Embed Size (px)
Citation preview
TRANSPORT and ROAD RESEARCH LABORATORY
Department of the Environment Department of Transport
TRRL LABORATORY REPORT 912
THE EFFECT OF TRAVEL COSTS ON THE DESIGN OF HIGHWAY ALIGNMENTS
by
J Broughton BSc PI1D
Any views expressed in this Report are not necessarily those of the Department of the Environment or of the Department of Transport
Access and Mobility Division Transport Operations Department
"Transport and Road Research Laboratory Crowthorne, Berkshire
1979 ISSN 0 3 0 5 - 1 2 9 3
Abstract
1.
2.
3.
4.
CONTENTS
Introduction
The maximisation of the economic value of a highway scheme
2.1 The economic value of a highway scheme
2.2 An economic criterion for use in highway design
2.3 The implementation of the new criterion
The inclusion of traffic costs in MINERVA
3.1 The Traffic Cost Model
3.2 The new objective function used with MINERVA
An optimisation with traffic costs included
4.1 The test scheme
4.2 The vertical alignment which minimises the construction cost
4.3 The optimisation of the scheme to maximise its economic value
4.4 Summary of results for the test scheme
4.5 Limits on gradient
4.6 The cost implications of the new objective function
5. Conclusions
6. Acknowledgements
7. References
8. Appendix 1 :
9. Appendix 2:
9.1
9.2
Notation
The Traffic Cost Model
A model for calculating travel costs
The Growth Factor
Page
1
1
2
3
3
4
6
7
8
9
9
10
10
13
14
14
15
16
16
23
24
24
25
© CROWN COPYRIGHT 1979 Extracts from the text may be reproduced, except for
commercial purposes, provided the source is acknowledged
Ownership of the Transport Research Laboratory was transferred from the Department of Transport to a subsidiary of the Transport Research Foundation on 1 st April 1996.
This report has been reproduced by permission of the Controller of HMSO. Extracts from the text may be reproduced, except for commercial purposes, provided the source is acknowledged.
THE EFFECT OF TRAVEL COSTS ON THE DESIGN OF HIGHWAY ALIGNMENTS
ABSTRACT
The alignment chosen for a new road will affect the travel costs of all drivers using it. In current practice, travel costs are taken into account when making an economic evaluation of a new road but are often not included explicitly when designing the alignment of the road. This inconsistency may lead to the construction of a new road to a design which does not represent the best balance between the costs of travel and of construction.
In this report a simple method is described for allowing the expected travel costs of a new road to influence the design of the alignment. The con- version of an existing vertical alignment optimisation computer program to take account of travel costs is described.
The new program has been applied to a current highway scheme with a vertical alignment that had been optimised to minimise the scheme's construction cost. The highway considered passes through hilly terrain, and it is shown that an improvement in the economic value of the scheme can be achieved by introducing travel costs into the design process.
1. INTRODUCTION
The Highway Optimisation Program System 1 (HOPS) has been developed at TRRL to assist the highway
engineer in the choice of an optimal vertical alignment for a road design whose horizontal alignment has
been previously defined. The system has proved successful in reducing construction costs but it has been
felt that, in common perhaps with more conventional design techniques, there was insufficient scope for
taking account of the travel costs incurred by the future users of the road.
The aims of reducing construction costs and travel costs are often in conflict when designing the
alignment of a new road. The minimisation of construction costs will find the cheapest alignment which
is suitable for the use of traffic and which obeys such constraints as maximum permissible gradients. In
contrast, the minimisation of travel costs willprobably be achieved by a road of constant gradient joining
the end points. An optimal design must lie between these extremes, balancing increases in construction
costs above the minimum against savings in travel costs.
Withey 2 has studied the possibility that road users were penalised when the design for the vertical
alignment of a length of motorway was optimised specifically to minimise the costs of earthworks. The
motorway passed through gently undulating country and Withey concluded that "the savings of earthworks'
"costs exceeds any increase in vehicle operating costs by 20 to 1 at least and the decision not to include
vehicle operating costs in the optimisation ~rocedure is justified". However, Withey's test was based on a
motorway with a maximum permitted gradient of three per cent, and it was felt that his conclusion was
mainly applicable to roads with such low gradients. Some roads have much higher gradients and accordingly
a method has been devised for introducing travel costs into the alignment optimisation. This allows inter-
action between construction costs and travel costs, leading to a design that offers the best compromise
between economies in the two types of cost.
Drivers using a new road in the course of a journey will incur lower costs than if only the original
road network were available. Such savings in cost contribute to the overall benefit derived from a new
road, and the value of a proposed road is currently established by comparing the expected benefits with
the costs of construction, using the techniques of cost-benefit analysis. Thus, the optimal design for a new
road should achieve the best balance between costs and expected benefits, and the minimisation of
construction costs is only one aspect of the optimisation. It may be possible, once construction costs
have been minimised by 0ptimisation, to further improve the scheme by making changes to the alignment
which, in return for an increase in construction costs, offer a greater increase in traffic benefits. Such a
balance between costs and benefits is discussed in this Report, and a criterion for choosing a best design
is developed that embodies the economic principles of the COBA program 3. This leads to a convenient
form of optimisation procedure and the method has been incorporated in a specially modified version
of MINERVA 4, a computer program in the HOPS system which optimises vertical alignments to minimise
construction costs.
The modified version of MINERVA has been tested on a current road design for a rural by-pass
through hilly country. This test was made to illustrate the effect of the new optimisation procedure on
vertical alignment design. The results of the test are presented in this Report.
Although MINERVA optimises only vertical alignments, the same method of treating travel costs can
be applied in the simultaneous optimisation of the vertical and horizontal alignments of a new road. The
influence of travel costs on the design are greater when the horizontal alignment may be varied, as the
opportunity exists of shortening the road and "so reducing travel distances; however, this is not covered in
detail in this Report, which deals mainly with vertical alignment optimisation. The method has been
implemented in NOAH, a horizontal alignment optimisation program that has been developed at TRRL.
A test of NOAH on a major motorway scheme has been reported 5 and gives further evidence of the effect
of travel costs on highway alignment optimisation and on the economic evaluation of a scheme.
