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Light rail transit: Cost and output
JOURNAL
OF URBAN ECONOMICS
7, 20-30 (1980)
Light Rail Transit:
Cost and Output
HANS BREMS Department
of Economics, University
of Illinois
at Urbana-Champaign 61801
Received November 14, 1977; revised February 7, 1978 Growing concerns about environment, oil depletion, and traffic congestion call for a fresh look at msss transit-specifically non-polluting, low noise, nonoil-consuming, high-speed electric transit. This article attempts to relate cost to output, both for light rail and the alternative transit mode, i.e., the express bus. For that purpose a simple algebraic framework is developed, tailored to data obtained from a study of alternative transit modes in the Pittsburgh South Hills corridor. The data were prepared by De Leuw Cather and Company for the Port Authority of Allegheny County and were published by the U.S. Department of Transportation in a recent study of the state of light-rail art. The article finds the volume of output at which, considering cost alone, a mass-transit authority would be indifferent between light rail and express buses. At the end, noncost factors are considered.
1. INTRODUCTION Bus transit offers wide but low-speed coverage at minimum investment but high traffic cost and objectionable noise and pollution levels. At very heavy investment and no pollution, heavy rail offers high-speed but narrow coverage. At lower traffic cost and less noise than buses, and at lower construction cost per mile and wider coverage than heavy rail, light rail offers a nonpolluting compromise between these two extremes. Yet, as shown in Fig. 1, light rail on exclusive right of way may achieve high travelling speed-so essential in attracting ridership. For its cost-output study of light rail and express buses, this article will develop an algebraic framework using the following notation. 2. NOTATION
A = a = b = /? =
line-acquistion cost, dollars per line mile per year traffic cost, dollars per passenger-space mile time consumed by a train dwelling at a terminal, hours terminal dwelling time as a fraction of travelling time, pure number 20
00941190/80/010020-11$02.00/O
LIGHT
RAIL TRANSIT
21
C = cost, dollars per line mile per year d = distance between terminals, miles F = number of vehicles in fleet j = frequency, average number of trains starting a round trip per hour h = hours of operation of system per year j = average number of vehicle hours of operation per fleet vehicle per yf= k = vehicle capacity, passenger spaces per vehicle m = operation and maintenance cost, dollars per vehicle mile n = average number of trains operating wi = fraction of h during which i-thirds of fleet is operating, i = 1, 2, 3 P = line-acquisition price, dollars per line mile p = vehicle-acquisition price, dollars per vehicle r = rate of interest, pure number per year = travelling speed between terminals, miles per hour S T = time consumed by a tra.in between starting two consecutive round trips, hours = time consumed by a train travelling between terminals, hours t u = useful life of line or vehicle, years V = average number of vehicles per train W = output per fleet vehicle, passenger-space miles in both directions per fleet vehicle per year X = output per line mile, passenger spaces in both directions per year Y = output of line, passenger-space miles in both directions per year. 3. OUTPUT 3.1 Output of Line
Imagine a single line d miles long on which j trains are starting a round trip every hour for h hours per year. Each train has v vehicles and each vehicle k passenger spaces. Output is produced whether the vehicle is travelling in one or the other direction, hence output of the line in passenger-spacemiles in both directions per year will be Y = 2djhkv.
(1)
Define travelling speed between terminals as s = d/t.
(2)
If T is the number of hours consumed by a train between starting two consecutive round trips, then l/T is the number of times that, train can start a round trip per hour. An n/T is the number of times all n trains can do so. But that is the same thing as traffic frequency j = n/T.
(3)
HANS BREW
22 30
1 :~;;“‘s”~~,, 8 PHILADELPHIA RED ARROW SAN FRANCISCO
@
-lo
THE HAGUE% FRANKFURT
$AN
PHILADELPHIA CITY TRANSIT
0
P,TTsB$GH'wARK
COLOGNE oByN
HANNOVERO HAMBURG 0
I
20
40
GOTHENBURG OPITTSBURGH
0 MUNICH
FRANCISCO
I
0
I
1
80
100
I
60
EXCLUSIVEANDSEMI-EXCLUSIVERIGHT-OF-WAY IN PER CENTOF NETWORK 0
EXISTING
DATA:
@FUTURE
SYSTEMS
DE LEUW, CATHER AND COMPANY CONVERTED TO MILES.
FIQURE
(1976).
SYSTEMS 84.
KILOMETERS
1
The time consumed by a train between starting two consecutive round trips is twice the sum of travelling and terminal dwelling time: T = 2(t + b). Let terminal
dwelling
(4)
time be in proportion
to travelling
b = @t,
time: (5)
where p is a parameter equalling, say, &. Now insert (5) into (4), (4) into (3), (3) and (2) into (1) and find output of line Y = hknsv/(l in passenger-space miles in both directions
+ p)
(6)
per year.
