Transport Models
TAG Unit 3.1.2

June 2005

• general principles of transport modelling;
• an outline of spatially detailed transport models; and
• an outline of spatially aggregate models.

In the past much transport modelling, particularly on the highway side, has concentrated on what are essentially supply effects, relating to networks. In such cases, apart from allowance for general background growth, the demand for travel is assumed fixed. Since the publication of the 1994 SACTRA Report, this assumption has been considered untenable in most cases, and the presumption is that demand will be potentially affected by any proposed policy/scheme. It is, of course, in the nature of multi-modal studies that the total demand by mode should not be assumed fixed.

The focus on what may be termed 'variable demand modelling' requires an understanding of the basic principles of transport economics, and this is the main topic discussed in this section. The terms 'supply' and 'demand', which are taken from economics, are increasingly being used in the transport context, and it is useful to define them, and the related concept of an equilibrium system, at the outset.

In classical economics it is conventional to treat both supply and demand as functions of cost, but to 'invert' the normal graph by plotting cost on the vertical axis, as in Figure 2.1. The notion that travel demand T is a function of cost C (as shown in the Figure) presents no difficulties: the term 'demand' model implies a procedure for predicting what travel decisions people would wish to make, given the generalised cost of all alternatives. These decisions include choice of time of travel, route, mode, destination, frequency/trip suppression.

Figure 2.1 Demand/Supply Equilibrium

However, if these predicted travel decisions were actually realised, the generalised cost might not stay constant. This is where the 'supply' model comes in. The classical approach defines the supply curve as giving the quantity T which would be produced, given a market price C. However, it is more straightforward to conceive of the inverse relationship, whereby C is the unit cost associated with meeting a demand T. Since this is exactly what is required for the transport problem, this interpretation is adopted here. The supply model reflects how the transport system responds to a given level of demand: in particular, what would the generalised cost be if the estimated demand were 'loaded' on to the system? The most well-known 'supply' effect is the deterioration in highway speeds, as traffic volumes rise. However, there are a number of other important effects, such as the effects of congestion on bus operation, overcrowding on rail modes, and increased parking problems as demand approaches capacity.

Since both demand and supply curves relate volume of travel with generalised cost, the actual volume of travel must be where the two curves cross, as in Figure 2.1 - this is known as the 'equilibrium' point. A model with the property that the demand for travel must be consistent with the network performance and other supply effects in servicing that level of demand is referred to as an 'equilibrium model'.

Although the term demand is often used as if it related to a quantity which was known in its own right, it must be emphasised that the notion of travel demand always requires an assumption about costs, whether implicitly or explicitly defined. The actual demand which is predicted to arise as a result of a strategy or plan is assumed to be the outcome of the equilibrium process referred to above.

Of course the level of demand will reflect the demographic composition of the population, together with other external changes (e.g. effects due to land-use, income, car ownership etc.). However, when assessing the impact of a policy, which means essentially changing the supply curve, the demand curve is held constant. Hence, the testing of strategies can be viewed as a comparison of two (or more) equilibrium points, using a common demand curve, but with each equilibrium point associated with a different supply curve. This is demonstrated in Figure 2.2.

Figure 2.2 Appraisal in the Base Year

Over time, the population and land-use will vary, and this will lead to different demand curves, each related to a particular point in time. In addition, in making forecasts, there may be different views on how the future population and land-use will develop, so that different assumptions (often termed 'scenarios') may be required for the same year. The demand model therefore needs an interface with external 'planning' or 'land-use' data (in particular, forecasts of car ownership) to reflect how the scenario assumptions affect total travel demand.

There is, in fact, some debate about the extent to which the 'external' changes and the transport changes can really be separated - in particular, transport changes may give rise to land-use changes, and the demand for car ownership is likely to be in some way conditioned by the availability and cost of travel opportunities. The majority of transport models do assume independence: it is the particular characteristic of the group of models termed 'land-use/transport interaction' models discussed in Land-Use/Transport Interaction Models (TAG Unit 3.1.3) that they attempt to link the two elements explicitly.

• the equilibrium demand for travel for a given scenario and a given strategy is a function of 'external' elements associated with the scenario and the equilibrium generalised cost arising from the strategy; and
• underlying this equilibrium is a general demand function dependent on the scenario and driven by generalised cost and a supply function dependent on the strategy being considered.

There are thus two types of representation required - demand and supply - plus a procedure to achieve equilibrium. All three components present some difficulties, and these are further discussed in the remainder of this Section. Central to the whole modelling process is the notion of 'generalised cost' which is defined next.

Generalised cost is usually a linear combination of the various components of a journey. The components of generalised costs vary by mode.

For car, generalised cost is usually taken to be a combination of: (a) in-vehicle travel time; (b) operating costs (related to distance travelled); (c) parking 'costs' (which notionally include time spent searching and queuing for a space and walking to the final destination); and (d) tolls or congestion charges. Money costs are usually converted to time units using a value of time.

For goods vehicles, the components are usually similar, except that different vehicle operating costs and values of time are used.

For public transport users, generalised cost is usually a combination of: (a) walking time from the origin to a stop or station (usually weighted relative to in-vehicle time by a factor of about two); (b) waiting time for the service (again, usually weighted relative to in-vehicle time by a factor of about two); (c) fare; (d) in-vehicle time; (e) penalty representing the inconvenience of changing between services; and (f) walking time to the destination (again, usually weighted relative to in-vehicle time by a factor of about two). Again, money costs are converted to time units using a value of time.

For transport modelling purposes, the components are those perceived by travellers, often referred to as 'behavioural' values. Thus, car operating costs are usually taken as fuel costs, and car parking costs and public transport passenger interchange penalties may contain elements to ensure that the model better reflects actual behaviour. Goods vehicle operating costs, by contrast, are likely to include all resource costs, including the time costs of the driver valued at an average wage, although variations may be adopted to reflect, in effect, drivers' perceptions of their resource costs.

As noted above, the demand model predicts what travel decisions people would wish to make, given the generalised cost of all alternatives. In principle, these decisions can include choice of time of travel, route, mode, destination, frequency/trip suppression, and most of these decisions require a further definition of their dimensions. For example, choice of time of travel requires the modeller to define whether it is desired to predict the precise departure time, say, or merely to distinguish whether the journey is made in the peak or off-peak period, and destination choice requires a definition of the level of spatial representation.