2. THE MAXIMISATION OF THE ECONOMIC VALUE OF A HIGHWAY SCHEME
The economic value of a highway design depends on a wide range of factors and so it is generally complicated
to evaluate. In Section 2.1 the methods currently used are summarised, and in Section 2.2 the use of these
in highway design is described. A simplified form of the current economic criterion for selecting the best
from a group of designs is developed. This is applied in Section 2.3 to the question of alignment optimisation,
and it is shown that in this case only construction and travel costs need be considered when finding the
alignment which maximises the economic value of the scheme.
The definition currently used for the economic value of a new road excludes any effect which at
present cannot be costed in a generally accepted manner, so social and environmental factors are excluded.
These are, however, often significant and if suitable methods of costing become available then it will be
possible, in principle, to include them in the framework described below, to affect the alignment optimisation
directly. In the absence of such methods, various strategies can be adopted to take account of these effects
indirectly during optimisation.
2
2.1 The economic value of a highway scheme
The purpose of constructing a new road is to redistribute traffic over the surrounding road network
in order to achieve a series of objectives, including alleviation of traffic congestion, a reduction in the
number of accidents, shorter travel times and reduced vehicle operating costs. The economic merit o f the
new road is currently based on the expression of the last three of these items in financial terms, and the
benefit to the community for each item is then calculated by comparing the original situation with that
resulting from the construction of the new road. For example, the benefit to vehicle operating costs in
each future year is calculated on the basis of predictions for that year of traffic flows over the original
network and over the extended network which includes the new road. Total vehicle operating costs are
calculated for the two networks, and the benefit is estimated by calculating the reduction in cost.
The benefits will naturally be spread over a number o f years, and the method of discounting is used
to express future benefits in terms that are compatible with the construction costs. A discount rate r is
laid down, and any benefit obtained i years after construction is divided by (l+r)i; thus, the later a benefit
is experienced the less it is worth in comparison with the cost o f construction. The discounted benefits
are then accumulated over a lengthy period, usually the first thirty years o f operation, to give the "overall
benefit" of the scheme.
This overall benefit is set against the cost of constructing the new road, to see whether it represents
a satisfactory return on the capital investment. Two important economic indicators are:
Net Present Value (NPV) = Overall Benefit - Construction Cost
NPV/C Ratio = Net Present Value/Construction Cost
The NPV/C ratio of a highway scheme measures the benefits expected by the community as a
proportion of the capital investment, and it plays an important part in determining the ranking of the scheme
within the road programs in Britain. In particular R, a minimum acceptable value for the NPV/C ratio, is
specified by the Government and any scheme whose ratio is less than the current value o f R will be rejected
or postponed, unless there are overriding considerations.
2.2 An economic criterion for use in highway design
It is normal when designing a new road for the engineer to consider a number o f different alignments,
one of which will eventually be chosen for construction. This choice will be influenced by many criteria
including economic and technical factors, environmental considerations and the views of local residents.
The economic criterion that has been recommended is based on the Incremental NPV/C Ratio, an index
used to compare the economic merits o f two alternative alignments. In order to define this ratio, suppose
that alignments s and t are to be compared, where t is more expensive than s and where construction costs
and overall benefits are Cs and Ct, Bs and Bt respectively. The Incremental NPV/C Ratio that compares s
and t is
IR = ( ( B t - Ct) - ( B s - Cs) ) / ( C t - Cs) . . . . . . . . . . . . . . . (1)
and for t to be preferred to s this ratio must exceed R.
3
A procedure is recommended in the COBA manual 3 which operates in several stages to choose the
best from a group of alignments. In the first stage, the alignments are ranked in order of increasing cost,
and the cheapest is taken as the standard for the next stage. In the next stage, the standard is compared
with successively more expensive alignments until either an alignment is found for which IR > R or it
has been shown that for all the remaining alignments IR < R. In the second case the standard is chosen as
the best alignment, in the first case the new alignment becomes the standard for the next stage, and
the procedure continues with a new set of comparisons.
A similar procedure can be developed directly from the definition of the Incremental NPV/C Ratio.
Suppose that u and v are two alignments to be compared using the Ratio, then if u is more expensive than v,
u is preferred to v if
so from (1)
and
IR ~ R
(Bu-- Cu) - ( B v - Cv) > R . ( C u - Cv)
( B u - C u . ( 1 + R ) ) > ( B v - C v . ( 1 + R ) )
Alternatively, if v is more expensive than u, u is preferred to v if
IR < R
so f r o m ( l ) ( B v - C v ) - ( B u - C u ) < R . ( C v - C u )
and (Bu - Cu. (1 + R) ) > ( Bv - Cv. (1 + R) )
Thus, irrespective of their relative construction costs, to choose between u and v it is necessary to evaluate
the function:
Overall Benefit - (1 + R) . Construction Cost . . . . . . . . . . . . . . (2)
and take the alignment which gives the higher result. This argument leads to a new method for choosing
the best among a group of alignments: select the alignment that maximises this function. The alignment
chosen by the COBA procedure will always maximise the function (2), so that the method developed
here will always have the same result as the COBA procedure but is quicker and more convenient.
It may be noted that if R is zero then (2) is equal to the Net Present Value of the design, and it is
the zero value that is currently used for scheme evaluation. Positive values for R have been used in the
past, and it is possible that positive values will again be used, so R will be treated as a variable in this section.
2.3 The implementation of the new criterion
The method developed in the preceding section can be used in an optimisation procedure. If all the
trial alignments generated during optimisation are compared, the one that maximises (2) will have the
greatest economic value, so that (2)is the objective function for an optimisation procedure to maximise
the economic value of a highway. One of the terms in (2) is the overall benefit, which depends on a range
of factors. It is shown in this section that under a particular condition many of these can be eliminated
from the objective function, leaving only those which can be calculated directly from the configuration of
the road and the traffic expected to use it.
4
The condition mentioned above is that the changes made to the alignment during optimisation should
not be so great as to alter the traffic flows predicted for the new road. This will be valid so long as the changes
made are insufficient to persuade drivers to change their choice of route. This condition will apply during
the optimisation of the vertical alignment of a scheme for which the horizontal alignment is fixed; it will
also apply during horizontal alignment optimisation where the re-alignment is limited to a narrow corridor.
It will be assumed henceforth that the range of alignment changes possible during optimisation is such
that the condition is satisfied. Most benefits are now effectively fixed during optimisation: in particular,
all the benefits from improved travel on the surrounding road network will be fixed. On the new road, the
number of accidents in a future year will not vary significantly with alignment changes, as predicted by the
relationship used in COBA3: this relationship takes no account of vehicle speeds and predicts 0 .4-1.2
personal injury accidents per million vehicle-kilometres on different classes of rural highway. As traffic
flows have been assumed to be independent of alignment, the number of accidents will depend only on
the overall length, and any change in the costs attributable to these accidents will be negligible in proportion
to other travel costs and benefits.
