The quality variable represents all quality dimensions other than frequency. The
variable is continuous and has been defined according to the following reference
points:
Currently, it tends to be the local authority that is responsible for quality measures
in regard to shelters, so the question could be asked whether operators engage
themselves in providing quality improvements by investing in infrastructure. In view
of the fact that on a number of routes in our sample, quality level 2 had already
been achieved, it was however necessary also to introduce a higher quality level to
see if the proposed change in subsidy system would provide incentives to these
operators to invest more in quality than they currently do.
One of the insights gained from the first round of interviews was that decisions on
tendered services are often made on the back of the commercial service product.
This applies for example to fares, where the fares charged on tendered evening
services are often the same as those charged during daytime. It may also apply to
quality if the vehicles used to operate commercial services are the same as those
used to operate tendered services on the same route. The model therefore applies
to commercial services only, where the operator has the discretion to set all three
key variables that are used in the model. The implications for currently tendered
and currently marginal services were assessed at a later stage, outside the model.
The model can be set up for any time period; in practice the operators have
supplied data for four-week periods.
The estimation of the demand and cost functions is described in Sections 3.3 and
3.4 below.
3.3.1 Overview
The purpose of the demand function in the overall optimisation model is to specify
the sensitivity of bus demand to the key variables that are at the discretion of the
operator.
A key element of the present study was to set up the model for a number of routes
with various characteristics (e.g. urban, interurban, rural) in order to determine
whether the impact of the proposed change in subsidy system would vary across
different types of route. The model should therefore be flexible and be able to
accommodate various route type characteristics.
A second key element of the study was that the modelled results would not be used
on their own but would be discussed in follow-up interviews with operators. In view
of this, it was essential that the model was able to take local circumstances into
account, for example the local sensitivity of demand to price changes, and the local
potential for patronage growth.
It was therefore decided not to use national or other general parameters on the
sensitivity of bus demand to price, frequency and quality, but to specify the demand
model in a flexible way and to determine the appropriate demand sensitivities in
discussions with the operators (as part of the second round of interviews).
3.3.2 Functional Form of the Demand Model
Given that the model is an optimisation model, a constant elasticity model could not
be used. In such a model, raising prices would (assuming inelastic demand)
always be profitable, so that the modelled profit-maximising price would tend to
infinity and therefore be unrealistic.
This left a choice between two other main functional forms for the demand model:
Of these, the negative exponential form was chosen for use in the study. A
negative exponential form has an elasticity proportional to the absolute fare level
which previous experience in modelling bus demand has shown to be broadly
appropriate. By contrast, a linear demand curve has in the past been found to be
too sensitive to fare. There is in fact a tradition of using negative exponential
demand functions in bus modelling, for examples LT Planning works on this basis.
However, the exact form of equation (2) was amended to model more appropriately
the sensitivity of demand to frequency changes, and to appropriately model the
quality variable.
Additional flexibility in regard to the frequency elasticity was required in view of the
comments made by bus operators about certain routes having limited growth
potential. On these routes, increasing frequency might not result in substantial
patronage growth but it may still be possible to lose substantial numbers of
passengers if frequency is reduced to unacceptably low levels. The flexibility was
introduced by introducing an additional variable j in the following way:
By varying the variable j, the difference between the upside and downside potential
of demand in response to frequency changes can be varied (for example by
introducing B in a quadratic rather than linear form).
The quality variable Z has been introduced in a way that prevents the model from
selecting quality levels higher than three, since such values are undefined. In view
of the robustness of the optimisation results, it was necessary to introduce a
continuous function rather then simply cut off the quality variable at level three.
The desired property was largely achieved in the following way:
This introduces quality as a quadratic form with a peak at Z=4, i.e. demand actually
falls when selecting a quality level of excess of four, and is almost flat when the
value of four is approached. In combination with quality costs rising more than
proportionately with increasing quality, a situation where the optimisation model
selects overly high quality levels is thereby avoided.
3.3.3 Parameter Estimation
For each route, the parameters of the demand function have been estimated taking
two different perspectives into account:
- According to how demand might be expected to respond to price and
frequency changes on the basis of evidence from earlier studies; and
- According to how the relevant operator believes demand will respond to
changes in price, frequency and quality.
Since the existing evidence on the responsiveness of demand to quality is not very
well developed, operator views of the sensitivity of demand to quality were used in
all cases.
