1.1.1 Where an elasticity model is appropriate the functional form and parameter values need to be selected. The simplest functional form - an 'own-cost' elasticity model - assumes that changes in the demand for travel between two points can be adequately estimated purely by a function of the change in costs between the two places.

1.1.2 Own cost elasticity models assume that the demand for travel between two points is purely a function of the change in costs on that mode between the two places. The strength of that function can vary for different trip lengths.

1.1.3 However, if costs do indeed change, the relationship between change in demand and change in costs can take a number of forms, but only exponential and power formulations, and a composite of the two forms (called a Tanner function), will be considered here. With a power formulation the proportionate change in trips is related to the proportionate change in costs, as shown in the equation below. With an exponential formulation, on the other hand, the proportionate change in trips is a function of the absolute change in costs. Other, more complex, relationships are described in VDM Appendices (Unit 3.10.5).

1.1.4 For most applications the Power relationship below which is a simple own cost elasticity model due to its constant elasticity value is recommended:

T _ij = g _ij ^* ₀T _ij ^* (G _ij /₀G _ij ) ^A

Where:

1. T_ij is the forecast number of trips between zones i and j

2. G_ij is the forecast disutility or generalised cost

3. g_ij is the forecast growth rate relative to an earlier or base year

4. ₀T_ij is the number of trips in the earlier or base year

5. ₀G_ij is the disutility or generalised cost in the earlier or base year

6. A is the elasticity, which should be negative and is the same for all trips in the same user class.

1.1.5 This is a well-behaved formulation that is simple to apply, and is base independent: that is, it is guaranteed to give the same results if forecasts are produced from one year to another directly or via an intermediate year. It assumes that a proportionate change in trips is related to a proportionate change in costs. As the parameter A is constant the implied elasticity is the same for all lengths of trip within the same user class (i.e. it is "distance neutral").

1.1.6 This formulation can easily be set up using the matrix manipulation facilities available in modern transportation modelling suites, or in some modelling suites combined directly with the assignment process. The facility is also available within the DIADEM modelling framework.

1.1.7 An alternative formulation is the "Exponential" relationship. In this case the effective elasticity increases with increasing trip cost, and hence for study areas where there are a wide variety of trip lengths the effective elasticites could vary markedly. Thus the exponential approach should only be considered in the case where the study area is small and urban, and where a general elasticity approach is being combined with a logit choice mechanism to jointly represent the individual demand mechanisms. Most logit mechanisms in the variable demand hierarchy share this exponential function characteristic, but some have a more benign effect since trip re-distribution, for example, can be constrained to avoid changing the overall number of trips. In that case trip re-distribution in the face of changing travel costs effectively adjusts the proportions of trips of different length to compensate for the changes. Similarly mode-choice models estimate shares rather than absolute numbers.

1.1.8 The "Exponential" formulation assumes that the proportional change in trips is a function of the absolute change in costs:

T_ij = g_ij ^* ₀T _ij ^* exp {B^*(G_ij - ₀G_ij)}

Where the elasticity of demand with respect to generalised cost U is B*Gij with B negative. This is an own cost elasticity which is not simple due to the elasticity not being constant.

1.1.9 These equations can be used in two ways. They can be based or pivoted on a base year, where items 4 and 5 in paragraph 1.1.4 (base trips and costs) are known from empirical data, and the product g_ij ^* ₀T_ij represents what is referred to as the Reference Case Matrix, see VDM Scope of the Model (Unit 3.10.2). Alternatively, the equations can be formulated to compare costs between alternatives for the same year, where the 'earlier' year costs and trips are derived from the other scenario. More details on how these equations are to be used are given in VDM Appendices (Unit 3.10.5). This should be read in conjunction with the guidance given with the software employed to undertake the transport modelling (see also VDM Convergence Realism and Sensitivity (Unit 3.10.4).

1.1.10 Both these formulations are closely incremental in nature, allowing the number of trips in the system to change up or down. This is in contrast to most of the individual demand-response mechanisms that are set out as share formulations where the total number of trips is fixed (say by TEMPRO all-mode forecasts) and merely allocated to one choice or another (e.g. to different modes or destinations).