Thus, the only benefits which can be costed and which depend significantly on the alignment are
reductions in the costs and delays incurred by individual vehicles in driving along the road. Lower gradients
will lead to reduced vehicle operating costs and shorter journey times, and the effect of the latter is costed
using a value for the occupants' time per vehicle. The time and operating costs for a single vehicle form
the travel cost for the journey of that vehicle along the new road, and so of the various items contributing
to the objective function (2) it is only travel and construction costs that vary with alignment changes.
The summary in Section 2. t of the method for assessing the benefit to vehicle operating costs
resulting from a particular alignment for the new road leads to a simple form of objective function, derived
from (2). Let N be the original network of roads, to which the road is to be added. Then ]n a future year
the benefit will be:
Vehicle operating cost for the network N before construction of new road
- (Vehicle operating cost for N after construction + Vehicle operating cost for new road)
= Reduction in cost over N due to new road - Cost for new road.
The reduction in cost over the network N that results from the new road will not vary with alternative
alignments if the condition mentioned above is satisfied. Consequently, for a particular alignment the
contribution of vehicle operating cost savings to the overall benefit is the discounted sum of the savings
over N for any alignment minus the discounted sum of vehicle operating costs for that alignment. The
reduction in time costs can be expressed similarly. Thus, if BY is the discounted sum of travel cost savings
over the original network N, the contribution of travel cost savings to the overall benefit is:
Contribution to Overall Benefit = BY - Discounted sum of travel costs for alignment
and BY is constant for all alternative alignments. The discounted sum of travel costs for a road is often
referred to as the "traffic cost" for that road.
5
The other economic benefits (accidents, reductions in noise, etc) for the complete network do not
depend significantly on the alignment chosen, so their discounted sum BZ will remain constant. Hence,
for all alternative alignments the overall benefit can be expressed:
Overall Benefit = (BY + BZ) - Traffic cost o f alignment . . . . . . . . . . . (3)
so (2) can be re-written:
Objective Function = (BY + BZ) - [Traffic Cost + (1 + R) . Construction Cost] . . . . (4)
As BY and BZ are constant for all the alignments that will be generated during an optimisation, the
alignment which maximises (4) will minimise the revised objective function:
Objective Function = Traffic Cost + (1 + R ) . Construction Cost . . . . . . . . . (5)
Consequently, the traffic cost is the only component of the overall benefit that needs to be con-
sidered when designing the road to maximise its economic value. This simplifies the calculations greatly.
Equation (1) of Section 2.2 can also be simplified. Let Ts and Tt be the traffic costs of alignments
s and t respectively, then from (3) the increased benefit o f t compared with s is
Bt - B s = T s - Tt
so that the Incremental Ratio is
IR = [ ( B t - B s ) - ( C t - C s ) ] / [ C t - C s ]
= [ ( T s - T t ) / ( C t - C s ) ] - 1 . . . . . . . . . . . . . . . . . ( 6 )
The objective function (5) was developed from the Incremental NPV/C Ratio, but it possesses an
inherent logic. It considers that money spent on road-building has a different utility from money spent on
travel: the expenditure o f £1 on improving a road is justified only if the reduction in traffic cost exceeds
£(1 + R). The lower the value chosen for R the greater is the emphasis placed on reducing vehicle operating
costs and raising average speeds in the selection process.
The definition used for the economic value o f a road excluded all effects which at present cannot be
costed, so these are not represented in the objective function. It is sometimes important for one or more
o f these effects to influence the optimisation, and several methods are available. The most direct method
is to specify appropriate standards and then to constrain the alignment at each sensitive point to ensure that
the optimised alignment achieves all standards. An example is the restriction of the level o f the vertical
alignment at sensitive points so that the visual intrusion of the road will be acceptable. Another example
is the use of level restrictions near a residential area to limit the nuisance of traffic noise by ensuring that
the road is in cutting.
3. THE INCLUSION OF TRAFFIC COSTS IN MINERVA
In the previous section the objective function used for alignment optimisation was expanded to include the
traffic cost o f a new road in order to maximise its economic value. In this section it is shown how this
6
objective function has been implemented in MINERVA 4, a vertical alignment optimisation program from
the HOPS suite of computer programs.
3.1 Tile Traffic Cost Model
The conventional version of MINERVA calculates the construction cost of the road, which is one of
the two parts of the objective function (5): the Traffic Cost Model has been developed for incorporation
into MINERVA to provide the other part. Experience with MINERVA has shown that of the various
components of the construction cost of a road, only the costs of earthworks and bridges change significantly
during vertical alignment optimisation, so these are the only parts of the construction cost treated in the
calculations and included in the tabulated results.
The Traffic Cost Model operates in two stages. In the first stage the total of the first year travel
costs is calculated, using the engineer's prediction for the first year traffic flows, and in the second stage
this total is multiplied by a "growth factor" to give the traffic cost. The Model has been designed to be
simple to use and rapid in operation, as in a typical optimisation it will need to process hundreds of trial
alignments.
In the first stage, the travel costs of representatives of three classes of vehicle - cars, light goods and
other goods vehicles - are calculated from simulated journeys along the road: the costs are obtained from
empirical equations relating vehicle costs and average speeds to the gradients included in the vertical
alignment, and the flows forecast for the three classes in the first year are then applied to obtain the total
of the first year travel costs. It is necessary to assume equal traffic flows in either direction, as the
observations from which the empirical equations derive were averaged over both directions. This is
described more fully in Appendix 2, where in addition the growth factor G is developed as a suitable means
for summarising in one variable the user's forecast of future traffic growth and changes in the real cost of
road travel. It also takes account of the discounting process, which emphasises the early years of operation
and so reduces the effect of the uncertainties that result from forecasting the future. G provides an
estimate of the traffic cost for the new road from the travel costs predicted for the first year of operation:
Traffic Cost = G. Total of the first year travel costs
The costs as first calculated by the Traffic Cost Model are based on the price levels which prevailed
at the date for which the Model's cost equations were prepared (mid-1973). Construction costs are
invariably calculated using design year prices, so these costs must be multiplied by an inflation index I.