The previous demand elasticities were drawn from the work that ITS did for
LEK/CfIT. In Table 3.2 of their final report, the following relevant demand
sensitivities are included:
Table 3.2 - Demand Responses from ITS Work for LEK/CfIT
Although the ITS result suggests inelastic demand (in terms of price) for all route
types under consideration, using these elasticities in our optimisation model was
found to be problematic. When assuming a price elasticity of -0.3 (as in the results
for the large radial route above), the theoretical profit maximising price will, given
the demand function used, produce a predicted profit-maximising price of two to
three times the current price. It was not felt to be sensible to use such a price as
our base position and estimate the impacts of a move from FDR to IPP against
such a benchmark.
An important result from our interview programme was that although operators
generally perceive the short run elasticities on their routes as inelastic, they do not
feel this to be the case in the long run. They believe that they can earn more profits
in the short run by increasing fares above the current levels but feel that due to the
additional patronage falls that they would suffer in the longer term, increasing fares
would not be consistent with building a sustainable business. We therefore believe
that when modelling operators' profit maximising decisions, it is the long run
elasticity values that are the relevant ones to use.[7] In view of this, the model has
been set up with a long-run demand elasticity for small price changes (10 per cent)
of around - 0.9. This compares with lower values used by LEK, who used a
medium run fare elasticity value of -0.8.
The ITS frequency elasticities more accurately reproduced operators' current profit
maximising decisions (taking into account the fact that typically, operators run more
buses than would be profit maximising due to the threat of entry), so these were
therefore retained. It was eventually decided to set the power variable j at 1.5 for
all routes and to calibrate the i variable so as to achieve the elasticity values above
for small frequency changes.[8] This way, a reasonable balance was achieved
between the upside and downside sensitivity of demand to frequency changes.
In the absence of sufficiently robust theoretical evidence on the sensitivity of
demand to quality, operators' views on the impacts of quality measures on demand
were used to arrive at the assumptions contained in Table 3.3. We note that these
quality elasticities are significantly below those assumed by LEK. However, in view
of the fact that we were seeking to predict how operators would respond to the
change in subsidy system, and since operators will base their response on their
own views, we considered it to be important that the assumptions used reflected
operators' views as much as possible.
Table 3.3 shows that decreasing returns to quality have been assumed. This
assumption was necessary in order to make the optimisation model work; if this
assumption had not been made, the model could have produced optimal quality
levels well in excess of 3, for which no quality dimensions had been defined. We
believe that the assumptions in Table 3.3 are, nevertheless, a reasonable reflection
of operators' views of quality improvements. It should also be noted that
investment in bus lanes etc. was not considered as part of the present study as
operators would not undertake this in any case. In the context of the quality levels
as defined above, such investments would have resulted in levels of well over 3,
with substantially higher patronage gains.
3.4.1 Introduction
Like the demand model, an essential feature of the cost model had to be that it
should be based on the actual costs on the modelled bus routes. This would
ensure that the modelled results be realistic for the routes under consideration,
maximising the value of the second interview round with operators.
The cost model has been built on the basis of CIPFA principles. These principles,
which are widely used in the bus industry, imply that costs are allocated to one of
the following three cost drivers:
- vehicle hours;
- vehicle kilometres; or
- peak vehicle requirement.
The model consists of two main elements:
- an analysis of current costs on the route, allocating them to vehicle hours;
vehicle kilometres and peak vehicles; using current levels of vehicle hours,
vehicle kilometres and peak vehicles, unit costs are calculated for each of the
cost drivers.
- an estimation of the number of vehicle hours, vehicle kilometres and peak
vehicles needed to run a given modelled timetable, and calculation of the total
costs using the unit costs estimated above.
These two elements are discussed in the two subsections below.
3.4.2 Current Cost Drivers
A number of cost categories can be entered into the model (which may vary since
operators' accounting methods vary) and each category can then be allocated to
vehicle hours, vehicle kilometres or peak vehicles.
Some operators provided data showing their costs broken down between time
related costs, distance related costs or maximum peak vehicle requirements or
overhead related costs, in which case these costs have been allocated to the
relevant cost driver. Other operators provided a more detailed breakdown of costs
(staff costs, maintenance, depreciation etc.). In these cases, the costs have been
allocated as follows:
- to vehicle hours: staff costs, engineering costs, overhead costs;
- to vehicle kilometres: fuel costs, tyres costs; and
- to peak vehicles: depreciation, insurance, licence costs.
On the basis of the levels of vehicle hours, vehicle kilometres and peak vehicles
needed to operate the current timetable on a route (data supplied by operators),
unit costs for vehicle hours, vehicle kilometres and peak vehicles were then
estimated.[9]
Fuel costs were treated separately since current net fuel cost levels would no
longer be correct if FDR were to be abolished. Fuel costs were therefore made
dependent on the level of FDR: if this is reduced or abolished, current fuel costs
and the cost per vehicle kilometre are automatically adjusted upwards.