1.1.11 Whilst the formulation is relatively easy to set up, there are some issues that must be dealt with when considering the parameter values to assign to a demand segment.

1.1.12 'Own cost' elastic assignment modelling in congested Urban Areas should be undertaken at a peak hour unless there are significant variations in demand, or congestions levels are high in which case the modelling should cover linked time-periods, sub-divided into time slices and sub-periods.

1.1.13 The size of the parameter value will reflect the number of responses that the elasticity formulation is acting as proxy for. For instance, if the elasticity formulation is taking the place of all responses then it will be larger than if it is acting as proxy for only one or two responses. The table below sets out the recommended starting values for the elasticity of demand with respect to journey time.

Table 1: Derived long-term journey time elasticites for different uses (derived from 1997 DMRB Vol 12 Section 2 Part 2 Table C2 and its para C13)

Purpose	Time elasticity - High modal competition	Time elasticity - Low modal competition	Time elasticity - High modal competition including time-switching	Time elasticity - Low modal competition including time switching	Trip frequency effect (only)
HB Work	-0.22	-0.14	-0.48	-0.30	-0.04
Employer's Business	-0.60	-0.35	-0.96	-0.55	-0.15
Essential Other	-0.47	-0.26	-0.65	-0.36	-0.12
Discretionary Other	-0.35	-0.20	-0.50	-0.28	-0.10
Note: The values are based on car journey time elasticities - equivalent generalised cost elasticities would be about 10-50% higher, depending on the value of time and average network speed. Short-term elasticities are 28%, 8% & 5% less for HBW, Employer's Business and Discretionary purposes.

1.1.14 Equivalent journey cost elasticities can be calculated from the above table by dividing the elasticities by the proportion of the total generalised cost made up of journey time. For instance, if a model assigns on the basis of generalised cost (t+kd), the appropriate time elasticity must be multiplied by a factor (1+kv) where v is the average speed in the base year in kilometres per minute if journey time (t) is in minutes and distance (d) is in kilometres. Values of Time and Operating Costs (Unit 3.5.6), can be used to provide the relevant factors for given combinations of purpose, forecast period and congestion level if standard values of time are being used. In practice, the generalised cost elasticities will be between 10% and 50% higher than the values shown in the table above with values at the lower end for Employer's Business trips, urban areas and later forecast years.

1.1.15 If an exponential formulation is used then the above values will need to be subsequently divided by the mean generalised cost to give the equivalent parameter value.

1.1.16 The estimated generalised cost elasticities (and associated parameter values if an exponential model is used) may need to be adjusted so that the fuel cost elasticity estimate from the model reflects the national overall estimate of -0.3 (see Variable Demand Modelling - Convergence Realism and Sensitivity (Unit 3.10.4)).

1.1.17 Where possible, the trips should be split by trip purpose (and any other known major variation such as willingness to pay or movement type). If this is not possible, for instance where only a single private vehicle user class is available, then they should be split by time-period. VDM Appendices (Unit 3.10.5) shows how, by using the national car driver journey purpose mix for each period of the day (from NTS), the above elasticities can be converted to elasticities for all trips by time period. If local data suggests a significantly different mix of purposes by time-period, then the local proportions can be substituted for the national ones.

1.1.18 Care should be taken when dealing with intra-zonal trips. Because most assignment models do not output intra-zonal costs (since intra-zonal trips are not assigned) there may be problems with using incremental models where there are observed intra-zonal trips in the base year trip matrix. It is desirable that robust estimates of intra-zonal costs should be estimated in these instances. These could be some function of the inter-zonal costs, for example half the minimum inter-zonal costs for that zone. Further advice is given in section VDM Scope of the Model (Unit 3.10.2). Power function elasticity models will be particularly sensitive to very small intra-zonal costs, and this is one reason why they should be avoided when this is the case.

Appendix 2: Functional Forms for VDM

2.1 Detailed Advice on Functional Forms of VDM

2.1.1 There are various mathematical functions that can provide a suitable relationship between travel demand and the disutility or generalised cost of a trip. These all offer broadly similar behaviour, but have subtly different mathematical properties. Section 1.1 of Appendix 1 discussed the equivalent subtle differences of power functions and exponential functions for own cost elasticity models, both of which can provide a convenient downward sloping relationship as shown in Figure 1 in Section 1.3 and the parameters can be adjusted to give an elasticity of any required strength.