Clearly, the more detail we require, the more complex becomes both the specification of the demand model and the process of achieving equilibrium. It is important, therefore, to tailor the level of detail to the requirements of the problem. In addition, while the components of choice referred to above (i.e. time of travel, route, mode, destination, frequency/trip suppression) are the types of response which are most commonly modelled, it may not be necessary to model each component separately.

For example, if the emphasis is on reducing peak highway travel, it may not be considered important to know whether the reduction has been brought about by trip suppression, mode switch, trip redistribution or time of day switch. Although it may still be considered more reliable to model the separate mechanisms, the fact that they are not considered necessary for the final output opens up the possibility of simplification.

In the simplified diagram presented in Figure 2.1, the demand for travel T was viewed as a one-dimensional quantity, dependent on another one-dimensional quantity cost, C. In practice, in order to represent the essential spatial component of transport, it is necessary to distinguish movements, at some level of detail, based on the area of origin and the area of destination, and, in most cases, by the mode that is used. For a consistent account, it is likely to be necessary to distinguish at least the modes car, public transport, and non-motorised or 'slow' (walk/cycle), and further subdivisions (e.g. between bus and rail) may be required in some cases. Other dimensions, such as time of day, may also be required.

Hence, it is necessary to deal with multi-dimensional arrays, or matrices, representing demand for alternative travel opportunities. In general, each of these opportunities may have its own (generalised) cost. Thus the demand model has to establish a mapping between the matrix of costs and the matrix of resulting demand. In principle, any coherent procedure for achieving such a mapping (e.g., a set of 'rules') could be used, but it is highly convenient to do this by means of specific mathematical forms.

A central concept in transport economics is the so-called 'elasticity of demand', which is a measure of the sensitivity of the response to changes in cost (and other variables). This can be given an exact mathematical definition, and it can be calculated for any demand model, though in most cases it is dependent on a particular point on the demand curve. For any chosen travel quantity, it represents the percentage change in demand which results from a percentage change in cost, assuming that all other costs remain at their current level (this is the economists' familiar ceteris paribus condition). When the cost refers to the same travel quantity as the demand, it is termed an 'own' elasticity, and when it refers to a different quantity, it is termed a 'cross-elasticity'.

It is possible to develop simplified demand models by making assumptions about the elasticities. Apart from the limiting case where all elasticities are zero (i.e. demand is constant, and therefore unaffected by cost changes), the simplest assumption which will yield a demand model is that all cross-elasticities are zero, and all own elasticities are constant (though not necessarily with the same value). Note that, because demand increases when costs fall, own elasticities will be negative, and, for most practical cases, cross-elasticities will be positive (or zero).

The assumption that cross-elasticities are zero results in a major simplification of any demand model, since it allows each travel quantity to be modelled independently. Whether it is an acceptable simplification depends, naturally enough, on whether the true cross elasticities can be expected to be small⁽¹⁾ . In most cases, there will be little empirical evidence, and it will have to be a matter of judgement. There are many cases, however, where it could not be justified. For example, the demand for travel by public transport would not be expected to be independent of the price of motoring. On the other hand, at greater levels of detail, the demand for travel by public transport between two particular zones in the peak might be assumed to be independent of the price of travel by car between two other zones in the off-peak.

The assumption that elasticities can be treated as constant will, in general, be more reasonable (provided, of course, that the values used are appropriate). Constant elasticity models can be viewed as local approximations to a fully-specified demand function. It follows that they should produce acceptable forecasts provided the costs do not change too much from the current position. In practice, this is likely to be the case for the majority of strategies, though some of the more 'radical' may not fit into this category.

The problem with elasticity models, in this respect, is that if there are a large number of categories to be forecast, then it is usually more convenient to calculate the demand curve by means of an explicit mathematical function than to have a large table of elasticities to apply. Hence, demand models based on elasticities are best suited to 'sketch planning' where we are trying to forecast a small number of aggregate quantities.

So far the demand function has been discussed in terms of the number of 'transport alternatives'. The most common way of dealing with this problem is by means of 'choice models' which predict the proportions of an overall total demand which will be allocated to each alternative. In most cases, these proportions are defined 'conditionally', using a specified hierarchy. Thus one model might define the proportions of a given total demand for public transport that go to different destinations, while another might define the proportion of total travel which makes use of different modes. There are well-known rules which must be followed when specifying and constructing such hierarchies of choice models (see, for example, Ortúzar and Willumsen (1994), section 7.4). A central property of choice models is that cross-elasticities are non-zero more or less by design: since in general the selection of one option can be considered to be a rejection of others, there is an inherent interdependence between the options which is explicitly recognised by choice models.

In addition to these dimensions of choice, however, there are other distinctions which relate essentially to the travellers. Different persons have different basic demand for travel. For example, employed persons need to get to work, retired people have more free time etc. Choices are likely to be different between those who have access to cars and those who do not, those who face different levels of pricing (e.g. children and Old Age Pensioners), those who have different levels of income etc. In addition, even for the same person, responses may be different according to the purpose for which the journey is made: this relates both to the inherent need for the journey and to institutional constraints on the timing of the journey.

In principle, therefore, there is a case of having separate demand functions for different categories of purpose and person-type (often referred to as 'segments'). How far this is worthwhile depends on two key questions: the extent to which responses are different between segments, and the extent to which segments grow at different rates over time. If neither of these are significant, it is unlikely to be worth making the distinction.

Changes in the distribution of person-type segments between the base and forecast years will have repercussions on total demand, as will changes in zonal populations. These can be assumed not to affect the functional form of the demand curve per se, but to affect its 'location' or scale (see paragraph 2.12.1).

The purpose of the supply curve in a transport model is to reflect the way in which costs of travel vary according to usage of particular facilities e.g. a highway link or a train. As usage rises costs generally rise also, typified by the congestion that occurs as a road or turning movement approaches capacity. (Note that rising usage does not always lead to a rise in the public transport costs as experienced by users, as the response of the operator may be to increase levels of service). The supply curve is an integral part of the assignment (route choice) stage of the transport modelling process.

The primary purposes of assignment models are to provide travel cost information to demand models, enable spatially detailed analyses of problems to be undertaken, and to provide information for operational, environmental, economic and financial appraisals. The relative importance of each of the above functions will vary according to the requirements of the study being undertaken. A model for transport strategy development has most need of a cost generator, whereas one being used for development of a transport plan will need a good quality operational and environmental forecasting capability.