This takes account of the change in prices between mid-1973 and the date at which the engineering unit
rates apply to ensure that traffic and construction costs are compared on a common level of prices. Thus
Traffic Cost (design year prices)
= I.G. Total of the first year travel costs (mid-1973 prices)
A value for the product I.G is one of the items of data that must be input to the Traffic Cost Model.
Note that, in order to simplify the calculations, all costs are measured at prices current at the time of
design but are discounted to the year of construction, the time at which the cost of the road is paid. It is
assumed that traffic will begin to use the road in the following year. 7
Table 1 presents as an example the values of G obtained from different estimates of traffic growth
between 1975 and 2005. These are given by Tanner 6 in a national forecast of traffic growth, and are his
"low, middle and high forecasts" which, for 2005, range from 505 x 109 to 594 x 109 pcu kilometres.
Three different rates of increase in the cost of travel, as measured in constant costs, are also used. The
values of G are only illustrative and for a particular scheme there may be reasons for raising or lowering
them significantly.
TABLE 1
Typical values of the growth factor
Traffic Growth from ref 6)
Cost increase (per cent pa)
0
1
2
Low
11.84
13.06
14.29
Middle
12.22
13.50
14.77
High
12.36
13.65
14.95
This shows that despite the wide variety of conditions covered, G lies between 11.84 and 14.95, and
the smallness of this range is largely the result of the discounting process. If there were no traffic growth
and zero cost increase the value of G would be 10.37. These figures indicate that when using this model a
simple estimate of G, within the range indicated, could well prove adequate.
It should be noted that two important changes I/ave occurred since the work reported here was
carried out: official estimates of future traffic growth have been lowered, and the discount rate has been
reduced from 10 per cent to 7 per cent. These have countervailing effects on the calculation of G, but
the net effect is to increase values to the range 17.3-19.5.
3.2 The new objective function used with M I N E R V A
It is generally true that the greatest possible reduction in the traffic cost of an alignment that can be
achieved by changes to its vertical alignment will be only a small proportion of the original value. If the
absolute minimum traffic cost is subtracted from the objective function then precision will be improved,
for computers can store values with only limited accuracy; moreover, a better indication will be given of
the scope for reducing the traffic cost.
The vertical alignment that will probably minimise the traffic cost is a ramp of constant gradient
joining the fixed end-points, although in practice this is unlikely to be built as it would have a very high
construction cost. The traffic cost for this alignment is termed the "ramp cost". Now, suppose that
alignments s and t have construction and traffic costs Cs and Ct, Ts and Tt respectively, then from (5) s is
preferable to t if
Tt + ( I + R ) . C t > T s + (1 + R ) . C s
8
The ramp cost TR can be subtracted from both sides:
( T t - T R ) + ( I + R ) . C t > ( T s - T R ) + ( 1 + R ) . C s
so that the objective function:
Objective Function = (Traffic Cost - Ramp Cost) + (1 + R) . Construction Cost . . . . (7)
gives the same optimal alignment as does the earlier function (5), but with less risk of computational
inaccuracy. Thus, the difference between the traffic cost and the ramp cost is calculated by the Traffic
Cost Model and is used in the modified version of MINERVA to maximise the value of a new road.
4. AN OPTIMISATION WITH TRAFFIC COSTS INCLUDED
The special version of the program MINERVA described in Section 3 has been used to optimise an existing
design for a current road scheme. The design was also optimised by the normal version of MINERVA to find
the vertical alignment which minimised the construction cost. Comparison of the two optimal alignments
will demonstrate the effect of including the traffic cost in the optimisation of this vertical alignment.
4.1 The test scheme
The scheme is a by-pass through hilly terrain in South West England. It includes 10.7 km of dual
carriageway trunk road, each carriageway comprising two lanes, and a 500 metre long viaduct. Experience
with MINERVA suggested that in this case the components of the construction cost which would vary
significantly during vertical alignment optimisation were the costs of the viaduct and the earthworks, so
these were the only constructional items included in the calculations. The design standards specified for
this scheme take the normal values for a road of this type, except for the maximum permissible gradient:
this has been set by the designer to 5.1 per cent on account of the hilly terrain.
The ground longitudinal section is shown in Figure 1, and the engineer's alignment is superimposed.
Two optimised alignments are also included in Figure 1 and will be discussed later.
The items of traffic data needed by the program include two-way flows of vehicles estimated for the
first year of operation, divided into categories of cars, light goods and heavy goods vehicles. The data
available consisted of 16 hour flows for the peak month of August, and to obtain annual flows these
were multiplied by the "Annual Traffic Multiplier" recommended in the COBA manual. The resulting
flows during the first year of operation for the different categories were:
Cars 3 666 000 vehicles
Light Goods 177 000 vehicles
Heavy Goods 243 000 vehicles
A growth factor of 11.84 was chosen, this being the lowest value of Table 1, and an inflation index
of 1.35 was selected to allow for price rises between mid-1973 and the time of design, mid-1975. Their
combined effect ig to predict that:
Traffic Cost (1975 prices)
= 16.0. Total of the first year travel costs (1973 prices)
4.2 The vertical alignment which minimises the construction cost
The normal version of MINERVA was used to find the vertical alignment for the test scheme which
minimised the construction cost: this will be referred to as the Minimum Construction Cost alignment
(MCC), and it is shown in Figure 1. It can be seen that considerable changes have been made to the engineer's
alignment and in particular the height of the viaduct has been increased.
The traffic cost was not considered in designing MCC: nonetheless, the traffic cost of this alignment
can be calculated by the Traffic Cost Model. The Model uses equations that relate velocity and fuel
consumption to gradient to calculate for each category of vehicle the fuel, time and miscellaneous costs
of a typical vehicle. These are then aggregated to give the total of the first year travel costs, as shown in
Table 2.
TABLE 2
Traffic cost calculated for minimum construction cost alignment
Cars
Light Goods
Heavy Goods
Cost per vehicle (averaged over both directions)
Fuel Cost (p)
18.49
53.80
100.62
Time Cost (p)
11.96.