3.2 Output Per Fleet Vehicle Define average output per fleet vehicle as W = Y/F
= hknsv/[(l
in passenger-space miles in both directions
+ P)F]
(7)
per fleet vehicle per year. All
LIGHT
RAIL TRANSIT
23
transit authorities must live with the fact that their full fleet F is needed for only a fraction of the hours of operation. Now n is average number of trains operating and v average number of vehicles per train, hence nv is average number of vehicles operating. That number must be less than or equal to number of vehicles in fleet : nv 5 F.
(8)
The average number of vehicles operating, nv, is a weighted average of the numbers operating at different times. Consider only three possible fractions, i.e., f, $ and g of fleet operating. Define wi as the fraction of hours of operation h during which i-thirds of the fleet is operating, where i = 1, 2, and 3 and w1 + ~2 + w3 = 1. In other words, wi are the weights of the weighted average of the numbers of vehicles operating at different times, nv = + F far the fraction WI of hours per year nv = 3 F for the fraction w2 of hours per year nv = F for the fraction w3 of hours per year, so the average number of vehicles operating will be nv = ($wl + $w2 + w,)F.
(9)
hnv/F = j,
(10)
In (7) call the ratio which is a measure of vehicle utilization equalling vehicle hours of operation per fleet vehicle per year. Use (10) to write average output per fleet vehicle as W = Y/F = jks/(l + p), (11) which is a measure of vehicle productivity
to be needed in Sec. 4.3 below.
S.S Output Per Line Mile Define average output per line mile as X E Y/d = hknsv/[ (1 + P)d] in passenger spaces in both directions
(12)
per year.
4. COST 4.1 A Linear Cost Function As a first approximation
let us estimate a linear cost function C=A+aX
(13)
in dollars per line mile per year. Output X per line mile may be expanded
HANS BREMS
24
by expanding traffic. But at given average vehicle hours of operation per fleet vehicle per year j, at given vehicle capacity k, at given travelling speed s, at given dwelling fraction B, output per fleet vehicle (11) will be given, and traffic can be expanded only by expanding fleet F. Up to an upper bound, let such expansion occur at constant traffic cost a per unit of output X per line mile. The upper bound is determined by safety considerations and technology : At high-speed operations and automatic block signalling systems, train intervals (‘Lheadways”) cannot be lower than 90 to 120 seconds. At a 120-second headway, what would output X per line mile be like? Well, let hours of operation per year be h = 3900; let vehicle capacity be k = 115 passenger spaces per vehicle; and let average number of vehicles per train be v = 2. Output per line mile will then be X = 53.8 million passenger spaces in both directions per year. In what kind of dollars should we measure cost? Interest and depreciation on line and vehicles should be calculated on the historical cost of acquisition, so we follow De Leuw, Cather, and Company [2] in measuring them in 1975 dollars. Operating and maintenance costs, on the other hand, should be those of a typical year of operation during useful life, so we follow De Leuw, Cather, and Company [2] in measuring them in 1985 dollars. 4.2 Line-Acquisition
Cost
De Leuw, Cather, and Company [2], estimated the price of a new line in dollars per line mile and found : TABLE
1
Line-acquisition price P, millions of dollars per line mile
Busway
Light rail
8.61
11.05
Acquisition price P may be allocated among individual years of useful life by multiplying it by T + l/u, where r is the rate of interest in a pure number per year, u is useful life of the line in years, and l/u the depreciation coefficient in a pure number per year. Let r = 0.08 and u = 30, then r + l/u = 0.1133, and line-acquisition cost is: TABLE
2
Line-acquisition cost A = (r + l/u) P, millions of dollars per line mile per year
Busway
Light rail
0.98
1.25
LIGHT
&!I Trafic
RAIL TRANSIT
25
Cost: Vehicle Acquisition
Traffic cost includes, first, vehicle-acquisition cost and, second, cost of operation and maintenance. We begin with vehicle-acquisition cost. The vehicle-acquisition price of a light-rail vehicle was estimated by De Leuw! Cather, and Company (1976), and the conventional 40-foot urban bus currently used by most U.S. mass transit operations is priced around $100,000. TABLE
3
Vehicle-acquisition price p, dollars per fleet vehicle
Express bus
Light rail
100,000
400,000
Acquisition price p may be allocated among individual years of useful life by multiplying it by r + l/u as before. As before, let T = 0.08. County of San Diego Cl], estimated useful life u to be 15 years for buses and 30 years for light-rail vehicles. Consequently r + l/u = 0.14667 for buses and 0.11333 for light rail-vehicles, and vehicle-acquisiton cost will be: TABLE
Vehicle-acquisition cost (r + l/u)p, dollars per fleet vehicle per year
4 Express bus
Light rail
14,667
45,332
To find vehicle-acquisition cost in dollars per passenger-space mile we must estimate the right-hand side of Eq. (ll), jrCs/(l + p). Let an austere U.S. transit system operate for 15 hours a day, 5 days a week, 52 weeks a year. In that case h = 3900 hours per year. Assume that w1 = w2 = co3= 5. Then according to (9) nv = QF, and according to (10) j = 2600. Define a passenger space as 5 square feet of vehicle floor space, then k = 67 and 115, respectively, for express buses and light rail. Measured in miles per hour, travelling speed s was estimated by De Leuw, Cather, and Company at 6 to 22 and 16 to 25, respectively, for express buses and light rail, or a simple average of 14 versus 20. The main reason for the difference is probably acceleration ability. Measured in feet per second2, the service acceleration rate was estimated at 0.3 to 3.2 and 4.5 to 5.1, respectively for express buses and light rail, or a simple
HANS BREMS
26 average of 1.8 versus side of Eq. (11) :
4.8.