• a computerised representation of the network (and for public transport the services that operate on the network);
• a mechanism to calculate viable routes through the network (path build);
• an origin/destination (OD) matrix of travel demand;
• a mechanism for loading OD demand onto the alternative routes available; and
• a mechanism for ensuring that supply and demand are in equilibrium at the end of the assignment model process.

The following paragraphs provide an overview of the types of transport supply representations that should be considered for use within a study. The key variable is the level of aggregation to which the representation of road traffic and public transport supply will be subjected. In terms of road traffic, the range available is from explicit representation of all roads and junctions with significant traffic levels, through to area-wide representation of speed/flow relationships using a single curve.

Aggregate approaches have benefits in terms of model run times (and hence the ability to tests a wide variety of alternative proposals) and in terms of the level of demand/supply convergence that can be achieved. However, aggregation reduces the ability of the model to reflect accurately all of the routeing opportunities that might arise from a change to the highway network, and inherently increases the margin of error associated with the estimation of travel costs.

Aggregate approaches are thus more suited to transport models whose purpose is to assist in the design of transport strategies, where a wide range of alternatives approaches will need to be tested and optimised. Spatially detailed representations are most applicable where the aim is the development of plans to solve individual problems.

The weaknesses inherent in aggregation of the supply representation can be reduced by taking a hierarchical approach to model formulation. In this configuration, the upper tier is the demand model with a spatially aggregate supply representation. The lower tier is a detailed network assignment model. The linkages need to ensure that the detailed model characteristics can be compressed to form the supply representation for the upper tier model, where travel demand forecasts are estimated. Demand forecasts from the upper tier model can in turn be disaggregated to the level of the detailed model zoning system, allowing their detailed routing implications to be tested and understood.

2.5 Highway Supply Curves

• area speed/flow relationships with sets of fixed route alternatives;
• link-based speed/flow relationships using 'notional' highway links;
• link-based speed/flow relationships using 'real' highway links; and
• detailed network representations with junction turning movements explicitly modelled.

The above list is in descending order of level of spatial aggregation, relating to both zone size and to supply. Each of these alternatives to highway supply representation is discussed below. It should be noted that the above are not discrete alternatives, and that combinations of their main features are possible in order to meet the needs of particular modelling exercises.

• capacity restraint (the modelling of congestion);
• multi-routeing (the spread of trips across alternative routes); and
• equilibrium - generally seen as the fulfilment of Wardrop's First Principle which states that under equilibrium conditions no driver can reduce generalised cost by changing route.

All of the methodologies described below possess these features.

Multiple time periods can also be an important feature of assignment models, and generally representations of peak and off-peak periods are a requirement of a demand modelling process.

Modelling two peak periods and an average interpeak period would be standard practice. However, there are some circumstances where that would not be appropriate. For example, in the case of a very large area to be modelled, and a large number of options to test, a model which treated three periods of the day separately may take much longer to run than the 14-hour period generally available over night. In this sort of case, some compromise is necessary and modelling the day as a whole (that is, the period from the start of the morning peak period to the end of the evening peak period) might be a way forward. Note that this would not mean necessarily that congestion effects could not be represented; they could be crudely modelled by use of averaged speed/flow relationships.

Area speed/flow curves with fixed routes are the most aggregate form of highway supply representation, and are best suited to studies whose purpose is to develop transport strategies rather than plans. In this methodology one or more area speed/flow curves are defined for each of the (generally large) zones in the model, as a means of representing congestion effects. The unit for flow in this context is pcu-kilometres, as the speed/flow 'links' do not have defined lengths.

For each OD pair a set of fixed routes is defined in terms of the distance travelled on each of the area speed/flow links. The route set is generally established so as to reflect distinctly different travel opportunities. Because this is the most aggregate form of supply modelling the representation of intra-zonal as well as inter-zonal movements is essential. Routes for intra-zonal movements are defined in the same way as for inter-zonal movements.

A variant on this methodology is to combine area speed/flow curves with representation of motorway and/or strategic highway links as separate units of capacity, an approach that is definitely required if differential pricing policies such as motorway charging are to be tested.

Area speed/flow curves and the associated set of alternative OD routes are most accurately and economically generated using a road traffic assignment model of the same area. Thus this form of aggregate supply modelling is best suited to the hierarchical modelling concept described above. The process of generation of the aggregate representation of supply can be fully automated and if necessary repeated for test options that involve highway infrastructure changes.

Speed/flow curves can be estimated by the application of upward and downward factors to the base year detailed assignment model trip matrices. For each matrix factor, speeds and flows can be accumulated for the links making up each zone, giving a series of points on a curve. Routes between OD pairs can be generated through an analysis of the routes output by the detailed assignment model.

The use of fixed routes within this methodology is necessitated by the fact that the supply representation does not constitute a network of 'physically' connected nodes and links, and thus a path building process is impractical. The trip loading process is generally an integral part of the demand model, with choice of route occurring at the bottom (most sensitive) point of the choice hierarchy. The combination of fixed routes and a high degree of spatial aggregation means that this form of representation of transport supply can achieve a high degree of convergence with a relatively low number of demand/supply iterations (see below for more on convergence).

In contrast to the above this approach uses conventional assignment modelling techniques in the context of an aggregated approach to supply representation. Conventional node and link network definitions combined with path building and trip loading procedures are employed. A full Wardrop equilibrium assignment can be achieved. The aggregation ensures that run times are relatively short and that a high degree of convergence is achieved.

• spider links - connecting centroids of adjacent zones, with capacity set at the combined level of all of the roads that cross the zone boundary in question;
• links representing 'amounts' of highway capacity, such as an urban central area within an inter-urban model.

As with the fixed route area speed/flow approach described above, it is common practice for this form of aggregate representation of highway supply to contain a network of explicitly coded motorway and other strategic highway links. These are coded in a manner similar to that for a conventional node/link assignment model, leaving the aggregate links to represent all other capacity.