2.1.2 Most of the mechanisms required in a variable demand model allocate trips between a set of choices, giving rise to the slightly more complex relationship of Figure 2 in Section 1.3. Here again a range of mathematical functions, most based on powers or exponentials, or both, can recreate the desired relationship. VDM Appendices (Unit 3.10.5) discusses the detailed functional forms of VDM models and the derivation of many of the forms, while for example Ortuzar and Willumsen (2001) provides even more detail.

2.1.3 The multinomial logit choice function is one of a number of possible formulations of "random utility" models in which a random component is added to the deterministic utility of choice p as follows:

U_p = ∑_n β_n x_n + ε_p

Where the utility (or disutility) U_p of choice p is calculated as the sum of:

the generalised cost G_p = ∑_n β_n x_n of choice p, with the set of cost components x_n weighted by coefficients β_n, summed over all components relevant to choice p as explained in VDM Scope of the Model (Unit 3.10.2). (For example, if G is measured in units of time, x might be the money cost of a journey and β the inverse of Value of Time), and
a random component ε_p used to represent variations in the situation or tastes of individual travellers, or modelling errors, or unobserved elements of the alternative choices. (In the most general case this random component can depend on both the traveller and on the choice alternative).

A choice-specific calibration constant (i.e. constant specific to the mode or areas used in calibration) could be added to the generalised cost function to adjust the calculated choice to the observed value.

A random utility model assumes that the alternative with the maximum utility (or minimum disutility) is chosen, so that a probabilistic model results.

2.1.4 The assumed statistical distribution of the error terms or residuals ε_p determines the exact mathematical formulation. For example, assuming one particular distribution for the random components, that they are Independent and Identically Distributed (IID) extreme value variables, leads to the widely-used multinomial logit model:

T_p = T_tot exp(-λU_p)/{∑_q exp(-λU_q)}

2.1.5 Conventionally, different Greek symbols have been used for this sensitivity parameter λ according to the mechanism it is applied to (for example α for trip frequency, and β for mode split), but usage varies and here we will use the λ formulation for all applications, distinguishing between the different mechanisms of variable demand by a subscript.

2.1.6 The elasticity of demand in this formulation is -λU_p(1-T_p/T_tot), so that the elasticity scales with U, and tends to be larger for longer trips for a given value of λ and larger for choices with a small share of the total. If those implications are inappropriate for the model area a different functional form or a series of calibration areas should be used to produce a model with suitable implications.

2.1.7 Other forms such as the power function or the Tanner function, which have been described in relation to 'own cost' elasticity models, or formulations assuming a normal distribution of error terms (Probit models) are possible but little used in modelling for scheme appraisal. However, different formulations of the logit model which have less restrictive statistical assumptions are also possible and are being investigated in current research (See VDM Appendices (Unit 3.10.5) for more details).

2.1.8 The remainder of this Advice focuses on hierarchical and multinomial logit. The logit formulation (and its nested variants) can be used, in slightly different formulations, for each of the mechanisms of the variable demand model. These are discussed in turn below.

Trip Generation and Frequency

2.1.9 In the trip generation stage the numbers of trips of different kinds made from each zone can be predicted as a function of the numbers of different types of households and inhabitants in the zone. For the purposes of this advice, however, we are interested primarily in how the externally estimated demand from each zone might respond if travel costs change, i.e. trip frequency change. The growth factors derived for fixed OD matrices usually assume that travel costs would be unchanged from the base situation. If so they are known as a reference case and are a suitable starting point for variable demand modelling. If as part of that modelling trip generation is to be made responsive to changes in travel cost (see Section 1.4) the requirement is to make trips in any category elastic to changes in cost and thus model trip frequency.