3.4.3 Costs to Run the Model Timetable
The second element of the cost model is an estimation of the number of vehicle
hours, vehicle kilometres and peak vehicles required to run a given timetable, and a
calculation of the resulting costs using the unit costs estimated above.
To this end, the model requires a number of base inputs, such as the length of the
route, the average round trip time and the maximum round trip. Values of vehicle
hours, vehicle kilometres and peak vehicles were then calculated using the
following formulae:
Vehicle Hours = avtime * bph * hpd * boardpen * adj
Vehicle Kilometres = 2 * routelen * bph * hpd * adj
Peak Vehicle Requirement = maxtime * bph * adj
Where:
avtime is the average round trip time
bph is buses per hour
hpd is hours per day
boardpen is the boarding time penalty
adj is an adjustment factor
routelen is the length of route
maxtime is the maximum round trip time
The adjustment factor is necessary for various reasons, including such factors as
waiting times at termini and driver training, but also because of issues related to the
necessary simplifications to the model. For example, the timetable could only be
represented in the model as an average frequency per hour. If in fact some buses
terminate short of their final destination, or frequency varies during the day, the
modelled number of vehicle hours, vehicle kilometres and peak vehicles will be
different from the actual ones. Applying an adjustment factor ensured that the cost
model correctly replicated the cost of operating the current timetable on the route.
Our approach assumes that the factors influencing the adjustment factor would
remain unchanged during the optimisation process. However, a number of
important issues arising from the proposed subsidy change (such as the possibility
of some services terminating outside the busy section) were discussed in detail
with the operators during the follow-up interviews.
A boarding time penalty was introduced in the vehicle hours formulae to avoid a
zero marginal cost per passenger in the model, which could lead to undesirable
results when optimising. A time penalty of three seconds per passenger was
used.[10] As boarding time delays are already implicit in the average round trip time
and therefore counted twice, the adjustment factor was set to compensate for this
double counting effect.
3.4.4 Additional Elements in the Cost Model
A number of additional elements were introduced to the cost model to facilitate the
optimisation process. The first additional element is the explicit representation of
quality. Since quality is one of the three key variables examined, it was essential to
have it represented in a realistic way both in regard to its costs and its revenues.
The costs of providing quality have been derived from ITS (2002),[11] in which
estimates of the amortised weekly unit costs of providing quality are contained.
Using these figures, a quality cost function was estimated dependent on the level of
quality provided, the number of stops on the route and the number of peak vehicles
required. To ensure realistic optimisation results, the level of quality was specified
as a quadratic, implying that modest quality improvements are relatively cheap but
that more substantial quality improvements become progressively more expensive.
The second additional element introduced in the cost function was a relief cost
function. The idea of relief costs was also based on the ITS work and assumes
that the operator is required to run relief buses if services become overcrowded.
Although strong assumptions are required to estimate a relief cost function, the
inclusion of such a function in the model is essential to avoid the model producing
profit-maximising results with unrealistically high load factors.
An approach was chosen whereby the relief cost function imposes penalties if the
average load factor selected by the model is significantly higher than the current
average load factor. The current average load factor is influenced by a large
number of factors, including for example the ratio between peak and off-peak
traffic. If traffic is reasonably evenly spread throughout the day and week, it will be
possible to achieve higher average load factors than in a situation where there are
very strong peaks. Whilst a move to IPP may result in patronage growth and
thereby produce small increases in load factors, large modelled increases in load
factor will not be realistic.
The penalties are imposed by multiplying the costs as calculated in Section 3.4.3
by a factor that increases as the ratio of modelled to current load factors
increases.[12] Initially, the penalties need to be very small but then have to become
progressively more severe. This was achieved by modelling the relief cost
multiplication factor as a cubic function, the shape of which is shown in Figure
3.1.13
Figure 3.1 - Form of Relief Cost Function
3.5: Profitability Calculation
The model includes a separate sheet where the profit and loss account is shown.
The revenues are obtained by multiplying the average fare per passenger (which
includes the concessionary fares reimbursement) by the number of passengers, to
which then the total IPP (if any) and any current tendered revenue are added. The
total IPP plays an important role in the model but the current tendered revenue
does not; this is included for information purposes only.
The costs are composed of the core costs as calculated in Section 3.4.3, to which
the quality costs and the relief costs are then added.
The model shows the absolute level of profit and the return on sales achieved.
3.6: The Optimisation Model
The optimisation model, shown in Figure 3.2, uses the Solver routine in Microsoft
Excel and is able to maximise profits by manipulating the price, buses per hour and
quality variables, subject to a number of constraints.