The process of coding these networks is not well established, and the theoretical basis for aggregating areas of highway capacity into single links has not been clearly defined. For example, capacities of highways at the point where they cross zone boundaries may not actually encompass the limiting factor in terms of travel between two zones. Useful guidance on this type of approach is given in TRL Project Report PR/TT/092/97. This has not been formally published but can be obtained from the DfT's ITEA Division. A complex process of trial and error calibration could well be required to get the model to perform in a satisfactory manner. The modelling of intra-zonal movements, necessary with large zones, also presents theoretical and practical difficulties for this approach.

The uncertainty about the theory of this approach to highway supply representation means that generation of aggregate supply representations from detailed assignment models of the study area is difficult to automate, such that the process can be repeated with confidence where major highway supply changes are to be tested. This limits the potential for use of this type of supply modelling in a hierarchical model structure.

Assignment models in this context are typified by relatively small zones and a highway network that represents all main roads. (Note that, in this context, small zones are defined as ones in which the traffic impacts of intra-zonal trips can be assumed to be negligible.) Capacity restraint is represented by highway link-based speed/flow relationships. This type of application requires multi-route modelling and equilibrium assignment procedures. This is because of the importance of such models in the appraisal of strategies and schemes that affect the capacity of specific links in the highway network. Model convergence is measured using stability and proximity criteria (Design Manual for Roads and Bridges (DMRB), Volume 12.2.1). Aggregate outputs are available as they are for the more strategic models, but spatially more detailed outputs such as corridor flows and journey times are also available.

A detailed zoning system and a network that includes all roads that carry significant volumes of traffic characterises this approach to highway supply modelling, generally known as congested road traffic assignment modelling. Multi-routeing and equilibrium assignment are essential features. Capacity restraint is affected through the explicit modelling of junctions, taking account of physical turning capacities, signal timings and the interaction of conflicting traffic movements. Link speeds are generally fixed, that is, all delays are assumed to be as a result of conflicts at junctions. Use of link-based speed/flow procedures is sometimes made in the peripheral parts of the network, to provide realistic routeing into and out of the area of junction modelling.

If the model tends towards the transport strategy development end of the spectrum, in some circumstances junction representation is usually a simple extension of link based speed/flow modelling procedures described in the previous section. In fully specified congested assignment models, used for development of detailed transport plans, the junction modelling procedure is used to represent the interaction between junctions. For example, where modelled queue lengths exceed the available queuing capacity, these models represent the effects that this will have on the workings of the upstream junctions. Similarly, the effects of bottlenecks in the network in 'metering' the flow of traffic to downstream junctions are also represented.

Convergence is measured using stability and proximity criteria as defined in Values of Time and Operating Costs (TAG Unit 3.5.6) of Traffic Appraisal in Urban Areas (DMRB 12.2.1). While aggregate outputs are readily available from congested assignment models, it is the ability of these models to produce a wide variety of detailed junction performance information that distinguishes them from other types. For example, possible outputs include: flows on links by direction; main turning movements at main junctions; total delays at junctions; delays for main turning movements; and queues at junctions.

However, it should be noted that even with a well-converged model, the queue and delay information can display considerable instability from iteration to iteration and great care is required to avoid over-interpretation of the model output.

• aggregate approaches involving sets of fixed route alternatives (generally one option for each mode);
• link-based representations involving an aggregated representation of public transport services coded onto aggregate highway and rail network link definitions;
• link-based based representations involving an aggregated representation of public transport services coded onto networks definitions containing 'real' highway and rail links; and
• detailed service definitions coded onto networks coded as 'real' highway and rail links.

Again the above should not be viewed as discrete alternatives; they provide a continuum within which the needs of a particular study can be addressed.

The above list is in descending order of level of spatial aggregation, relating to both zone size and to supply. The fixed route approach is a direct parallel to the most aggregate form of highway modelling described earlier. The fixed routes are definitions of the path taken by passengers for each OD pair, defined in terms of distance travelled on each strategic network link, with fare and frequency measures also defined. The passenger paths are most accurately and economically generated using outputs from a detailed public transport assignment model of the study area, and therefore the approach is best suited to a hierarchical model structure. Where a radical change to public transport supply is to be tested, it is often convenient to code this into the detailed model first, and re-run the passenger path generation process.

Options involving simplification of the public transport supply representation to meet the requirements of an aggregate highway network, or to reduce coding effort, require skilled judgement on the part of the modelling practitioner. There is a danger that the resultant representation of public transport services will have lost some important characteristics of the public transport system, particularly those relating to interchange potential and cost. The approach to be taken to service aggregation therefore requires detailed consideration at the model design stage.

Disaggregate coding of all public transport services onto a network that contains all relevant highway and rail links provides the best basis for representing public transport travel costs and routeing opportunities. Such a model requires an initial high effort with respect to service coding, though the work involved here can easily be over-estimated and should be offset against the design effort and skilled modelling inputs required for a satisfactory aggregate approach. In order to obtain benefit from the detailed service representation small zones are required, and this can add significantly to model run times. However, it should be noted that public transport models that do not involve the modelling of capacity restraint (crowding) have no iterative procedures, and hence the path-build and loading take place only once.

• sub-modal choice;
• multi-routeing;
• capacity restraint (crowding effects); and
• operator response to patronage change.

Sub-mode choice can be carried out as part of the route choice process within the assignment model, or as part of the overall demand model hierarchy. In the above examples the fixed route approach would involve sub-mode choice as a demand model stage. For the other approaches a decision would need to be made as part of the overall model design. In some studies there may be a need to explicitly model the different modes used on different legs of a multi-modal trip. The capabilities of the alternative approaches to this will need to be carefully assessed.

Multi-routeing is important where there are a number of viable alternative passenger paths through a network - e.g. parallel rail routes or bus rail competition (where sub-mode choice is a feature of the assignment model). Multi-routeing is also important where zonal aggregation means that the walk element of the start and end of trips is poorly represented, and hence allocation of all trips to a single route opportunity would significantly affect loadings on alternative routes.

It is general practice in the UK for public transport models to ignore the potential impacts of crowding upon route choice and perceived costs, though a notable exception occurs in models of London. Where this significant simplification is unacceptable, the assignment process needs to form part of an iterative process under which wait times and/or perceived journey times are re-calculated between runs of the assignment model, with the iterations carrying on until a converged position is achieved. However, as is the case with the LTS model of London, the resulting model run times can become very large. However, where crowding exists. or could occur as a result of some strategies or plans, it may be important to represent it in the model to ensure that decisions are robust.