2.1.10 The elasticity function could be a power function or an exponential function as described in Section 5.1. However, if logit is used for the other mechanisms, a similar exponential function is generally used to adjust trip frequency. In this case, the function operates simply as an elasticity with respect to disutility or generalised cost, since the relevant choice is to travel or not to travel, and the disutility of not travelling remains constant:

T_i = ₀T_i exp(-λ_freq G_icomp)/exp(-λ_freq ₀G_icomp)

Where T_i is the number of trips from origin zone i, prefix 0 denotes the base values, λ_freq is the choice sensitivity parameter for the trip frequency stage and the generalised cost G_icomp is the composite cost or disutility calculated across the trip origins.

Composite Costs

2.1.11 In the above and equivalent equations the disutility (or generalised cost) G_icomp must be calculated to represent the "average" perceived cost, or compound cost, across all alternative choices available at lower levels in the hierarchy. Thus for trip frequency the compound cost of travel for trips from a given zone must be calculated across all available choices of destination zone, mode and time-period if the latter is included. G_icomp is intended to provide an estimate of the likely average cost from zone i and incorporates the probability of making each choice, to give a "logsum" cost as described in Section 1.9. This calculation is applied to each category of travel (e.g. trip purpose by SEG) in each origin zone separately. When a scheme is introduced, only trips from those zones where an appreciable proportion of trips experience the scheme, or its surrounding effects, will be noticeably affected.

Trip Distribution

2.1.12 Distribution models spread the generated trips over the available destinations, depending on the generalised cost of reaching that destination. Early demand modelling based distribution on distance rather than cost, and often as a simple negative power function of distance. These were known as "gravity" models in analogy with the gravitational attraction between masses, but use of logit functions based on utilities or generalised costs is now almost universal in demand models. In addition to cost, distribution also depends on some measure of the attraction of a zone, estimated in terms of the numbers of "opportunities" such as jobs or retail floorspace in the zone.

2.1.13 Most distribution models are designed to guarantee that the total number of trips from the origin zone (or to the destination zone) is equal to the total number of trips for that zone forecast at the trip generation/frequency stage. If the design guarantees that property for both origins and destinations the model is known as doubly constrained.

2.1.14 Some distribution models do not depend on travel costs and merely estimate future OD matrices directly from a base-year matrix and the future row and column totals. These methods are known as Fratar or Furnessing and are used to provide reference case growth factors for movements between zones to reflect the forecast growth in zonal trip ends. Apart from that role in providing inputs to variable demand modelling they are not relevant in the context of multi-stage demand modelling.

2.1.15 The general form for a doubly-constrained distribution model is:

T_ij = a_i^*b_j^*O_i^*D_j^*f(G_ij)

Where:

T_ij is the number of trips from zone i to zone j,

O_i is the total number of trips originating in zone i and

D_j is the total number of trips ending in zone j

T_ij depends on the travel disutility or cost via the deterrence function as f(G_ij),

Which in most models is a logit function

exp(-λ_dist G_ijcomp)/{∑_kexp(-λ_dist G_ikcomp),

Where G_ijcomp is a composite cost calculated across the available modes and time periods, if these choices are to be calculated after distribution

a_i and b_j are balancing factors which are only used when the model is singly or doubly constrained (see Section 1.17) to ensure that ∑_j T_ij = O_i (ie there are O_i trips originating in zone i), and ∑_i T_ij = D_j (ie there are D_j trips ending in zone j), and are calculated at each iteration of the constraining routine as a_i = 1/∑j B_j D_j f(G_ij) or b_j = 1/∑i a_I O_I f(G_ij)

(In the above equation, T_ij is proportional to O_i, the total number of trips originating in zone i,T_ij is also proportional to D_j, the total number of trips ending in zone j, or alternatively in a singly-constrained model the facilities available in zone j (numbers of jobs, retail floorspace, etc) so that the number of trips ending in j depends also on the competing attractivities of other zones. (See VDM Scope of the Model (Unit 3.10.2) for conversion from Production/Attraction matrices to O/Ds))

Mode Choice

2.1.16 For mode choice, trips between each origin-destination pair of zones are allocated to the available modes according to the composite disutility or generalised cost of travel by that mode:

T_ijn = T_ij exp(-λ_mode G_ijn)/{∑_mexp(-λ_mode G_ijmcomp)}

if mode choice is the only demand response; and

T_ijn = T_ij exp(-λ_mode G_ijncomp)/{∑_mexp(-λ_mode G_ijmcomp)}

if mode choice is a more sensitive response than distribution. T_ijn is the number of trips choosing mode n from a set of modes m and λ_mode is the choice sensitivity parameter for the trip mode stage. The composite cost G_ijncomp is calculated across the time periods in a way that weights the average according to the probability of choosing that period. The summation is across all available modes m, including n. However, if mode choice is less sensitive than distribution, the composite cost G_imcomp must be calculated to forecast an overall modal split for each origin zone. If there is more than one public transport mode it is usual to use a nested or hierarchical model, with a higher level split between car and public transport (and possibly slow modes also). The allocation to the different public transport modes (and between walk and cycle if modelled) is then made at a lower level (see Section 1.9) or possibly in assignment.

Time of day choice

2.1.17 Macro time period choice (or the allocation of trips between broad time periods) assuming this is the most sensitive response takes the form:

T_ijms = T_ijm exp(-λ_time G_ijms)/{∑_texp(-λ_time G_ijmt)}

Where T_ijms is the number of trips between zones i and j by mode m in time period s. G_ijmt is the disutility or generalised cost of travel between zones i and j by mode m in time period t, which may typically be peak and inter-peak and λ_time is the choice sensitivity parameter for the time period stage. However, if it is above mode choice and distribution, it would take the form:

T_is = T_i exp(-λ_time G_is)/{∑_t(-λ_time G_it)}

Where T_is is the number of trips in zone i in time period s and G_is is the disutility or generalised cost of travel in zone i in time period s.

2.1.18 It should be noted that the sequence of responses given in this section is arbitrary and should not be taken as the recommended structure.

2.1.19 Research is underway into the modelling of micro time period choice to improve the robustness of current models. Research results can be found at www.dft.gov.uk/pgr/economics/rdg/.

Appendix 3: Example Estimation of Modal Split

3.1.1 Modal split for those travellers who have a car available is likely to be substantially different from the overall split across all travellers. The method described below is very approximate, but it gives a general indication of how important the alternative public transport is likely to be (or for any alternative mode, since the principle is the same).

3.1.2 First, it is necessary to estimate the generalised costs by the car and non-car modes (bus in this example) for trips affected by the scheme being assessed. This may involve several different groups of traffic movements, in which case the estimate should be made for an "average" journey in each group (though, for the purposes of this exercise, the judgement of an "average" or "typical" journey can be very approximate). The generalised costs are as follows:

Car Travel: G = 2*A + 60*D/V + D*VOC/(occ*VOT) + PC/(occ*VOT)

where A is the access time at both ends of the trip: since this will generally be walked, it is weighted by 2; D is the mean journey distance in kms; V is the mean traffic speed in kph; occ is the mean car occupancy; VOC is the mean car operating cost in pence per km; and VOT is the value of time per person in pence per minute; PC is half the mean car parking cost in pence. G is measured in minutes in the calculations described here.

Bus travel: G = 2*Walk + 2*Wait + 60*D/V + Fare/VOT + I

Where Walk is the sum of the time spent walking to the stop or station at the origin end of the journey and the time spent from the destination stop or station to the actual destination, and Wait is the mean time spent waiting for the service, which will be half the service interval for frequencies of two per hour or better, rising to a nominal 15 (minutes for less frequent services. Walk and Wait are weighted by 2 since this time is costed highly. V is the mean journey speed, including stopping. The mean fare appropriate to those travellers likely to choose between car and public transport (i.e. probably not concessions) should be for a single journey divided by the mode-specific value of time as for car. If access is by Park and Ride it will be necessary to add in the car access generalised cost, including half any parking fee, and if access at either end of a rail journey is by bus the extra generalised cost should be calculated as described here and added on. If the journey requires interchange between services the extra wait time (and walk time if relevant) should be included and an interchange penalty of, say, 6 minutes per interchange added.