18.23
21.96
Misc. Cost (p)
10.77
21.83
42.76
Total Cost (£)
0.412
0.939
1.653
First year two-way flow
3 666 000
177 000
243 000
Total
First year Traffic Cost
(£)
1 511 700
165 800
401 550
2 079 050
~The ramp cost, as defined in Section 3.2, is calculated to be £1 978 120 in the first year, and this differs
from the total of the first year travel costs by only 5 per cent, despite the hilly terrain. The difference
between the two is now multiplied by the growth factor and the inflation index, to give the value required
by the objective function (7):
£ ( 1 6 . 0 x ( 2 0 7 9 0 5 0 - 1978 1 2 0 ) ) = £ 1 614879
The construction cost for MCC is calculated to be £3 285 686, so the value of the objective function is:
£(1 614 879 + ( 1 + R ) . 3 2 8 5 6 8 6 ) = £ ( 4 9 0 0 5 6 5 + R . 3 2 8 5 6 8 6 )
The same method gives an objective function for the engineer's alignment of £(5 739 629 + R . 3 985 521).
This shows that the new alignment costs £699 835 more than the engineer's design to build, and is economically
superior for all probable values of R.
4.3 The optimisation of the scheme to maximise its economic value
A series of optimisations was carried out on the scheme described in the previous section, using
different values of R: R was defined in Section 2.1 to be the minimum acceptable value of the NPV/C ratio.
This value is normally predetermined for the highway engineer, but it is useful to see how the effect of the
10
traffic cost on the optimisation varies with R: values of 0.0 and 0.2 were used. The latter value is recomm-
ended in the COBA manual and the former value will lead to the maximisation o f the Net Present Value,
as noted in Section 2.2. The weighting of the construction cost in the objective function (7) declines as
R is reduced, so that the alignment obtained with R = 0.0 would be expected to present the greater
variations from MCC.
The various costs of MCC are compared in Table 3a with the costs for the engineer's alignment and
the alignments optimised using the objective function (7) with R = 0.0 and R = 0.2: these last two align-
ments are referred to as A1 and B1 respectively. The construction costs o f all three are very similar and
only with R = 0.0 is there a significant reduction in the traffic cost.
TABLE 3a
Summary of costs with original traffic forecast
Alignment
Engineer's Alignment
MCC
A1
B1
Traffic Cos t -Ramp
Cost (£)
1 754 109
1 614 879
1 561 650
1 614 039
Earthwork Cost (£)
1 984 534
1 236 188
1 213 270
1 213 938
Viaduct Cost (£)
2 000 987
2 049 498
2 079 653
2 050 126
Construction Cost (£)
3 985 521
3 285 686
3 292 923
3 264 064
No te: MCC - alignment with minimum construction cost
A1 - alignment optimised with objective function (7) and R = 0.0
B1 - alignment optimised with objective function (7) and R = 0.2
The Construction Cost does not include costs that do not depend on the vertical alignment.
The Traffic and Ramp Costs are discounted over 30 years.
Alignment A1 shows some small changes from MCC, but it is only over the central section o f the route that
the changes are significant. These can be seen in Figure 1 : A1 takes a higher, flatter line between chainages
8 500 and 11 000, but otherwise changes are only slight. B1 follows MCC closely, and is not shown in
Figure 1 : the fact that it is 0.7 per cent cheaper than MCC is due to a minor inefficiency in the optimisation
procedure used in MINERVA.
The objective functions o f A1 and B1 are compared in Table 3b with the objective function for
the MCC alignment, evaluated with R -- 0.0 and R = 0.2.
11
TAB L E 3b
Objective function with original traffic forecast
MCC
Alignment optimised with particular value of R
R= 0.0
Objective Function
4 900 565
4 854 573 (A1)
per cent of MCC value
99.06
Objective Function
5 557 702
5 530 916 (B1)
R= 0.2
per cent of MCC value
99.52
Note: Objective Function = Traffic Cost - Ramp Cost + (1 + R). Construction Cost These costs are taken from Table 3a.
It can'be seen that the reductions in the objective function are slight, and this leads to the conclusion
that the alignment that minimises the construction cost very nearly maximises the value of the scheme.
One reason for this may be that on this scheme traffic flows, and hence the traffic cost, are relatively low
for this class of road, and indeed the total traffic forecast is at the lower end of the range for which a two-
lane dual carriageway is appropriate.
Consequently, an additional series of computer runs was carried out in which the traffic flows were
brought near to the road's capacity by doubling them. The design data used in the earlier runs were still
applicable as the same class of road was still to be designed, so the traffic data alone of the data input to
MINERVA were modified. Some factors not represented in the program, such as standard of junction,
may have needed improvement to accommodate the increased traffic, but all of the items that affect the
optimisation process were unchanged.
In the new series of runs, values of 0.0 and 0.2 were again used for R. The optimal alignments will
be referred to as A2 and B2 respectively, and are shown in Figure 2. The various costs of A2 and B2 are
shown in Table 4a: the traffic cost of MCC has been re-evaluated with the new data.
TABLE 4a
Summary of costs with doubled traffic forecast
Alignment
MCC
A2
B2
Traffic Cost -Ramp
Cost (£)
3 229 758
2 730 044
2 707 475
Earthwork Cost (£)
1 236 188
1 394 265
1 423 391
Viaduct Cost (£)
2 049 498
2 121 432
2 126 515
Construction Cost (£)
3 285 686
3 515 697
3 549 906
Note: MCC - alignment with minimum construction cost
A2 - alignment optimised with objective function (7) and R = 0.0
B2 - alignment optimised with objective function (7) and R = 0.2
The Construction Cost does not include costs that do not depend on the vertical alignment. The Traffic and Ramp Costs are discounted over 30 years.
12
The values of the objective function obtained with the doubled traffic forecast are shown in Table 4b,
and these show that with the higher traffic forecast the road should no longer be built to the cheapest design.
It is now desirable to invest in a more expensive road which will offer greater long-term benefits through
reduced travel costs.
T A B L E 4b
Objective function with doubled traffic forecast
MCC
Alignment optimised with particular value of R
R= 0.0
O~ective Function
6 515 444
6 245 741 (A2)
per cent of MCC value
95.86
ONective Function
7 172 581
6 967 363 (B2)
R = 0.2
I per cent of MCC value
97.14
Note: Objective Function = Traffic Cost - Ramp Cost + (1 + R). Construction Cost These costs are taken from Table 4a.
As a final proof of the economic advantage of A2 and B2 over MCC, Incremental NPV/C ratios are
calculated in Table 5. For a given value of R, a more expensive design is preferred if the ratio of the
increase in NPV to the increase in construction cost exceeds R: this is true in both cases, so that with
either value of R the alignment optimised with the new objective function is definitely superior to the
Minimum Construction Cost alignment.