Let @ = &. We can now find the right-hand TABLE
5 Express bus
Vehicle use j, hours of operation per fleet vehicle per year Vehicle capacity k, passenger spaces per vehicle Travelling speed s, miles per hour Terminal dwelling-time fraction p, pure number Output per fleet vehicle jks/ (1 + p), millions of passenger-space miles in both directions per fleet vehicle per year Now divide
the result
of Table
4 by that
TABLE
Light rail
2600
2600
67 14
115 20
0.10
0.10
2.22
5.44
of Table 5 and find:
6
Vehicle-acquisition cost (1 + 0) (r + l/u)p/ jks, dollars per passenger-space mile
Express bus
Light rail
0.0066
0.0083
Becailse the light-rail vehicle has twice the useful life U, has a 1.72 times higher vehicle capacity k, a,nd has a 1.43 times higher travelling speed s, its vehicle-acquisition cost in dollars per passenger-space mile is merely 1.26 times that of a bus. 4.4 Trufic
Cost: Operation and Maintenance
De Leuw, Cather, and Company (1976) estimate costs of operation and maintenance in dollars per vehicle mile as follows : TABLE
7 Express bus
Light rail
Operation and maintenance costs m, dollars per vehicle mile : Maintenance of rights-of-way Energy General and administration Vehicle maintenance Conducting transportation
0.037 0.101 0.628 0.258 0.852
0.191 0.209 0.532 0.216 0.650
Total, m, dollars per vehicle mile
1.876
1.798
LIGHT
RAIL TRANSIT
27
So the operation and maintenance cost of propelling a light-rail vehicle 1 mile is 0.96 times that of propelling a bus. Maintenance of rights-of-way is 5.16 times as high; energy 2.07 times as high; general and administration 0.85 times as high; vehicle maintenance 0.84 times as high; and conducting transportation 0.76 times as high. The reasons for the lightrail superiority in conducting transportation are, first, that at the same money wage rate in dollars per hour, wage cost in dollars per vehicle mile will be lower for the vehicle travelling more miles per hour. Second, the driver’s wages in dollars per vehicle mile of a two-vehicle train are only one-ha.lf of those of a one -vehicle train. Such costs of operation and maintenance in dollars per vehicle mile are easily translated into dollars per passenger-space mile : Divide Table 7 by vehicle capacity k and fmd : TABLE
8 Express bus
Light rail
0.0280
0.0156
Operation and maintenance costs m/k, dollars per passenger-space mile
So because a light-rail vehicle carries 1.72 times as many passenger spaces, the operation and maintenance cost of propelling a light-rail passenger space 1 mile is merely 0.56 times that of propelling a bus passenger space one mile. 4.5 Overall Tra&
Cost
Traffic costs are vehicle-acquisition cost plus operation and maintenance cost, so let us add Tables 6 and 8 and find overall traffic cost: TABLE
a = (1 + P)(r + llu)pljks
9 Express bus
Light rail
0.0346
0.0239
+ m/k,
dollars per passenger-space mile
The overall traffic cost of propelling a light-rail passenger space 1 mile, then, is merely 0.69 times that of propelling a bus passenger space 1 mile. 4.6 Linear Cost-Function
Estimates
Our Sets. 4.2 through 4.5 have now determined
the parameters :
AI = 0.98 millions of dollars per line mile per year Az = 1.25 millions of dollars per line mile per year
HANS
28
BREMS
al = 0.0346 dollars per passenger-space mile cz2= 0.0239 dollars per passenger-space mile, of our linear cost function
(13) for express buses and light rail c = A1 + UlX
(14)
c = A2 +azx,
(15)
respectively, shown graphically in Fig. 2. Light rail has higher line-acquisition cost but lower traffic cost than express buses. At precisely which output per line mile X would express buses and light rail have the same cost C? Set the right-hand sides of (14) and (15) equal and find X = (AZ - Al)/(ar
- az) = 2,5.2 million,
(16)
which is feasible, i.e., slightly less than one-half of the upper bound X = 53.8 million found in Sec. 4.1. Considering cost, alone, then, a transit authority would be indifferent between express buses and light rail at an output per line mile equalling X = 25.2 million. At that output, what would traffic frequency f be like? Take (1) and (12) together and express the latter as j = X/(2hkv). (17) C IXPRESSBUS: C = 0.98 + 0,0346X MIGHTRAIL:
C = 1,25 + 0,0239X
OUTPUTPER LINE MILE, MILLIONS OF PASSENGER SPACES IN BOTHDIRECTIONSPER YEAR FIQURE 2
LIGHT
RAIL TRANSIT
29
Apply the numerical values of h and k used in Sec. 4.3 and let number of vehicles per train be v = 1 for buses and v = 2 for light rail. For buses, Eq. (17) will then give us .f = 48.2 corresponding to a bus starting a round trip every 1.24 minutes. For light rail f = 14.0 corresponding to a two-vehicle train starting every 4.29 minutes. 4.7. The Sensitivity