3.1.3 Taking as an example the following cost components:

	Car trip	Bus trip
Access walk - both ends (mins)	3	10
Wait time (mins)	NA	8
In-vehicle time (mins)	30	50
Fuel, fares and other costs (pence)	160	250
Park cost - half of charge per one-way trip(pence)	0	NA
Interchange penalty (mins)	NA	5
Mean occupancy	1.3	NA
Generalised cost (mins)	51.4	122.2

Then the generalised cost in the last line is obtained via the calculations above, using a Value of Time of 11 pence/minute.

3.1.4 The notional modal split is calculated as:

P_PT = exp(-λ_modeG_PT)/{exp(-λ_modeG_PT) + exp(-λ_modeG_car)}

Where P_PT is the proportion of travellers choosing public transport, G_PT and G_car are the generalised costs of travel by public transport and car respectively, and λ_mode is the mode choice sensitivity parameter, (which can be given a value of say 0.04), unless there is local knowledge of the prevailing value. This gives 5.5% to bus, but note that this calculation does not include any mode-specific constant beyond that implied by any mode specific value of time: it is not possible to generalise about the values of these constants, but they are generally found to reduce the share to public transport, so the 5.5% estimated in this case is likely to over-estimate the use of bus by car-available travellers.

3.1.5 Consider a highway improvement that saves 1.5 minutes in journey time for road traffic. The bus has to stop periodically and with acceleration and deceleration cannot make full use of the higher road speeds. Assume it gains only half this time, 0.75 minutes. Then in the "after" case, the car generalised cost falls to 49.9 minutes, and the bus generalised cost to 121.5 minutes. The mode split to bus falls to 5.4% whereas, if it had been assumed that the bus journey time had not changed, the reduction in car generalised cost would reduce the mode split to bus to 5.2%. Thus in this example inclusion of the effect on bus times means that the modal share hardly changes from the "before" situation.

3.1.6 However, although the effect on mode split is very small, this gain in journey time for bus travellers may account for an appreciable part of the total economic benefit. Car-available travellers account for only 5.4% of the total flow, and with only half the saving in travel time they account for only 2.7% of the benefit to car users. However, bus users who do not have a car available also gain this benefit, and they are likely to be much greater in number than the car-available bus users.

3.1.7 Such detailed assessment of the impact on public transport, in the absence of a full public transport modelling will be largely confined to circumstances where no public transport alternative is being proposed but the impacts on public transport of the scheme (or the congestion in the without scheme scenario) are thought to be significant.

Appendix 4: Incremental Model Formulation

4.1.1 When specifying an incremental hierarchical logit model, scaling parameters as provided in section 1.11.15 could be used. These parameters refer to the probability of nests of alternatives or composite alternatives. They reflect the ratios of the lambdas for different response mechanisms as you move up the model structure. The scaling parameters are applied to the logsums of the composite or nested alternatives. They should have a value between 0 and 1 if the responses have been included in the correct order in the model, such that the sensitivity of the responses changes down the hierarchy from lower to higher.

4.1.2 The standard incremental multinomial logit model is given as:

mathematical formula

where

	is the forecast probability of choosing alternative i
	is the reference case probability of choosing alternative i (calculated from the input reference demand)
	is the scaling parameter (always =1 for the bottom level of the hierarchy)
	is the change in the utility of alternative i

For the choice at the bottom level of the hierarchy the change in utility is given by

where

	is the reference generalised cost and
	is the forecast generalised cost, skimmed from the latest assignment
	is the spread or dispersion parameter (defined by the user); it should be negative

For choices above the bottom level of the hierarchy the change in utility is the composite change over the alternatives in the level below:

mathematical formula

4.1.3 This model formulation can be used for mode choice, time period choice and singly constrained distribution.

A modified version of the logit model is used for doubly-constrained distribution as follows:

mathematical formula

where

	is the forecast number of trips travelling from zone i to zone j
	is the reference case number of trips travelling from zone i to zone j
	is the number of trips travelling from zone i
	are destination-based constants, normalised so that is equal to the number of zones

Note that destination constraints are summed over all person types within a purpose, and across all modes and time periods, if those choices have been modelled.