TABLE 5
Incremental NPV/C ratios comparing the optimised alignments with the minimum construction cost alignment
Alignment
A2
B2
Reduction in Traffic Cost
(1)
499 414
522 283
Increase. in Construction Cost
(2)
230 011
264 220
Increase in NPV
(3) = (1) - (2)
269 703
258 063
Incremental NPV/C Ratio
= (3)/(2)
1.173
0.977
Note: The doubled traffic forecast was used during the optimisations that produced A2 and B2.
4.4 Summary of results for t i le test scheme
In Section 4.3 two series of tests were described in which the vertical alignment of a highway design,
previously optimised to minimise construction costs, was re-optimised to take account of the expected
benefits to future road users. In the first series the engineer's traffic forecast was used, which predicted
low traffic flows relative to the design capacity. These tests showed that the alignment that minimised
construction costs very nearly maximised the economic value of the road.
13
The traffic forecast was doubled for the second series of tests, bringing the traffic flows near to the
design capacity. More expensive alignments were found which offered sufficient increases in benefits to
warrant the extra investment; these cost approximately £0.25m more to build than the cheapest alternative,
but increased the NPV by £0.25m. The values of the Incremental NPV/C Ratio were approximately 1.0,
and so the extra expenditure would be justified.
The most significant changes made by including traffic costs in the optimisation occurred between
chainages 7 000 and 12 000. This section of the route is shown in Figure 3, and the four optimised
vertical alignments are shown with the alignment which minimised construction cost. This figure shows
that as the significance of the traffic cost increases relative to the construction cost, the optimal alignment
tends to become flatter, with lower gradients and longer curves. Alignment B 1 was not shown in Figure 1
as it differs so little from the MCC alignment: these differences can be seen between chainages 9 000
and 10 200 in Figure 3.
4.5 Limits on gradient
The gradients that may be incorporated in the design of a new highway are normally subject to
nationally defined limits. The reasons for this policy include the desire for uniform design standards
throughout the highway network and the need to avoid the excessive operating costs incurred on steep
stretches of road. An alternative approach that has been suggested would include travel costs in the
design process, relying on these to reduce steep gradients to a more acceptable value.
The results from the test scheme show that the steepest gradients on a vertical alignment optimised
with an objective function that includes travel costs were not significantly lower than the values obtained
by optimising with construction cost alone. Consider, for example, the long inclines at either end of the
route. It will be recalled that the designer specified a maximum permissible gradient of 5.1 per cent, this
being slightly greater than the steepest gradient in his design, and the alignment which minimises construction
cost has a steepest gradient of 5.03 per cent. Thus the optimisation has not been affected by the limit on
gradient, and could have produced steeper gradients if this would have lowered the construction cost
further. The influence of heavy traffic reduced the steepest gradient to 4.98 per cent, and the gradient of
neither incline was reduced by more than 0.1 per cent. On the other hand,if a limit of, say, 4 per cent
had been imposed then the optimised alignment would be far more expensive to build, and the tests show
that the extra expense would not be justified by the corresponding reduction in travel costs.
This suggests that a national system of gradient limits could lead, if too strictly applied, to instances
of excessive investment in roads built to an uneconomically high standard. The inclusion of traffic costs
in vertical alignment design may not significantly lower gradients relative to the cheapest alignment but
it does provide a consistent criterion for choosing gradients which gives due weight to the interests of
road users.
4.6 The cost implications of the new objective function
The results presented above demonstrate that the construction cost of a vertical alignment optimised
with the new objective function (7) tends to be higher than for an alignment optimised to minimise construction
costs. Since the evaluation of a road scheme takes account of the traffic benefits and construction costs of
the scheme, it is perfectly possible for the alignment with the greatest economic value to be significantly
more expensive than the alignment with the minimum construction cost (MCC). However, unless the
14
method described above has been used there is no way of knowing whether an alignment which is more
expensive than MCC is optimal, and it is quite possible that the increase in traffic benefits has been carried
too far.
Consider, for example, three vertical alignments for the by-pass scheme studied in this section.
One alignment (MCC) has been optimised to minimise the construction cost, another (!32) has been
optimised to maximise its economic value by minimising the new objective function, and a third, (D),
has been designed by manual methods and gives even lower traffic costs but is more expensive to build
than B2.
TABLE 6
Comparative costs for the three alignments
Alignment MCC B2 D
Construction Cost (£m) 3.286 3.550 3.700
Traffic Cost - Ramp Cost (£m) 3.230 2.707 2.600
Note: Costs for MCC and B2 taken from Table 4a, costs for D are hypothetical.
If B2 had not been developed, D would be compared with MCC and, since this comparison gives an
Incremental NPV/C Ratio of +0.52 using equation (6), alignment D would be preferred. However, D
actually is sub-optimal since its construction cost has risen beyond that of B2 to an extent that cannot be
justified by the increased traffic benefits. This is shown clearly when D is compared_with B2; the
Incremental NPV/C Ratio is -0.28, so that B2 - the cheaper alignment - is preferred.
In this case the existence of a scheme which is more expensive to build than MCC but does maximise
the economic value of the road has demonstrated that an even more expensive alignment is not the best
one to build.
5. CONCLUSIONS
A method has been described for designing the alignment of a new road that explicitly takes account of
the travel costs of the drivers who will use the road. Details have been given of the implementation of this
method using MINERVA, an existing computer program from the Highway Optimisation Program System
that optimises the vertical alignment. A current highway scheme passing through hilly country was studied,
and the effect of including travel costs during alignment optimisation was demonstrated. This example
showed that if the road is heavily trafficked then extra investment can be justified in order to achieve
a flatter profile and reduce travel costs. It was shown that the new method will sometimes lead to a cheaper
road than would be designed using a minimum cost criterion, since the new method is consistent with the
criterion used for comparing alternative routes for a new road.
The scheme studied provided some evidence that the current system of gradient limits on new
highways should be replaced by the inclusion of travel costs quantitatively in the design process.
15
6. ACKNOWLEDGEMENTS
The work described in this report was carried out in the Access and Mobility Division of the Transport
Operations Department of TRRL.
7. REFERENCES
I. DEPARTMENT OF THE ENVIRONMENT, HIGHWAY ENGINEERING COMPUTER BRANCH.
Highway Optimisation Program System. London, May 1974 (Department of the Environment).
. WITHEY, K H. The optimisation of the vertical alignment of the M5 motorway from Chelston to
Blackbrook. Department of the Environment, TRRL Report LR 473. Crowthorne, 1972 (Transport
and Road Research Laboratory).