Related to the crowding issue is the potential for public transport operators to respond to rising or falling patronage levels. This effect can be included as a fare or frequency response or some combination of the two. The relationship between patronage and service levels is complex and currently not fully understood. It is reasonable to assume that bus operators in a competitive market will respond so as to maintain the equilibrium between operating costs and revenues that can be assumed to exist in the base situation. However, factors such as real wage changes and technology developments could influence applicability of this assumption. For the rail mode, which is subject to much greater regulation and long lead times for vehicle purchase and infrastructure development, the operator response is likely to be slower and less certain.

Taking account of operator response allows the potential for stemming the long term 'spiral of decline' in patronage and service to be investigated, alongside the potential for establishment of 'virtuous circles'. Models in which this feature is included have demonstrated that the impact is significant. However, including this feature in the model complicates the software and can greatly increase model run times. A simpler approach is to reflect operator response by adjusting input service levels as part of option development. Where operator response is included, care is required to ensure that the model does not generate a 'hidden' investment scenario, especially where this might result in increases in grants or subsidy payments.

As noted earlier, at equilibrium the demand for travel must be consistent with the network performance and other supply effects in servicing that level of demand. In other words, if the estimate of demand is loaded on to the supply, the resulting costs should exactly generate the estimate of demand which is loaded. (Note that this equilibrium is different from assignment equilibrium, discussed earlier in this TAG Unit.)

The importance of the need to find the points of equilibrium with some accuracy must be emphasised. The demand/supply diagrams shown in Figures 2.1 and 2.2 have been drawn with false origins for the sake of clarity. However, the diagram drawn as in Figure 2.3 is likely to be a better representation of reality. The benefits are actually, in essence, a very small quantity derived as the difference between two large quantities which have a certain degree of error associated with them. In order to derive the benefits accurately, it is essential that the equilibrium points are found accurately for both the do-minimum and do-something cases. Failure to recognise this fact and adopt modelling procedures which enable equilibrium to be found with accuracy could easily result in erroneous decisions being taken.

Figure 2.3 A Less Distorted View of Appraisal

In practice, with the exception of the very simplest models, there are no direct ways of calculating the equilibrium solution, and it is necessary to set up iterative procedures. Although a well-conceived iterative system should converge to a unique solution, the nature of the method is likely to produce only approximate equilibria, both because of inherent computational inaccuracy (e.g. rounding) and the desire to limit computing time.

In addition, it is very often the case that the iterative system is not well conceived: at best, this can mean that it takes a very long time to converge, at worst, that convergence is not obtained at all.

In order to address this issue, it is necessary to develop criteria for satisfactory model convergence. For a detailed model, the total number of demand estimates may be very large, and while it would be possible to test each element for stability, the natural desire for compromise means that criteria may be defined at relatively aggregate levels. However, while the procedure may appear to converge according to these criteria, in reality stability is not being achieved at more detailed levels.

If the aim of the model is merely to give broad levels of magnitude, a high degree of convergence may not be important (provided, of course, that the iterative process has not simply 'got stuck'). However, for detailed comparison of options, it is essential that the accuracy of convergence is substantially greater than the difference between the options. In other words, the chance of erroneously concluding that option A is preferred to option B because of inadequate convergence must be minimised. This is particularly important when there is a tendency for successive iterations to oscillate around the true solution, as is very often the case.

The simplest form of iterative procedure is known as the 'cobweb': at any stage in the iterative sequence, this simply takes the 'current' demand, 'loads' it on to the supply, calculates the resulting costs, inputs these to the demand model to get a new 'current' demand, etc., as illustrated in Figure 2.4. It is well-known that such procedures have no general guarantee of convergence, and in those cases where they should converge, convergence is very slow. This is because, at each iteration, all previous estimates are discarded. In principle, some averaging procedure which makes use of previous estimates is always to be preferred.

Figure 2.4 The Cobweb Method of Seeking Equilibrium

• Establish that a unique solution (equilibrium point) exists. For most transport problems this is likely to be the case, though it may be difficult to prove.
• Design a 'search' procedure ('algorithm') which consistently improves the estimate of the solution with each iteration. This is a specialist topic. The most straightforward procedure in common use is the so-called 'Method of Successive Averages' (MSA): see, for example, Ortúzar and Willumsen (1994). However, although this is generally guaranteed to converge, the rate of progress may be extremely slow.

On theoretical grounds, the preferred approach is to set up some kind of 'objective function' which is then minimised, yielding a result which coincides with the equilibrium solution. For a very limited range of demand and supply functions, this can be done using commercially available software. Such an approach provides much greater control, in terms of allowing appropriate convergence statistics to be designed, while at the same time offering substantial advantages in terms of computational efficiency. Unfortunately, specifying the appropriate objective function for an arbitrarily defined modelling system is a highly complex task, though there is at present great interest in developing the approach.

The best general advice that can be given is to investigate the sensitivity of the conclusions to the number of iterations. For example, assuming that options A and B have been independently judged to have 'converged', how does the comparison of A and B alter under the following numbers of additional iterations?

The output can also be subjected to a statistical analysis, with a view to estimating the error range of the current estimate in the light of subsequent iterations.

If the conclusions are generally unaffected by increasing the number of iterations for either option, or (which is effectively the same) the difference between the options is significant after taking account the error obtaining for each, then they can be considered secure. If this is not the case, further attention will be required to the level of convergence. While in the simplest case, this may merely mean increasing the number of iterations, in more serious cases it may require a reassessment of the iterative procedure.

A practical problem in carrying out such analysis is that the software may not have the facility to start off at a previous 'intermediate' estimate. If the iterative process is terminated after N iterations, and then has to be restarted at zero in order to obtain an estimate for N+1 iterations, the kind of sensitivity analysis envisaged here will be prohibitive. A 'warm start' procedure is thus essential, and software should be designed or chosen with this explicitly in mind.

So far, the focus has been on the specification of the demand curve at a particular point in time, and the problem of estimating equilibrium with different assumptions about supply. However, as already noted, the demand curve will shift over time, reflecting exogenous factors such as demographic and land-use changes. There are also some technical issues relating to other changes, such as the value of time. These matters will now be discussed.

As will be described in subsequent sections, a basic procedure underlying the vast majority of practical transport models is a 'trip generation' stage, which relates the general volume of travel (separately, in most cases, for a number of distinct journey purposes) to person-type characteristics. Regardless of how these relationships are derived, they tend to be applied at the zonal level: in this way they are sensitive to the (forecast) numbers of persons of different types in the zone.