of the Cost-Indiference
Point
The cost-indifference point is sensitive to changes in the four parameters Al, AZ, al, and az. Changes in the slopes al and a2 are particularly interesting. Assume express buses to be diesel-powered. Assume light-rail vehicles to be propelled by electricity generated by coal-fired, hydroelectric, or nuclear power plants. A higher relative price of oil would then raise the slope al but leave the slope a2 unaffected. The effect upon the cost-indifference point would be described numerically as follows. In (16) take the partial derivative of X with respect. to al. Multiply by the ratio al/X and arrive at the elasticity of X with respect to al : ax a1 _-
A2 =
aa, x
-
Al
a1
----=
(a1 - a212x
a1
- _____ = - 3.23, al - a2
(18)
which is valid in the immediate neighborhood of the cost-indifference point. The elasticity (18) is negative and says that raising the slope al by 1% will reduce X by 3.23%, in other words move the cost-indifference point 3.23y0 to the left. Next, let light-rail vehicles be propelled at. first by electricity generated by oil-fired power plants. A conversion to presumably cheaper nuclear power would then reduce the slope a2 but leave the slope al unaffected. The effect upon the cost-indifference point would be described numerically as follows. In (16) take the partial derivative of X with respect to ~2. Multiply by the ratio an/X and arrive at the elasticity of X with respect to a2 8X a2 a2 A2 - A1 a2 -z---z _= a1 - u2 au, x (a1 - c&2)2 x
2.23,
(1%
which is valid in the immediate neighborhood of the cost-indifference point. The elasticity (19) is positive and says that reducing the slope a2 by 1% will reduce X by 2.23%, in other words move the cost-indifference point 2.23% to the left. 5. CONCLUSIONS The empirical base of this article is the Pittsburgh South Hills corridor project L-33,and conditions elsewhere may be quite different. With this reservation our conclusions are the following.
30
HANS BREMS
At large outputs cost considerations do favor light rail, but its cost superiority is moderate. Even at its upper bound, light rail still costs 89% of express buses and saves merely 0.31 million dollars per line mile per year. At such moderate cost superiority, factors other than cost may become prominent. On the one hand, a light-rail line is a larger investment than a busway, and light-rail vehicles are a longer investment than buses. In an uncertain world larger and longer investments seem less attractive than smaller and shorter ones, hence an express-bus system may become preferable even in ranges lying to the right of our cost-indifference point. On the other hand, buses are noisy. De Leuw, Cather, and Company [2] estimate interior and exterior noise levels. Measured in dBA-weighted decibels the interior noise of a diesel bus at 35 miles per hour and a Boeing light-rail vehicle at 40 miles per hour were 80 and 67 to 70, respectively. Exterior noise at 50 feet from vehicle was 84 and 76, respectively. Lower tolerance thresholds of the noise, carbon monoxide, hydrocarbons, and nitrogen oxide generated by express buses may make a light-rail system preferable even in ranges lying to the left of our cost-indifference point. ACKNOWLEDGMENTS For a demonstration in 1977 of what De Leuw, Cather, and Company call a “highperformance light-rail system” the author is indebted to the Gothenburg Transport Authority and its Vice-President in charge of planning, Mr. Ragnar Domstad. For critical reading of an earlier draft, the author is indebted to John F. Due of the University of Illinois, to Vukan R. Vuchic of the University of Pennsylvania, to Mr. K. N. Andersen of the Copenhagen Transit Authority, and to an anonymous referee of this Journal. For a grant administered by the College of Commerce of the University of Illinois, enabling him to devote the full summer of 1977 to research, the author expresses his gratitude to Investors in Business Education.
REFERENCES 1. County of San Diego, Public Works Agency, Department of Transportation, “Investigation of a Light Rail System for the San Diego Region,” San Diego, California (1975). 2. De Leuw, Cather, and Company, “Light Rail Transit: State of the Art Review,” prepared for the U.S. Department of Transportation, Urban Mass Transportation Administration, Superintendent of Documents, Washington, D.C. (1976). 3. Port Authority of Allegheny County, “Comparative Analysis Study of Alternative Transit Systems : South Hills Corridor,” Allegheny County, Pennsylvania (1976).