. DEPARTMENT OF THE ENVIRONMENT, HEMA DIVISION. COBA. London (Department of
the Environment).
. DAVIES, H E H. Optimising Highway Vertical alignments to minimise construction costs: Program
MINERVA. Department of the Environment, TRRL Report LR 463. Crowthorne, 1972 (Transport
and Road Research Laboratory).
5. DAVIES, H E H and J BROUGHTON. Horizontal alignment optimisation: test of program NOAH
on a motorway scheme. Department of the Environment Department of Transport, TRRL Report LR 894. Crowthorne, 1979 (Transport and Road Research Laboratory).
. TANNER, J C. Forecast of vehicles and traffic in Great Britain: 1974 revision. Department of the Environment, TRRL Report LR 650. Crowthorne, 1974 (Transport and Road Research Laboratory).
. EVERALL, P F. The effect of road and traffic conditions on fuel consumption. Ministry of Transport, RRL Report LR 226. Crowthorne, 1968 (Transport and Road Research Laboratory).
. DAWSON, R F Fand P VASS. Vehicle operating costs in 1973. Department of the Environment, TRRL Report LR 661. Crowthorne, 1974 (Transport and Road Research Laboratory).
1 6
8 o
B 8 E ~ '~.
•
N'.~ o-o
(,=. (,=. m m ~
w < < 2 r 3
t-
0 f"
co
0
, I I I I I I I 0 0 (~ 0 0 0 0 0 0 0 0
0
CO
0 0 0
0 0 0
i,,-
0 C3 0 0
D- Z W
o Z C~ o~
.,J
U.; W Z
Z
w
0
0 I.-
o o F- 0 ~L
0 W
p.
U.
0 0 0 tO
0 0 0 li3
0 0 0
0 0 0
0 0
(w) apm!:l-lV
0 0 0
o o o
0 . - -
{,. (,-.
• o • - , ~ o ~ ~ ~ . , E "F= o 'F= = . - , . - , ~ =
o-o o-o E ,,-, ~. r.. ~ r - ,.~ _00 " - t -
._m ._m ~ ._mp c
0
I>,
I I I I I 0 0 0 0 0
m
/
I t
I 0
I 0
I 0
( w ) apnl.!:l.lV
0
CO
0 0 0 CN
0 0 0
0 0 0 0
0 0 t,.}
I - -
8
c • ~ I.LI .(Z t,~
I - - Q. o
0 I - - 0 Z 0 I.LI r,-
Z f,.9 m - - I
0 ' }
1,1.
0 0 0 ~0
0 0
-- 0 L~
0 0
- 0
0 -- 0
0 CO
0 0
OC~
,,,.-. ~ '.,.-. '.,.-
"0
~0 o~ o~ ~ "~ 0 ~-
0 0 "0 "0
~ = = = = .~ ~
, - . _ , . _ , . _ , . _ , ~ o.,_. ._ ,.,,'," ,..,m = m =,"," . ~ ~
o - o o - o o - o o - o '~
._=._~ ~._~ ~._~~._= ~ ~ ~ =
• I ' ÷
i [ I -I
,o ,'/
0 0
0
I lo
\
, " ~ i ~ ' i 0 0 0 0 O0 r- , co i_~
/ f , ~ -
2, / ~" .o." /
/
( ~ ) aPm,!3,1V
0 0 0 £N
0 0 0
o
z
,,=,
(0 ILl
F- Z MJ
z
i
..I
0
0 MJ i
i
F- a. 0
~L
0 0 o 00
0 0 0
0 r~
A1
A2
B1
B2
BY
BZ
Bn
Cn
Tn
Dt
Et
Fv
G
I
IR
MCC
N
NPV
NPV/C
r
TR
Y
Zv
8. APPENDIX 1
NOTATION
Alignment optimised with R = 0.0 and original traffic forecast.
Alignment optimised with R = 0.0 and doubled traffic forecast.
Alignment optimised with R = 0.2 and original traffic forecast.
Alignment optimised with R = 0.2 and doubled traffic forecast.
Discounted sum of travel cost savings over original road network N.
Discounted sum of those economic benefits over new road network that are independent of the alignment.
Overall benefit of scheme n (n = s, t, u, v).
Construction cost of scheme n (n = s, t, u, v).
Traffic cost of scheme n (n = s, t).
Change in travel costs in year t, relative to year 1
Ratio of traffic flow in year t to first year flow.
Two-way flow of vehicles from class v in first year.
Growth Factor.
Inflation index between mid-1973 and date o f engineering unit rates.
Incremental NPV/C Ratio comparing two schemes.
Alignment with minimum construction cost.
Original road network.
Net Present Value of scheme.
Ratio of Net Present Value to Construction Cost.
Discount rate.
Ramp Cost - ie Traffic cost for the alignment of constant gradient joining the end-points.
First year total of travel costs (1973 prices).
Travel cost for representative of class v (1973 prices).
2 3
9. APPENDIX 2
THE TRAFFIC COST MODEL
The Traffic Cost Model calculates the traffic cost for any alignment in two stages: firstly, the total of
travel costs in the first year of operation is estimated, and secondly this is multiplied by a growth factor
and an inflation index to give the sum of travel costs throughout the life of the road, discounted to the year of construction.
9.1 A model for calculating travel costs
The method used to calculate the total of first year travel costs is described below. It is designed to
operate rapidly, as it will be needed hundreds of times in the course of a typical optimisation, yet it takes
account of all major factors determining travel costs.
Road vehicles are divided for the purposes of the model into three categories: cars, light goods and
other goods vehicles. Public Service vehicles are not treated in the Traffic Cost Model as they will normally
form an insignificant proportion of the traffic using roads to which the Model will be applied. The Model
calculates the total of first year travel costs for each category as the product of the predicted two-way
flow in the first year and the travel cost for a representative vehicle. The travel cost is estimated from a
simulated journey along the alignment, using equations that were derived from the observations of EveraU 7
. to predict the speed and fuel consumption of the representative vehicle over short stretches of road from
the gradient and design velocity; the observations were averaged over both directions. These predictions
are converted into costs using the methods described by Dawson and Vass 8. Within the Model, travel costs
are divided into three classes: time, fuel and miscellaneous costs; the latter comprise those items, other
than fuel and vehicle occupants' time, which Dawson and Vass list as marginal travel costs. They quote
equations relating travel costs to velocity and it is these, with the fuel and time cost components removed,
that are used to predict miscellaneous costs. Values of the vehicle occupants' time and cost of fuel 8 are used to calculate time and fuel costs.