The data requirements are discussed explicitly in Data Sources (TAG Unit 3.1.5) but the general range may be noted here. As a minimum, zonal populations and employment will be required, together with the number of households, usually broken down by level of car ownership. Some breakdown by age and employment status is desirable, as is the availability of driving licences. The level of detail should be at least as much as that required for the implementation of the demand model: thus, if it is decided that the demand model should distinguish between different levels of income, then the model will require forecasts of the population at these different levels.

As a general guidance, the number of distinctions made within the demand model tends to be low. Since models of trip generation are relatively easy to develop, there is a case for allowing for a greater level of 'segmentation'. At the same time, the DfT has developed forecasts at the local authority level which require data at a reasonable level of disaggregation: a priori, it makes sense to make use of these. Since the zones in the study area will typically be more detailed than local authorities, some method of spatial disaggregation will be required. In general, it will be desirable to 'control' the predicted growth to the DfT forecasts, at least at the modelled area or study area level.

The predictions about future growth in demand need to be expressed in the form of a forecast year demand curve, as illustrated in Figure 2.5. Consider a base year description of travel movements (set of matrices) assumed to be in equilibrium at point A - if the growth rates resulting from the socio-demographic changes just discussed are applied, the resulting set of movements, often referred to as a 'reference case' forecast, can be considered to be a forecast of what would happen if there were no changes in travel cost. This will represent a 'point' B on the future demand curve, corresponding to the base year costs. Hence, it can be seen as a way of locating the future demand curve, assuming a constant functional form.

Figure 2.5 The Shift in the Demand Curve Over Time

It is important to note that this 'reference case' is not intended to represent a realistic forecast of what might happen - i.e., it is not an equilibrium solution. Typically, the increased growth will lead to cost changes (e.g. greater congestion) through the supply relationships. Having located the future demand curve, there is a need to invoke the convergence process in order to derive the equilibrium solution at point C. Note that, in addition, there may be forecast changes in supply (essentially the distinction between 'do-nothing' and 'do-minimum').

In this context, cost forecasts relating to, for example, the price of petrol or public transport fares, can be viewed as changes in the supply functions. However, since the demand curves are invariably specified in terms of generalised cost, this leads to a somewhat problematic issue, which is bound up with the question of value of time.

Generalised cost, being a weighted combination of time and money, has no intrinsic units (though it is clearly possible to 'measure' it in terms of time units or money units). Unfortunately, in making the standard assumption that the demand functions do not change in form over time, it is necessary to address the question: "In what units are generalised costs assumed to be defined?" The result of defining it in time units would lead to different forecasts from what would be obtained by defining it in money units. The only case in which it does not matter is when there are no changes in the magnitudes of the time and money components, nor of the way in which they are weighted together (value of time): this is the least likely assumption.

Guidance in line with best practice would be to assume that generalised cost is defined in units of time, on the principle that time is more universal than money. It should be noted, however, that in terms of the demand model assumption of constancy into the future, this is an empirical question, and virtually no information is available to support or reject it.

At the same time, there is a general presumption that the value of time will rise with income. Although this is a controversial topic in its own right, it is usually assumed that values of time should increase over time in line with the forecast growth in GDP. The DfT issues recommendations about what assumptions should be made for changes in value of time into the future, and though these recommendations relate strictly to their use in appraisal, they may be taken as representing reasonable practice for modelling as well.

Taken together, the implied consequence is that money costs will represent a smaller part of generalised cost in the future, so that the sensitivity (elasticity) to money costs will decline. While this has some plausibility, extrapolation to the longer term, where the implied overall growth in GDP may be substantial, may produce results which are difficult to accept. Forecasts over substantial periods of time should always therefore be scrutinised for plausibility. If the results are judged unreasonable, different assumptions may have to be made. The most neutral assumption would be that that the rise in money costs is 'more or less' in line with GDP, since this restores the balance between cost and time which would have existed at the time of estimating the demand model.

The concept of generalised cost is also used within highway assignment models as the criteria for route choice, whether this is modelled stochastically or deterministically. Although ideally the definition of generalised cost should correspond with that in the demand model, conventional practice in assignment has been to use a weighted average of time and distance, and allow the weighting to be determined on the basis of the plausibility of the predicted paths chosen.

With some effort, it is possible to interpret the weights in terms of a trade-off between in-vehicle time and car operating costs (the latter calculated on the basis of a simplified form of the recommended COBA relationships): given this, the weights could be changed to reflect changes in operating costs (e.g., fuel prices) and values of time. However, this is not recommended practice: applying it naïvely can lead to implausible routes being chosen. The standard best practice is to keep the calibrated weights constant for future year assignments.

Although this leads to a possible incompatibility between route choice and other elements of the demand model, it is judged that this is the lesser of two evils. A possible complication, when equilibrium assignment techniques are being used, is that the chosen paths between any pair of zones do not all have the same generalised cost, when this is measured in terms of the demand model definition. In this case, the recommended approach to calculating the cost to use in the demand model is to take the flow weighted average of the different costs over all the paths used.

A general description of the demand model is set out in the IHT's Guidelines on Developing Urban Transport Strategies: Section 6.3 for 'aggregate models', and Section 6.4 for 'disaggregate models'. The material is only summarised below.

The trip generation stage requires, for each modelled purpose, a prediction of the number of trips leaving each zone (Productions and Origins) and, in some cases, entering each zone (Attractions and Destinations). For those cases where the number of trips entering each zone is predicted by the distribution model, the trip generation stage is still required to estimate a measure of 'relative attraction'. This stage corresponds with the 'tour frequency' module in disaggregate applications.

In principle, the level of trip making could be sensitive to network conditions ('accessibility'), but there are very few practical applications of this. The default assumption is that trip rates do not depend on accessibility but only on socio-demographic characteristics.

• obtain available demographic forecasts;
• disaggregate as required (spatially, and by person-type);
• apply car ownership forecasts to disaggregate by car availability;
• apply estimated fixed trip rates to zonal quantities in each category;
• aggregate as appropriate.