OF URBAN ECONOMICS
7, 20-30 (1980)
Light Rail Transit:
Cost and Output
HANS BREMS Department
of Economics, University
of Illinois
at Urbana-Champaign 61801
Received November 14, 1977; revised February 7, 1978 Growing concerns about environment, oil depletion, and traffic congestion call for a fresh look at msss transit-specifically non-polluting, low noise, nonoil-consuming, high-speed electric transit. This article attempts to relate cost to output, both for light rail and the alternative transit mode, i.e., the express bus. For that purpose a simple algebraic framework is developed, tailored to data obtained from a study of alternative transit modes in the Pittsburgh South Hills corridor. The data were prepared by De Leuw Cather and Company for the Port Authority of Allegheny County and were published by the U.S. Department of Transportation in a recent study of the state of light-rail art. The article finds the volume of output at which, considering cost alone, a mass-transit authority would be indifferent between light rail and express buses. At the end, noncost factors are considered.
1. INTRODUCTION Bus transit offers wide but low-speed coverage at minimum investment but high traffic cost and objectionable noise and pollution levels. At very heavy investment and no pollution, heavy rail offers high-speed but narrow coverage. At lower traffic cost and less noise than buses, and at lower construction cost per mile and wider coverage than heavy rail, light rail offers a nonpolluting compromise between these two extremes. Yet, as shown in Fig. 1, light rail on exclusive right of way may achieve high travelling speed-so essential in attracting ridership. For its cost-output study of light rail and express buses, this article will develop an algebraic framework using the following notation. 2. NOTATION
A = a = b = /? =
line-acquistion cost, dollars per line mile per year traffic cost, dollars per passenger-space mile time consumed by a train dwelling at a terminal, hours terminal dwelling time as a fraction of travelling time, pure number 20
00941190/80/010020-11$02.00/O
LIGHT
RAIL TRANSIT
21
C = cost, dollars per line mile per year d = distance between terminals, miles F = number of vehicles in fleet j = frequency, average number of trains starting a round trip per hour h = hours of operation of system per year j = average number of vehicle hours of operation per fleet vehicle per yf= k = vehicle capacity, passenger spaces per vehicle m = operation and maintenance cost, dollars per vehicle mile n = average number of trains operating wi = fraction of h during which i-thirds of fleet is operating, i = 1, 2, 3 P = line-acquisition price, dollars per line mile p = vehicle-acquisition price, dollars per vehicle r = rate of interest, pure number per year = travelling speed between terminals, miles per hour S T = time consumed by a tra.in between starting two consecutive round trips, hours = time consumed by a train travelling between terminals, hours t u = useful life of line or vehicle, years V = average number of vehicles per train W = output per fleet vehicle, passenger-space miles in both directions per fleet vehicle per year X = output per line mile, passenger spaces in both directions per year Y = output of line, passenger-space miles in both directions per year. 3. OUTPUT 3.1 Output of Line
Imagine a single line d miles long on which j trains are starting a round trip every hour for h hours per year. Each train has v vehicles and each vehicle k passenger spaces. Output is produced whether the vehicle is travelling in one or the other direction, hence output of the line in passenger-spacemiles in both directions per year will be Y = 2djhkv.
(1)
Define travelling speed between terminals as s = d/t.
(2)
If T is the number of hours consumed by a train between starting two consecutive round trips, then l/T is the number of times that, train can start a round trip per hour. An n/T is the number of times all n trains can do so. But that is the same thing as traffic frequency j = n/T.
(3)
HANS BREW
22 30
1 :~;;“‘s”~~,, 8 PHILADELPHIA RED ARROW SAN FRANCISCO
@
-lo
THE HAGUE% FRANKFURT
$AN
PHILADELPHIA CITY TRANSIT
0
P,TTsB$GH'wARK
COLOGNE oByN
HANNOVERO HAMBURG 0
I
20
40
GOTHENBURG OPITTSBURGH
0 MUNICH
FRANCISCO
I
0
I
1
80
100
I
60
EXCLUSIVEANDSEMI-EXCLUSIVERIGHT-OF-WAY IN PER CENTOF NETWORK 0
EXISTING
DATA:
@FUTURE
SYSTEMS
DE LEUW, CATHER AND COMPANY CONVERTED TO MILES.
FIQURE
(1976).
SYSTEMS 84.
KILOMETERS
1
The time consumed by a train between starting two consecutive round trips is twice the sum of travelling and terminal dwelling time: T = 2(t + b). Let terminal
dwelling
(4)
time be in proportion
to travelling
b = @t,
time: (5)
where p is a parameter equalling, say, &. Now insert (5) into (4), (4) into (3), (3) and (2) into (1) and find output of line Y = hknsv/(l in passenger-space miles in both directions
+ p)
(6)
per year.