The cost calculated by the Traffic Cost Model is thus proportional to the traffic flow, and also depends
on the design velocity and vertical alignment. For a particular optimisation traffic flow and design velocity
are constant, so that the calculations are sensitive only to changes in the vertical alignment. It was, however,
observed in Section 4.2 that the traffic cost for the MCC alignment (which included gradients of 5 per cent)
was only 5 per cent greater than for the alignment of constant gradient joining the end-points, with a gradient
of 0.3 per cent. This suggests that, with gradients subject to the limits applied to English trunk roads, the
Traffic Cost Model will be far more sensitive to the overall length of a scheme than to its vertical alignment.
This is confirmed by experience 5 with the horizontal optimisation program NOAH, where the large
reductions of traffic costs achieved were attributable mainly to shortenings of the alignment.
The equations predicting vehicle performance and fuel consumption represent implicitly many aspects
of traffic flow which would be difficult to model explicitly, such as congestion. In the tests reported by
Everall 7, the drivers of the instrumented test vehicles were forced to respond to a wide range of road
conditions during their journeys: also, they were instructed to vary their driving styles between journeys.
These variations were reflected in the observations made, and thence in the equations, so that a wide range
of factors will influence the results of Traffic Cost Model without being treated specifically.
2 4
9.2 The Growth Factor
In this section the method is described by which the traffic cost o f an alignment is calculated from the
travel costs of representative vehicles, using a growth factor which subsumes the range o f variables normally
required by this calculation.
In the previous section the estimation was described of the travel costs o f representative vehicles from
the three classes into which road vehicles are divided. Let Fv be the two-way flow of vehicles from class v
in the first year, and let the travel cost calculated by the model for the representative o f class v by Zv: this
will be at 1973 prices, so the total of first year travel costs calculated by the Traffic Cost Model at 1973
prices is:
Y = Y~ Fv . Zv V
An inflation index I is needed to account for declining values between 1973 and the design year, so that
at design year prices the total o f the first year travel costs is I.Y.
Costs and benefits for the new road must be expressed in constant prices, but it is nevertheless possible
that values for fuel and time, expressed in constant prices, will rise. For simplicity, a common rate o f change
is assumed for the three components of travel cost: if the value in year t relative to year 1,the first year o f
operation o f the road, is Dt then the travel cost for representative v in year t at design year prices is:
Dt . I . Z v . . . . . . . . . . . . . . . . . . . . . . . . . . (a l )
This simple model should be reliable for the early years, and the emphasis placed on this period by dis-
counting at the current discount rate will minimise any error. This argument applies equally to the split
of traffic between the classes in future years, which is assumed to remain constant. Thus, if Et is the total
flow in year t divided by the total flow in year 1 then the number of vehicles from class v in year t is:
E t . Fv . . . . . . . . . . . . . . . . . . . . . . . . . . (a2)
so that the total travel cost in year t is the sum over the classes v o f the product (a2) . (al) :
(Et. F v ) . ( D t . I . Z v ) = ( D t . E t . l ) . ~ Zv. Fv V V
-- D t . E t . I . Y
T, the traffic cost for the road, is the sum over the life of the road of these costs, discounted at rate r
to the year of construction:
T = E D t . E t . I . Y . ( I + r ) - t t
= I . Y . E D t . E t . ( 1 +r ) - t . t
G = ~ D t . E t . ( 1 + r ) - t t
25
is defined to be the Growth Factor, where the summation is evaluated over the life of the road, often taken
to be thirty years. Thus the Traffic Cost Model calculates the traffic cost of an alignment from the equation:
Traffic Cost -- Growth Factor (G). Inflation Index (I) . Total of First Year Travel Costs (Y)
where G and I are specified by the user and Y is calculated by the Model from the user's first year traffic
forecast, as described in the preceding section.
Values of the Growth Factor under a wide range of forecasts are presented in Table 1, evaluated with
a discount rate of ten per cent over a period of thirty years. The small range of values demonstrates the
stability of this approach, and shows how the effect of the inherent uncertainty of traffic forecasting is
reduced by the discounting process. The user should normally be able to select a value of G from the range
indicated by Table 1, without the need for elaborate traffic forecasts.
26
(1584) Dd0536361 1,500 8/79 HPLtd So'ton G1915 P R I N T E D IN E N G L A N D
ABSTRACT
The effect of travel costs on the design of highway alignments: J BROUGHTON BSc PhD: Department of the Environment Department of Transport, TRRL Laboratory Report 912: Crowthorne, 1979 (Transport and Road Research Laboratory). The alignment chosen for a new road will affect the travel costs of all drivers using it. In current practice, travel costs are taken into account when making an economic evaluation of a new road but are of ten not included explicitly when designing the alignment of the road. This inconsistency may lead to the construction of a new road to a design which does not represent the best balance between the costs of travel and of construction.
In this report a simple method is described for allowing the expected travel costs o f a new road to influence the design of the alignment. The conversion of an existing vertical alignment optimisation computer program to take account of travel costs is described.
The new program has been applied to a current highway scheme with a vertical align- ment that had been optimised to minimise the scheme's construction cost. The highway considered passes through hilly terrain, and it is shown that an improvement in the economic value of the scheme can be achieved by introducing travel costs into the design process.
ISSN 0305-1293
ABSTRACT
The effect of travel costs on the design of highway alignments: J BROUGHTON BSc PhD: Department of the Environment Department of Transport, T R R L Laboratory Report 912: Crowthorne, 1979 (Transport and Road Research Laboratory). The alignment chosen for a new road will affect the travel costs of all drivers using it. In current practice, travel costs are taken into account when making an economic evaluation of a new road but are often not included explicitly, when designing the alignment of the road. This inconsistency may lead to the construction of a new.road to a design which does not represent the best balance between the-costs of travel and of construction.
In this report a simple method is described for allowing the expected travel costs of a new road to influence the design of the alignment. The conversion of an existing vertical alignment optimisation computer program to take account of travel costs is described.
The new program has been applied to a current highway scheme with a vertical align- ment that had been optimised to minimise the scheme's construction cost. The highway considered passes through hilly terrain, and it is shown that an imgrovement in the ecofiomic value of the scheme can be achieved by introducing travel costs into the design process.
ISSN 0305-1293