In the absence of any existing data, this could be an onerous task. Population forecasts by age and sex are usually available for local authority areas, but in principle all the other tasks could be considered the remit of the modeller. ITEA Division of the DfT can provide much of the required information at the local authority level, and below via the TEMPRO program available on its web site. However, those data should be examined and any local anomalies and updated local projections identified. Currently, projections of households, forecasts of household car ownership and trip end forecasts for all modes are available. Although not directly available, both the car ownership and trip end forecasting procedures require a disaggregation by household and person types, and a published methodology is available. In general, then, apart from ensuring conformity with the local zoning system, all the essential information is in the public domain. Although model zones will usually be rather smaller than local authority areas, the published methodology could in fact be applied at a more detailed level, provided only that the necessary demographic data is available at that level. This greater detail is usually available to local authorities, and it will usually be acceptable to disaggregate the trip end forecasts for local authorities on a relatively mechanistic basis.

In cases where there are serious doubts about the suitability of the national relationships incorporated in the DfT forecasts, it would be open to the model team to collect household interview data with a view to developing car ownership models and/or trip generation models. However, this should be very much an exceptional case.

In contrast to trip generation, the trip distribution and modal split changes will have to rely principally on local data, and furthermore, the collection of such data is problematical. Such models would ideally be based on household data: however, except in the case of the largest conurbations, the cost of collecting sufficient household data for this purpose can be prohibitive. This means that substantial reliance has to be placed on surveys made in course of travel (typically, roadside interviews (RSIs), and on-board public transport surveys). Although aggregate information such as traffic counts, may be of use for model validation, it lacks the dimensions of origin, destination, trip purpose and person-type that are needed for model building.

In most cases, matrices will be built independently by mode: it is essential that they are prepared on a 'Production/Attraction' (P/A) basis. To achieve this, the purpose should be recorded for both ends of the trip. Note that failure to do this is common in data collection, and is potentially disastrous to the modelling of demand (TAM RSI form in DMRB Volume 12.1.1.6 meets the requirements).

For highway matrices, based on RSIs, the methodology can follow that given in DMRB; separate matrices will be built by purpose and time of day, as a minimum. While in principle it should be possible to disaggregate this by further person-type segments, in practice the restrictions imposed by the RSI data collection format may make this infeasible.

On the public transport side, the format is less restrictive, and the data may be collected either at/near bus stops or railway platforms, or, with the operator's permission, interviewers may ride with the passengers and collect information en route. In this later case, there are some sampling issues, since there will be a bias induced by the tendency to interview passengers who remain longer on the vehicle. It is, in any case, critical that the ultimate origin and destination are correctly recorded, not merely the boarding and alighting points of the journey.

In large conurbations, where the public transport system is denser, journeys are more likely to be complicated by the possibility of interchange. As much detail as is possible should be recorded, using the LATS on-board surveys as a model.

Unless there are external reasons for greatly increasing the effort applied in the collection of highway and public transport matrix data, it is likely, in the context of a single study, that the data collected will not cover all possible movements between the zones in the modelled area. This will almost invariably be true for private transport trips, although it may be possible to sample all trips by public transport, so avoiding the problem of having to synthesise any missing public transport trips. A decision then has to be made as to whether these 'unobserved cells' are zero by nature (because the actual amount of travel between the two zones is negligible, and likely to remain so in all conceivable scenarios), or are zero merely by the random process of sampling observations (or possibly through a design flaw or other mishap). In the latter case, it will be necessary to 'infill' in some way.

In line with the earlier discussion, there is nothing inherently inappropriate about attempting to do this, and traffic count data are in principle valuable information. The problems reside in the detailed assumptions that are made in carrying out this process. There is abundant literature on the general procedure of 'deriving matrices from counts': crucial to the success of the exercise is that a substantial amount of individual data is collected, relative to the volume data (see Ortúzar and Willumsen, 1994).

The process of infilling is essentially similar to the construction of the demand model itself. This requires a matrix of costs, which will need to be obtained from an assignment stage. Since this matrix of costs, at least on the highway side, will be influenced by the volume of demand assumed, there is an inherent iterative sequence in all these elements which contributes to the building of the demand model and the current equilibrium ('base situation').

In the case of non-mechanised modes, the prospect of obtaining an acceptable matrix of movements from interviews in course of travel is far worse, for understandable reasons. Since the majority of these journeys are short, they can in principle be collected from diary information collected on a household interview basis. However, unless the area of interest can be heavily restricted, the costs of obtaining adequate coverage are likely to be prohibitive. In most cases, therefore, the matrices for non-mechanised modes will be almost entirely synthetic, built up from simple distance relationships from sources such as the National Travel Survey.

In the multi-modal context, it is obviously important that the data for different modes, whether observed or synthesised, is compatible, i.e. that the implied modal propensities are credible. The aim is to produce a set of base matrices by mode (and other segmentations, in particular purpose) which are both compatible with the demand model of mode and destination choice and consistent with the observed matrices for each mode. This may require further iterations of, and/or adjustments to, the mode and destination choice models.

For the journey to work, the Census data matrices may provide a useful source, both of the volume and pattern of movements, and of the modal proportions (see Data Sources (TAG Unit 3.1.5) ).

The demand model itself will typically be of the logit type (which includes the traditional 'gravity' model for distribution, or destination choice), based on generalised cost. The iterative nature of the process for providing the base matrices poses potentially severe problems for model calibration: in practice, it is standard to derive the cost matrices compatible with an assumed 'reasonable' level of demand, and to then hold them constant while calibrating the demand model.

A particular issue for the calibration is the hierarchical structure between mode and destination choice. A structure of mode choice conditional on, or simultaneous with, destination choice is often adopted. However, there is, if anything, stronger empirical support for a structure of destination choice conditional on mode choice (see, for example, the LTS91 model for London and the CSTM3 model for Scotland). This is typically found when the choice is between public and private modes of travel. Whichever structure is adopted it is essential to ensure that the cost-sensitivity of the primary (upper level) choice is not larger than it is for the conditional (lower level) choice. The following advice is taken from DMRB Vol. 12 Section 1, Chapter 17: There are no overwhelming reasons for selecting a particular modelling procedure a priori, and decisions on modelling adequacy and sophistication must be based on information acquired through observation. Little information is readily available about the performance of modal split models used in past studies. Model validation is, however, an essential part of the modelling process and efforts should, therefore, be made to determine how well the model performs against observed data.