3.2 Output Per Fleet Vehicle Define average output per fleet vehicle as W = Y/F
= hknsv/[(l
in passenger-space miles in both directions
+ P)F]
(7)
per fleet vehicle per year. All
LIGHT
RAIL TRANSIT
23
transit authorities must live with the fact that their full fleet F is needed for only a fraction of the hours of operation. Now n is average number of trains operating and v average number of vehicles per train, hence nv is average number of vehicles operating. That number must be less than or equal to number of vehicles in fleet : nv 5 F.
(8)
The average number of vehicles operating, nv, is a weighted average of the numbers operating at different times. Consider only three possible fractions, i.e., f, $ and g of fleet operating. Define wi as the fraction of hours of operation h during which i-thirds of the fleet is operating, where i = 1, 2, and 3 and w1 + ~2 + w3 = 1. In other words, wi are the weights of the weighted average of the numbers of vehicles operating at different times, nv = + F far the fraction WI of hours per year nv = 3 F for the fraction w2 of hours per year nv = F for the fraction w3 of hours per year, so the average number of vehicles operating will be nv = ($wl + $w2 + w,)F.
(9)
hnv/F = j,
(10)
In (7) call the ratio which is a measure of vehicle utilization equalling vehicle hours of operation per fleet vehicle per year. Use (10) to write average output per fleet vehicle as W = Y/F = jks/(l + p), (11) which is a measure of vehicle productivity
to be needed in Sec. 4.3 below.
S.S Output Per Line Mile Define average output per line mile as X E Y/d = hknsv/[ (1 + P)d] in passenger spaces in both directions
(12)
per year.
4. COST 4.1 A Linear Cost Function As a first approximation
let us estimate a linear cost function C=A+aX
(13)
in dollars per line mile per year. Output X per line mile may be expanded
HANS BREMS
24
by expanding traffic. But at given average vehicle hours of operation per fleet vehicle per year j, at given vehicle capacity k, at given travelling speed s, at given dwelling fraction B, output per fleet vehicle (11) will be given, and traffic can be expanded only by expanding fleet F. Up to an upper bound, let such expansion occur at constant traffic cost a per unit of output X per line mile. The upper bound is determined by safety considerations and technology : At high-speed operations and automatic block signalling systems, train intervals (‘Lheadways”) cannot be lower than 90 to 120 seconds. At a 120-second headway, what would output X per line mile be like? Well, let hours of operation per year be h = 3900; let vehicle capacity be k = 115 passenger spaces per vehicle; and let average number of vehicles per train be v = 2. Output per line mile will then be X = 53.8 million passenger spaces in both directions per year. In what kind of dollars should we measure cost? Interest and depreciation on line and vehicles should be calculated on the historical cost of acquisition, so we follow De Leuw, Cather, and Company [2] in measuring them in 1975 dollars. Operating and maintenance costs, on the other hand, should be those of a typical year of operation during useful life, so we follow De Leuw, Cather, and Company [2] in measuring them in 1985 dollars. 4.2 Line-Acquisition
Cost
De Leuw, Cather, and Company [2], estimated the price of a new line in dollars per line mile and found : TABLE
1
Line-acquisition price P, millions of dollars per line mile
Busway
Light rail
8.61
11.05
Acquisition price P may be allocated among individual years of useful life by multiplying it by T + l/u, where r is the rate of interest in a pure number per year, u is useful life of the line in years, and l/u the depreciation coefficient in a pure number per year. Let r = 0.08 and u = 30, then r + l/u = 0.1133, and line-acquisition cost is: TABLE
2
Line-acquisition cost A = (r + l/u) P, millions of dollars per line mile per year
Busway
Light rail
0.98
1.25
LIGHT
&!I Trafic
RAIL TRANSIT
25
Cost: Vehicle Acquisition
Traffic cost includes, first, vehicle-acquisition cost and, second, cost of operation and maintenance. We begin with vehicle-acquisition cost. The vehicle-acquisition price of a light-rail vehicle was estimated by De Leuw! Cather, and Company (1976), and the conventional 40-foot urban bus currently used by most U.S. mass transit operations is priced around $100,000. TABLE
3
Vehicle-acquisition price p, dollars per fleet vehicle
Express bus
Light rail
100,000
400,000
Acquisition price p may be allocated among individual years of useful life by multiplying it by r + l/u as before. As before, let T = 0.08. County of San Diego Cl], estimated useful life u to be 15 years for buses and 30 years for light-rail vehicles. Consequently r + l/u = 0.14667 for buses and 0.11333 for light rail-vehicles, and vehicle-acquisiton cost will be: TABLE
Vehicle-acquisition cost (r + l/u)p, dollars per fleet vehicle per year
4 Express bus
Light rail
14,667
45,332
To find vehicle-acquisition cost in dollars per passenger-space mile we must estimate the right-hand side of Eq. (ll), jrCs/(l + p). Let an austere U.S. transit system operate for 15 hours a day, 5 days a week, 52 weeks a year. In that case h = 3900 hours per year. Assume that w1 = w2 = co3= 5. Then according to (9) nv = QF, and according to (10) j = 2600. Define a passenger space as 5 square feet of vehicle floor space, then k = 67 and 115, respectively, for express buses and light rail. Measured in miles per hour, travelling speed s was estimated by De Leuw, Cather, and Company at 6 to 22 and 16 to 25, respectively, for express buses and light rail, or a simple average of 14 versus 20. The main reason for the difference is probably acceleration ability. Measured in feet per second2, the service acceleration rate was estimated at 0.3 to 3.2 and 4.5 to 5.1, respectively for express buses and light rail, or a simple
HANS BREMS
26 average of 1.8 versus side of Eq. (11) :
4.8.
