choose arrival window h, and wewrite this as Xk,h. For each PAT segment k, evaluate the ‘‘average’’ utility for each arrival window h:

V kh ¼ �bðk � hÞJ � anh if h < k

¼ �ank if h ¼ k

¼ �cðh� kÞJ � anh if h > k

ð8Þ

Find h for which V kh is maximised, and allocate the entire demand for PAT segment k (Tk) to the ‘‘auxiliary’’(or ‘‘target’’) estimate:

Y k;h ¼ T k for h max; 0 otherwise ð9Þ
Combine with previous estimates Xk,h using ‘‘MSA’’ weights to get updated estimate
X 0k;h ¼ ð1� kÞ � X k;h þ k � Y k;h ð10Þ

Note that this maintains the property that, for each PAT segment k,P

hX 0k;h ¼ T k, i.e. all the base demand isallocated to some arrival window h.

ethod of Successive Averages’’


The ‘‘segmented arrival time matrices’’ X 0k;h need to be stored for the following iteration. However, for theassignment the PAT index k is not required. Thus we proceed by calculating:

X 0�;h ¼X

k

X 0k;h ð11Þ

This gives the total demand in each arrival window h, where the boundaries are [PATh1, PATh2]. From this wecan obtain the cumulative demand profile:

X½PATh2� ¼Xh

m¼1

X 0�;m ð12Þ

We now need to translate this back to departure time windows. For this, we need the values of n at the bound-aries of the arrival time windows, as opposed to the mid-points: again, these can be obtained by means ofstraightforward interpolation. Given these, for each arrival time window h, the corresponding departure timepoints are [PATh1 � n(PATh1), PATh2 � n(PATh2)]. This allows us to state the cumulative demand in terms ofthe departure time, and we can write this as Q[PATh2 � n(PATh2)].

We can now interpolate again to apportion the curve to the assignment timeslices, defined in terms ofdeparture times. The difference between the cumulative curve at the start- and end-points of each assignmenttimeslice r then gives the amount of demand Ar to be assigned in that timeslice.

In this way, all the demands X 0�;h are allocated to a departure timeslice. Note that some of the allocationsmay be outside the span of the peak assignments: these will not be assigned, but are assumed to have theappropriate journey time characteristics of the pre- or post-peak.

The temporally dependent assignments are now carried out, yielding new estimates Hr and the iterativesequence continues.

The stabilising properties of the MSA algorithm should ensure that the profile of n(s) demonstrates reason-

able continuity, even if true convergence is hard to achieve. An alternative might be to attempt a stochasticallocation over the possible arrival time windows, by means of a logit model, for example.

Note that if the gradient n 0 never reaches the critical value of (b/a), on the early side, or (�c/a), on the lateside, then no shifting will take place. In this case, the entire demand for PAT segment k will be allocated to thearrival time window for k. This will reflect the case when demand is never significantly in excess of capacity

Finally, as noted, the same approach could be used if we allowed a further demand segmentation by sche-dule parameters.

3.4. Illustration

The travel time choice module requires similar quantities to be calculated on both departure time and arri-val time windows, and the notation is potentially confusing.

At least for the purposes of illustration, it is helpful to consider the variables in tabular form, along thefollowing lines (the values for entries in the tables are arbitrary, but intended to be indicative): see Table 1.

Each assignment timeslice is characterised by its start- and end-points, assumed to relate to departure timest: the mid-point of timeslice r is the assumed average departure time for the timeslice, and the skimmed min-imum time H(t) is assumed to relate to this point. It therefore becomes possible to calculate the average arrivaltime for each timeslice, by adding the travel duration to the average time of departure. Note that the arrivalcolumn is still in terms of the departure time windows, and hence is written [sr].

The figure below plots the travel duration Hr against the mid-point of interval r. It then plots the same valueagainst the implied arrival time, calculated by adding the duration to the interval mid-point. This is therefore arepresentation of n(s) (see Fig. 1).

We now need to switch to a consideration of arrival time windows. For this we require the values of n(s) notat the points [sr], based on the assignment timeslices (as in Table 1), but at the mid-points of the arrival timewindows. The next stage is therefore to interpolate the points on the n(s) curve: this can usually be done withgood accuracy, as shown in Fig. 2.

In a similar way, we can set out in tabular form the essential features of the arrival time intervals, inTable 2.

Table 1Departure time intervals r (for assignment) [min]

From assignment

r Start tr1 End tr2 Mid-point Duration Arrival [sr]r

(0) 0 < 0 ff 0

1 0 15 7.5 20 27.52 15 30 22.5 21.5 443 30 45 37.5 23.5 61....

Travel duration plotted against departure and arrival times

0

5

10

15

20

25

30

35

40

0 20 40 60 80 100 120 140 160 180 200

time (departure or arrival)

dura

tion

f(Dep time)f(Arr time)

Θ(t)

ξ(τ)

Fig. 1. Travel duration H(t) and n(s).


Given the durations for each window, we can calculate the arrival time with the maximum utility (sepa-rately for each PAT segment k), and by means of the MSA procedure, obtain the current estimate X 0k;h ofthe demand for each arrival time window. These can be summed over PAT segments to give X 0�;h, here givenas a frequency. From this we can infer a cumulative arrival time distribution X(s), where s here refers to theendpoint of the interval. Again by means of interpolation we estimate the duration n in relation to the end-points of the windows, and hence the implied departure time associated with the end-points. Note that thedeparture column is still in terms of the arrival time windows, and hence is written [th].

The cumulative distribution can be plotted both against the arrival time window endpoint and, as an esti-mate of Q(t), against the implied departure time associated with the endpoint, as in Fig. 3.