Before optimising, a subsidy scenario should be chosen. By default, it is possible
to choose the current level of FDR (at 80 per cent of duty payable); IPP of 10
pence; or any combination in between (e.g. reducing the FDR from 80 to 40 per
cent and introducing a IPP of 5 pence). It is also possible to change the default
values, for example to evaluate the impact of increasing FDR from 80 to 100 per
cent, or to evaluate the impact of introducing IPP without reducing the level of FDR.
Certain constraints are imposed at every optimisation round. These are that the
price charged and the quality level should be positive. Other constraints can be
imposed at the discretion of the user. There are four possibilities:
- Maximise profit. In this case, it is possible, but not mandatory, to impose the
constraint that the frequency should be an integer. This constraint can be
imposed because in the vast majority of cases, the frequency per hour that
operators run is indeed an integer. However, it is not necessary to impose this
constraint, and allowing frequency to vary continuously allows us to evaluate
how the change in subsidy system will affect marginal incentives to increase
or decrease frequency;
- Maximise profit subject to bus frequency equal to a certain value. With this
option, the desired frequency can be entered and the model optimises by
holding frequency constant and manipulating just the other two variables. This
option is very important if the operator runs more buses than it otherwise
would have done to avoid profitable gaps into which competitors might enter.
It can also be used to evaluate possible strategies where the frequency is not
an integer, e.g. one buses per two hours (in which case the value to be
entered is 0.5);
- Maximise profit subject to price equal to a certain value. This option holds
price constant and manipulates frequency and quality. This option is useful if
the operator chooses to set prices below profit-maximising levels, again
possibly to deter entry or for political reasons;
- Maximise profit subject to quality equal to a certain value. With this option,
only fare and frequency are manipulated. This option can be used if there is
no scope for changing the quality level offered, for example if new buses have
just been introduced on the route which are not going to be modified in the
short or medium term.
The optimisation model automatically records the situation immediately prior to the
optimisation so that the situations before and after the optimisation can be readily
compared. It is also possible to fix a certain outcome as the base case, to which all
subsequent outcomes are compared. The model also records all outcomes on a
separate output sheet.
Figure 3.2 - Optimisation Model
5: LEK (2002) Obtaining Best Value for Public Subsidy for the Bus Industry, Final Report, p8.
6: CfIT have informed us that the safety net proposed by LEK works by compensating bus operators of
all rural and interurban services services for any loss of Day 1 profits that result from the switch from
FDR to IPP.
7: A recent detailed econometric study of the demand for local bus services in England concluded that
the most likely values of the fare elasticity for England as a whole are around -0.4 in the short run,
and -0.9 in the long run: Joyce M Dargay and Mark Hanly "The demand for local bus services in
England" Journal of Transport Economics and Policy, January 2002, pp.73-91, see p.90.
8: The combination of the i and j variables represent the frequency elasticity. Given a value for j of 1.5, the i variable was calibrated such that the combination of the i and j variables resulted for small frequency changes (e.g. 10 per cent) in the frequency elasticities in Table 3.2. These frequency elasticities were sourced from the ITS report to LEK and we expect them as such to be consistent with what one would expect. The principal impact of setting j at 1.5 (as opposed to 1) is to dampen the modelled patronage increases for large frequency increases. This dampening was necessary to ensure that the model resulted in profit-maximising positions at plausible frequency levels.
9: These unit costs are defined as follows: - Cost per hour: (current hours related costs)/current vehicle hours - Cost per kilometre: (current kilometre-related costs)/(current vehicle kilometres) - Cost per peak vehicle: (current peak vehicle-related costs)/(current peak vehicles). These unit costs are different from the unit costs that would result by dividing the total current costs
by the current level of each of the cost drivers.
10: M A Cundill and P F Watts Bus Boarding and Alighting Times TRRL Report 521, 1973.
11: Insititute for Transport Studies Achieving Best Value for Public Support in the Bus Industry - Final
Report to LEK and CfIT (Annex 2) 2002.
12: Given the total number of passengers and the total capacity of each bus, an assumption in regard to
average journey length is required to calculate the average load factor. Without detailed surveys,
the average journey length cannot be estimated in great detail (although bus operators tend to have
abroad idea). However, the fact that the relief penalty is based on the ratio of modelled to current
load factor means that the model is largely insensitive to the assumed average journey length.
13: An alternative would have been to introduce a conditional function that only imposes penalties if the
ratio of modelled to current load factor exceeds a certain value. However, this would have made the
optimisation results considerably less stable. The consequence is that relief costs penalty can be
(slightly) negative if the modelled load factor is lower than at present.
[ Previous ]
[ Contents ]
[ Next ]