Finally, as has often been pointed out, structures based on simple logit assumptions do not normally produce matrices of movements which accord closely with 'observed' data. In practice, either an incremental approach is adopted, or a sufficient number of constants (or 'K-factors') are added to the model specification to improve the fit. From a theoretical point of view, there is little difference between these two methods.

When the model distinguishes different time periods, the cost matrices are likely a priori to differ by time period, raising a question of which cost matrix should be used in the demand model. If a time of day choice component is also included (though this is virtually never done in a spatially detailed model), then, assuming, as is likely, that it is below the model(s) of mode and destination choice, there is a theoretically preferred approach: the cost matrices should be the 'composite' matrices over the available time periods. In practice, for reasons of simplicity, it is normally assumed that particular purposes are dominant in particular time periods, and the demand model is calibrated on the costs for the selected time periods only. For example, the Home-based Work model is calibrated on peak costs, while Home-based Other is calibrated on off-peak costs, as in LTS91. There are potential incompatibilities here, but they are usually not severe.

Note that prior to assignment, it is necessary to convert from a P/A basis to an O-D basis. This is described in more detail below.

There is much useful discussion of the development of spatially-detailed road traffic and public transport passenger assignment models in Sections 6.10 and 6.11 of the IHT's Guidelines on Developing Urban Transport Strategies, to which the reader is referred.

The calibration and validation stages for the supply element of transport models are even more closely intertwined than for the demand stage. It is frequently the case that the data used for assignment model validation is not truly independent. If the validation process shows the model to be deficient in some respect, then there is often no alternative but to use the validation data as part of any re-calibration process. This is particularly the case where a re-estimation of the matrix is deemed necessary and validation counts are the only available source of data.

• matrix validation against screenline and cordon counts, and against observed trip movements;
• network validation procedures such as the checking of coded link lengths and examination of inter-zonal paths; and
• assignment validation involving comparison of observed and modelled data for link flows; turning movements; traffic queues and journey times.

Although the assignment stage is a route choice modelling exercise, it is rare for observed route information to be available for the model validation process. Assignment models generally need to be run many times before all significant problems associated with network and matrix definitions are removed.

• matrix validation against screenline and cordon counts and observed trip movements;
• network, route and service validation which primarily involves checking that the modelled flow of public transport vehicles is consistent with roadside counts; and
• assignment validation, which involves comparing modelled and observed:
• passenger flows across screenlines and cordons, usually by sub-mode but sometimes at the level of individual bus or train services;
• passengers boarding and alighting in urban centres; and
• vehicle journey times.

Calibration of the model usually involves adjustments to the relative valuation of the generalised cost components, for example, walk time, wait time and interchange penalty. Where logit models are used within the multi-route and sub-mode choice processes, the degree to which they spread trips across competing routes can be adjusted. In common with highway assignment models, there is a tendency for independent validation data to be used within the calibration process, and thus the two stages become closely intertwined.

The management of the equilibrium process for spatially detailed models remains relatively unsophisticated. There is little experience of the use of 'objective functions' (as discussed in Section 1 above), though work with the NAOMI model has attempted to rectify this. Hence convergence is normally attempted by means of, at worst, a standard 'cobweb' (which may not converge), or, at best, some damping procedure applied to the demand estimates.

Within such an iterative sequence, it is often the case that relative arbitrary control procedures are applied: e.g., in relation to how many iterations of a particular process to carry out. The development of appropriate convergence statistics remains rudimentary.

One reason for this is that the demand and supply models for a spatially detailed system are intrinsically separate, and require quite an elaborate interface. In a multi-modal context, the demand models will operate on a person basis, and, as noted earlier, the matrices with which the model deals are in the P/A format. The supply model, on the other hand, is interested in the origin and destination of each particular movement, since the operation of capacity is essentially independent of at which end of the movement the trip is deemed to be 'produced'. Additionally, on the highway side, capacity relates to vehicles, not persons.

• Convert from P/A to O/D.

Typically this is done by applying purpose-specific and time-specific factors, relating to the proportion of outbound and return movements that occur in each time period by purpose. These factors may be assumed constant (usually derived from data which includes little or no spatial variation), or, in more exceptional circumstances, would be supplied, on a policy-specific basis, by a time of day choice model.

• Converting from person to vehicle basis, for the car mode.

Again, this is normally done by assuming a constant occupancy (possibly varying with purpose and/or time of day - this could be derived from the RSI data, or national data such as NTS). In some demand models, however, the choice between car driver and car passenger may be explicitly represented, in which case the vehicle matrices are aligned with the car driver matrices, and the occupancy is derived from the modelled ratio of passengers to drivers.

• Aggregating over purposes.

In most cases, the supply model will not be sensitive to purpose variation, though a possible exception occurs in the case of Business travel, where the value of time for route or sub-mode choice may play a role.

A final possibility for the interface is to allow, at the assignment stage, a modification of the demand model matrices to reflect more closely 'observations' of journeys on the networks, typically by means of sector-specific factors. This can be seen as a variant on the incremental demand procedures discussed earlier.

Given appropriate matrices, these are then passed for assignment to the supply models. The output is, essentially, matrices of generalised cost components (in-vehicle time, waiting time, money costs etc.), which need to be combined into appropriate matrices for the next iteration of the demand model. As noted earlier, there is often a potential incompatibility between the definitions of generalised cost in the supply and demand models, and this requires careful treatment where multiple routes between origins and destinations are allowed for in the supply model. The resolution depends, ideally, on the details of the assignment. From a practical point of view, the best approach is to use flow-weighted averages of the generalised cost components: however, not all software packages permit ready calculations of these quantities.

In the case of highway movements, whether operating costs are explicitly used in the supply model or need to be 'externally' calculated from the time, distance and speed characteristics of the chosen paths, it must be remembered that these are on a vehicle basis, and that the same factors used for the demand/supply interface need to be applied in reverse to place the generalised costs on a person-trip basis.

From the point of view of guidance, in the absence of an 'objective function' approach, it would be best to apply a volume-averaging technique to the demand estimate, prior to carrying out the demand/supply interface, and to test convergence on the cost component matrices.

Section 5.2 provides a summary of the forecasting data that ITEA division will be making available to modellers, and how these fit together to form a standard forecasting framework.

The recommendation is to apply appropriate zonal growth factors to the base matrices in order to derive a 'r