Let @ = &. We can now find the right-hand TABLE
5 Express bus
Vehicle use j, hours of operation per fleet vehicle per year Vehicle capacity k, passenger spaces per vehicle Travelling speed s, miles per hour Terminal dwelling-time fraction p, pure number Output per fleet vehicle jks/ (1 + p), millions of passenger-space miles in both directions per fleet vehicle per year Now divide
the result
of Table
4 by that
TABLE
Light rail
2600
2600
67 14
115 20
0.10
0.10
2.22
5.44
of Table 5 and find:
6
Vehicle-acquisition cost (1 + 0) (r + l/u)p/ jks, dollars per passenger-space mile
Express bus
Light rail
0.0066
0.0083
Becailse the light-rail vehicle has twice the useful life U, has a 1.72 times higher vehicle capacity k, a,nd has a 1.43 times higher travelling speed s, its vehicle-acquisition cost in dollars per passenger-space mile is merely 1.26 times that of a bus. 4.4 Trufic
Cost: Operation and Maintenance
De Leuw, Cather, and Company (1976) estimate costs of operation and maintenance in dollars per vehicle mile as follows : TABLE
7 Express bus
Light rail
Operation and maintenance costs m, dollars per vehicle mile : Maintenance of rights-of-way Energy General and administration Vehicle maintenance Conducting transportation
0.037 0.101 0.628 0.258 0.852
0.191 0.209 0.532 0.216 0.650
Total, m, dollars per vehicle mile
1.876
1.798
LIGHT
RAIL TRANSIT
27
So the operation and maintenance cost of propelling a light-rail vehicle 1 mile is 0.96 times that of propelling a bus. Maintenance of rights-of-way is 5.16 times as high; energy 2.07 times as high; general and administration 0.85 times as high; vehicle maintenance 0.84 times as high; and conducting transportation 0.76 times as high. The reasons for the lightrail superiority in conducting transportation are, first, that at the same money wage rate in dollars per hour, wage cost in dollars per vehicle mile will be lower for the vehicle travelling more miles per hour. Second, the driver’s wages in dollars per vehicle mile of a two-vehicle train are only one-ha.lf of those of a one -vehicle train. Such costs of operation and maintenance in dollars per vehicle mile are easily translated into dollars per passenger-space mile : Divide Table 7 by vehicle capacity k and fmd : TABLE
8 Express bus
Light rail
0.0280
0.0156
Operation and maintenance costs m/k, dollars per passenger-space mile
So because a light-rail vehicle carries 1.72 times as many passenger spaces, the operation and maintenance cost of propelling a light-rail passenger space 1 mile is merely 0.56 times that of propelling a bus passenger space one mile. 4.5 Overall Tra&
Cost
Traffic costs are vehicle-acquisition cost plus operation and maintenance cost, so let us add Tables 6 and 8 and find overall traffic cost: TABLE
a = (1 + P)(r + llu)pljks
9 Express bus
Light rail
0.0346
0.0239
+ m/k,
dollars per passenger-space mile
The overall traffic cost of propelling a light-rail passenger space 1 mile, then, is merely 0.69 times that of propelling a bus passenger space 1 mile. 4.6 Linear Cost-Function
Estimates
Our Sets. 4.2 through 4.5 have now determined
the parameters :
AI = 0.98 millions of dollars per line mile per year Az = 1.25 millions of dollars per line mile per year
HANS
28
BREMS
al = 0.0346 dollars per passenger-space mile cz2= 0.0239 dollars per passenger-space mile, of our linear cost function
(13) for express buses and light rail c = A1 + UlX
(14)
c = A2 +azx,
(15)
respectively, shown graphically in Fig. 2. Light rail has higher line-acquisition cost but lower traffic cost than express buses. At precisely which output per line mile X would express buses and light rail have the same cost C? Set the right-hand sides of (14) and (15) equal and find X = (AZ - Al)/(ar
- az) = 2,5.2 million,
(16)
which is feasible, i.e., slightly less than one-half of the upper bound X = 53.8 million found in Sec. 4.1. Considering cost, alone, then, a transit authority would be indifferent between express buses and light rail at an output per line mile equalling X = 25.2 million. At that output, what would traffic frequency f be like? Take (1) and (12) together and express the latter as j = X/(2hkv). (17) C IXPRESSBUS: C = 0.98 + 0,0346X MIGHTRAIL:
C = 1,25 + 0,0239X
OUTPUTPER LINE MILE, MILLIONS OF PASSENGER SPACES IN BOTHDIRECTIONSPER YEAR FIQURE 2
LIGHT
RAIL TRANSIT
29
Apply the numerical values of h and k used in Sec. 4.3 and let number of vehicles per train be v = 1 for buses and v = 2 for light rail. For buses, Eq. (17) will then give us .f = 48.2 corresponding to a bus starting a round trip every 1.24 minutes. For light rail f = 14.0 corresponding to a two-vehicle train starting every 4.29 minutes. 4.7. The Sensitivity