The change in composite utility for origin zone a is calculated using:

mathematical formula

4.1.4 The illustrative parameter values currently provided in Section 1.11 can be used in an incremental model structure as follows:

Suppose we assume the follow choices available

Single trip purpose (say commuting) split into:
Two person types (say car available and car not available)
Car available hierarchy (from top to bottom): frequency, mode choice, time period choice, distribution (doubly constrained)
Car not available hierarchy (from top to bottom): frequency, time period choice, distribution (doubly constrained)

Inputs

4.1.5 Inputs to the demand model are:

	reference generalised cost from zone i to zone j by mode m in time period t, trip purpose p, person type c
	corresponding forecast generalised cost, skimmed from latest assignment
	corresponding reference demand, defined via the user interface

In all the above, there is no data for the highway mode for the no-car person type.

Bottom level utilities

4.1.6 The first step is to calculate the change in utility for the lowest level of the hierarchy:

Where is the mode-specific distribution λ parameter.

Doubly-constrained distribution

4.1.7 Since the lowest level is a doubly constrained distribution model we need to find the balancing factors B_jp. This requires solving the set of equations given by:

mathematical formula

such that the destination trip end constraints are met:

The destination constraints are calculated from the reference demand matrix:

mathematical formula

Note that the destination trip end constraints depend on destination and trip purpose only.

The balancing factors are normalised so that

where N is the number of destination zones.

On the first iteration only of the demand model the origin trip ends are calculated from the reference demand matrix:

mathematical formula

For subsequent iterations they are obtained from the application of the conditional probabilities described below.

Composite Utilities

4.1.8 The change in the composite utility from the distribution, time period choice and mode choice stages is then calculated:

mathematical formula

The reference case probabilities are calculated from the input reference demand as follows:

mathematical formula

Conditional Probabilities

4.1.9 Having calculated the change in the composite utilities it is possible to calculate the conditional utilities for each level of the model.

mathematical formula

Updated Trip Matrix

4.1.10 The application of the conditional probabilities gives an updated trip matrix

and updated origin totals:

Application of Frequency Model

4.1.11 The frequency model is only applied after the above process has converged. This gives the final trip matrix from the demand model:

Appendix 5: Absolute Model Formulation

5.1.1 The illustrative parameter values currently provided in Section 1.11 can be used in an absolute model structure as follows:

5.1.2

Assumed nesting

Layer 1 (Highest):	Frequency
Layer 2:	Main Mode
Layer 3:	Macro Time Period
Layer 4 (Lowest):	Destination

Notation

Trip Origin	i	Trip Destination	j, k
Macro Time period	t, s	Main Mode	m, r
Trips	T	Generalised Cost	G
Distribution parameter	λ_dist	Attraction Factor	B
Composite Utility	U	Tree parameters	θ_time, θ_mode, θ_freq
Pivot (reference) Trips	₀T	Pivot (reference) Utlities	₀U

Composite utilities

5.1.2 The incremental composite utilities summed over the choices in the destination layer are given by:

mathematical formula

Initial values for the attraction factors Bj are needed (see notes given later about the destination choice probabilities).

5.1.3 The composite utilities summed over choices in the time period layer are given by:

This uses the scaling parameter θ_time which reflects the ratio of the lambda for macro time period to the lambda for distribution.

5.1.4 The incremental composite utilities summed over the main mode layer are given by:

These composite utilities are used to calculate the choice probabilities in the various layers as follows. Where required, base utilities can also be calculated from the same composite utility formulae given above, but using base values for the generalised costs and balancing factors.

Choice probabilities

5.1.5

Layer 1, Frequency:

mathematical formula

Note that this calculation makes use of a reference utility value.

Layer 2, Main Mode Choice (m):

mathematical formula

Layer 3, Macro Time Period Choice (t):

mathematical formula

Layer 4, Destination Choice (j):

mathematical formula

Notes

All distribution models satisfy the constraint:

For doubly constrained destination choice models B_j needs to be calculated to satisfy the additional constraint:

Some models employ area specific, mode specific, and time period specific constants and/or sensitivity parameters which vary by zone or zone pairs. Advice on these matters can be found in sections 1.7.14 to 1.7.16.

Updated: April 2009

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