The final requirement is to interpolate the cumulative demand points on the Q(t) curve to obtain an esti-mate at the endpoint of the departure timeslices. Once again, this can usually be done with good accuracy, asshown in Fig. 4.

We can now return to Table 1, above, and add two more columns, as in Table 3. Given the cumulativedistribution, we can obtain the proportion of demand Ar to be allocated to each assignment time-slice r.

Travel duration plotted against arrival time - original and interpolated

0

5

10

15

20

25

30

35

40

01 02 03 04 05 06 07 08 09 01 00 110 120 130 140 150 160 170 180 190 200

Arrival time

dura

tion

interpolatedoriginal

ξ(τ)

Fig. 2. Interpolation of n(s).

Table 2Arrival time intervals h (for time period choice) [min]

h (k) Start PATh1 End PATh2 Mid-point �sh Interpolated MSA Cumulative Xh Interpolated Departure [th2]hDurationnð�shÞ

DemandX 0�;h

Duration n(PATh2)

0

1 0 10 5 20 0.03 0.03 20 �102 10 20 15 20 0.05 0.08 20 03 20 30 25 20 0.06 0.14 20 104 30 40 35 20.6818 0.07 0.21 21.1364 18.8636....


3.5. Convergence

There is a distinction to be made between true convergence measures (which determine how close to equi-

librium we are) and stopping measures, which merely report on whether the algorithm is making progress.Here, we concentrate on the former.

The implication of the equilibrium condition (7) is that, if for any ij pair and PAT segment k,V k� ¼MaxhV kh, we should have

X

h

X k;hðV k� � V khÞ ¼ 0 8 ij; k ð13Þ

while the same quantity could be >0 if the process had not converged. Hence the overall quantity:
X
ij

X

k

X

h

X k;hðV Max � V khÞ ð14Þ

is an indicator of convergence, the smaller the better.

Cumulative demand

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

130 180time (Departure t or arrival τ)

vs Arrival Timevs Departure time

Ω( τ)

Q(t)

-20 30 80

Fig. 3. Cumulative demand Q(t) and X(s).

Cumulative demand against departure time: original and interpolated

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-15 15 30 45 60 75 90 105 120 135 150 165 180 195Departure time

interpolatedoriginal

Q(t)

0

Fig. 4. Interpolation of Q(t).


It is usual to scale such quantities to obtain a ‘‘d-like’’ quantity such as:

Pij

Pk

PhX ij

kh � ðVijMaxkh � V ij

khÞPij

Pk

PhX ij

kh � ðVijMaxkhÞ

ð15Þ

Table 3Departure time intervals r (for assignment) [min]

r Start tr1 End tr2 Mid-point �tr From assignment Arrival [sr]r Interpolate Hence Ar

Duration Hð�tÞr Cumulative Q(tr2)

(0) 0 < 0 ff 0 0.08 0.081 0 15 7.5 20 27.5 0.179487 0.0994872 15 30 22.5 21.5 44 0.3 0.1205133 30 45 37.5 23.5 61 0.447757 0.147757....


Note that while this is a single overall measure, it could be broken down to inspect the convergence of indi-vidual ij pairs and, within that, individual PAT segments k. It would also be possible to substitute the denom-inator by the quantity

Pij

Pk

Ph Xk,h: this would then give the average ‘‘gap’’, in units of utility (here,

minutes), between the optimum and the current position, and would give a more intuitive indication of theseriousness of any such gap.

While it is convenient to state the conditions in terms of the gradient of travel time, the treatment of discretePAT segments means that a certain absolute difference is required to induce shifting. By the nature of things, agiven absolute difference is more likely to occur for longer trips than for shorter, and it is therefore useful tocheck convergence for certain combinations of trip length (e.g., for a typical urban area, Inner to Central,Outer to Central, Through trips etc.).

These are intended as reasonable recommendations for the monitoring of convergence. In the course ofimplementing the algorithm, it may be appropriate to develop other indicators, including stopping measures.

4. Conclusion

A method has been described, essentially heuristic in concept, for reflecting the changes in travel (depar-ture) time which may occur within the peak period as a result of changes in the network (effectively a func-tion of the ratio of demand to capacity). While it makes use of general theoretical principles, the applicationis intentionally postulated in terms of tools which are reasonably available within current modellingpractice.

At the time of writing, the method has not been fully tested, and it may be anticipated that its convergenceproperties will be relatively weak. Nonetheless, it could provide a method for investigating a phenomenonwhich is well attested but poorly understood – the ‘‘flattening of the peak’’ as demand grows, and the ‘‘returnto the peak’’ which occurs when additional capacity is introduced. It should also provide improved estimatesof the average costs for use in other parts of the overall demand modelling process.

It is hoped that the relative simplicity of the approach will encourage other researchers to try and imple-ment it. In doing so, it will be critical to pay careful attention to the various conversion processes whichgovern the interface between supply and demand. Although these are not inherently complex, they are alsonot trivial.

Finally, it should perhaps be emphasised that research work is continuing, and that heuristic approachessuch as that outlined here are not ultimately to be considered as substitutes for approaches that are basedon rigorous theoretical principles. Nevertheless, there is a need for practical tools which will allow the inves-tigation of the profile of congestion during peak periods, and the method outlined in this paper is put forwardwith this in mind.

Acknowledgements

Part of the work reported here was developed in the course of a project for the UK Department for Trans-port, and their support is gratefully acknowledged. However, this paper is the product of the author’s viewsalone, and does not necessarily represent those of the Department.


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Practical modelling of trip re-scheduling under congested conditions

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