of the Cost-Indiference
Point
The cost-indifference point is sensitive to changes in the four parameters Al, AZ, al, and az. Changes in the slopes al and a2 are particularly interesting. Assume express buses to be diesel-powered. Assume light-rail vehicles to be propelled by electricity generated by coal-fired, hydroelectric, or nuclear power plants. A higher relative price of oil would then raise the slope al but leave the slope a2 unaffected. The effect upon the cost-indifference point would be described numerically as follows. In (16) take the partial derivative of X with respect. to al. Multiply by the ratio al/X and arrive at the elasticity of X with respect to al : ax a1 _-
A2 =
aa, x
-
Al
a1
----=
(a1 - a212x
a1
- _____ = - 3.23, al - a2
(18)
which is valid in the immediate neighborhood of the cost-indifference point. The elasticity (18) is negative and says that raising the slope al by 1% will reduce X by 3.23%, in other words move the cost-indifference point 3.23y0 to the left. Next, let light-rail vehicles be propelled at. first by electricity generated by oil-fired power plants. A conversion to presumably cheaper nuclear power would then reduce the slope a2 but leave the slope al unaffected. The effect upon the cost-indifference point would be described numerically as follows. In (16) take the partial derivative of X with respect to ~2. Multiply by the ratio an/X and arrive at the elasticity of X with respect to a2 8X a2 a2 A2 - A1 a2 -z---z _= a1 - u2 au, x (a1 - c&2)2 x
2.23,
(1%
which is valid in the immediate neighborhood of the cost-indifference point. The elasticity (19) is positive and says that reducing the slope a2 by 1% will reduce X by 2.23%, in other words move the cost-indifference point 2.23% to the left. 5. CONCLUSIONS The empirical base of this article is the Pittsburgh South Hills corridor project L-33,and conditions elsewhere may be quite different. With this reservation our conclusions are the following.
30
HANS BREMS
At large outputs cost considerations do favor light rail, but its cost superiority is moderate. Even at its upper bound, light rail still costs 89% of express buses and saves merely 0.31 million dollars per line mile per year. At such moderate cost superiority, factors other than cost may become prominent. On the one hand, a light-rail line is a larger investment than a busway, and light-rail vehicles are a longer investment than buses. In an uncertain world larger and longer investments seem less attractive than smaller and shorter ones, hence an express-bus system may become preferable even in ranges lying to the right of our cost-indifference point. On the other hand, buses are noisy. De Leuw, Cather, and Company [2] estimate interior and exterior noise levels. Measured in dBA-weighted decibels the interior noise of a diesel bus at 35 miles per hour and a Boeing light-rail vehicle at 40 miles per hour were 80 and 67 to 70, respectively. Exterior noise at 50 feet from vehicle was 84 and 76, respectively. Lower tolerance thresholds of the noise, carbon monoxide, hydrocarbons, and nitrogen oxide generated by express buses may make a light-rail system preferable even in ranges lying to the left of our cost-indifference point. ACKNOWLEDGMENTS For a demonstration in 1977 of what De Leuw, Cather, and Company call a “highperformance light-rail system” the author is indebted to the Gothenburg Transport Authority and its Vice-President in charge of planning, Mr. Ragnar Domstad. For critical reading of an earlier draft, the author is indebted to John F. Due of the University of Illinois, to Vukan R. Vuchic of the University of Pennsylvania, to Mr. K. N. Andersen of the Copenhagen Transit Authority, and to an anonymous referee of this Journal. For a grant administered by the College of Commerce of the University of Illinois, enabling him to devote the full summer of 1977 to research, the author expresses his gratitude to Investors in Business Education.
REFERENCES 1. County of San Diego, Public Works Agency, Department of Transportation, “Investigation of a Light Rail System for the San Diego Region,” San Diego, California (1975). 2. De Leuw, Cather, and Company, “Light Rail Transit: State of the Art Review,” prepared for the U.S. Department of Transportation, Urban Mass Transportation Administration, Superintendent of Documents, Washington, D.C. (1976). 3. Port Authority of Allegheny County, “Comparative Analysis Study of Alternative Transit Systems : South Hills Corridor,” Allegheny County, Pennsylvania (1976).