Journal of Advanced Transportation 18:1 pp. 77-111, 1984
Optimization of Transit-System Characteristics
J.Edward Anderson
The transit industry is facing declining ridership and increasing costs with no apparent end in sight. This paper takes the view that totally new solutions are needed. The approach is to examine the equation for total cost of a transit system per passenger-mile to determine how to configure a new system to minimize this quantity. Term-by-term analysis leads to derivation of a consistent set of optimum characteristic: Guideway costs are minimized by distributing the load in very small capsules. The fleet cost is minimized by increasing the average speed without increasing the cruising speed by use of offline stations, which in turn minimize energy use by permitting nonstop trips. Maintenance costs are minimized by designing a very light-weight, automated vehicle with very few moving parts. While this general configuration has been known for several decades, it has not been generally recognized that it can be derived by minimization of system costs, and that cost minimization is obtained simultaneously with service maximization. While a great deal of controversy surrounded this concept a decade ago, advances in technology make it fully practical now.
Introduction
The aim of the transit-system designer should be to choose the characteristics of his system in such a way that the life-cycle cost per passenger-mile is minimum subject to constraints on performance, safety, dependability, environmental impacts, etc. As federal capital grants decline in both percentage and amount, interest has increased in this important problem of cost effectiveness. Pushkarev (1982), based on a comprehensive study of American rail systems, concluded that the field needs a new, narrow-guideway, low-cost transit system that does not require snow melting, i.e., a much more cost-effective system.
The problem has rightly been approached by trying to shave cost of acquisition and support of more-or-less conventional systems. I have for many years tried to look at the problem from a fundamental view, Anderson (1978), (1979), (1981), aimed at clarifying the characteristics of a transit system, optimum in the sense described above. The purpose of this paper is to update and consolidate my research on transit-system optimization and to further clarify the selection of characteristics of a new, optimum transit system.
J. Edward Anderson is Professor of Mechanical Engineering at the University of Minnesota.
The Equation for Cost per Passenger-Mile
In Appendix A I have derived a general equation for the cost per passenger-mile of any transit system in two different ways, and in a form useful for general systems analysis. This derivation is a modification of the derivation given in Anderson (1981). The results are presented in equations (A-3), (A-4), and (A-5) in terms of fixed costs and variable costs. These equations are not repeated here because it is essential for the serious reader to follow the derivation if the results are to be fully understood.
Note that the fixed costs are independent of the number of passenger-miles traveled, and the variable costs are proportional to the number of passenger-miles traveled. If the variable costs per passenger-mile, denoted by VC, are greater than can be borne by a reasonable fare per passenger-mile, the system will have to be subsidized at all levels of patronage, represented by the term PM/YR/LMI (passenger-miles per year per lane mile). If the annualized fixed-facility costs per lane mile, represented by FC,are high, then, as is well understood, the patronage level must also be high to bring the total cost per passenger-mile to a reasonable level.
In the following paragraphs, each of the terms in the equation for cost per passenger-mile is analyzed in terms of the system characteristics required to minimize the total cost per passenger-mile. Many inventors have attempted such cost minimization by more or less intuitive approaches, the result of which has been a number of sub-optimal systems. The approach through study of the cost equations can bring together all of the pieces of analysis and determine if, by minimizing one term, another term doesn't increase by a larger amount.
Fixed Costs per Year per Lane-Mile
The cost per year is the annual payment on the total cost. The ratio of the annual payment to the total cost is the annualization factor
A = i_____
1 - (1 + i)-n
where i is the interest rate and nisthe lifetime of the equipment. The derivation of this equation can be found in any text on engineering economics. A little analysis shows that if the equipment is paid off in less than nyears, the present value of the sum of the payments at a given interest rate is the same as if it is paid off in n years; therefore, it is logical to take nas the lifetime of the equipment. Much debate has occurred in the economic literature over what value to use for i for public-sector expenditures, but currently a value of 0.07 is reasonable. Using this value, we have
n, yrs. 5 10 15 20 30 40
A0.2440.1420.1100.0940.0810.075
The lifetime of fixed transit facilities is usually taken as about 30 years, and it is seen that there is only a small economic advantage to longer life.
Minimum fixed cost per lane-mile is attained for a transit system if the way is shared with road traffic, in which case, because transit carries only 3% of the urban passenger-miles in the United States (UMTA 1983) the way is paid for by highway taxes. In many cases, using a shared roadway is the only practical economic alternative; however, then the speed is very slow during rush periods and there are many accidents. For example, from UMTA (1983), the average number of collisions per million vehicle-miles for the average U.S. streetcar is 180, and the corresponding number for nine U.S. rail rapid transit systems is only 3.6, a ratio of 50:1. The fact that the exclusive-guideway rail rapid transit systems usually operate in trains only partially explains this large ratio. For reasons of speed and safety, exclusive guideways are used when they can be justified. I take as my objective, therefore, the reduction of the cost of these systems, but without losing sight of the need to compare them with mixed-traffic systems.
The most important factor in the cost of the roadway or guideway of a transit system is the weight of the vehicles it must carry. According to Pushkarev (1982), the damage to roads has been found to be proportional to the fourth power of the weight of the vehicles carried. The cost of a transit guideway similarly increases with the weight of the vehicles, with the power relationship dependent on cross sectional shape and other factors. The important point is that guideway cost increases as vehicle weight increases. This leads to a fundamental idea that has been expressed by many people in many ways: if the capacity of a transit vehicle can be split up into smaller units so that the load is distributed in smaller increments over the guideway, the required guideway weight per unit of length and the corresponding cost per unit of length reduce. Typical guideways for elevated rapid rail systems weigh in the neighborhood of 2000 pounds per foot. By using the above simple principle, we have found a way to reduce the guideway weight to only 140 pounds per foot. Heavier guideways not only cost more directly, they require larger equipment in manufacture and erection and more space, all of which add to the cost.
Based on guideway costs alone, the use of smaller vehicles traveling at shorter headways is a trend to be encouraged. Some analysts have argued that once the headway is down to one or two minutes, there is no point in reducing it further. Their analysis is based on the effect of reducing headway on reducing waiting time and hence on increasing patronage. The effect is very small. But a headway of one minute at say 30 miles per hour corresponds to a nose-to-nose vehicle spacing of half a mile. To obtain a line capacity equivalent to, say, two freeway lanes of automobile traffic, roughly 4000 people per hour, at one minute headways requires an average of 67 people per vehicle or train. If this capacity could be provided in units 1/60th as large, traveling at nose-to-nose spacings of 44 feet, the loading on the guideway is substantially reduced, and so is its cost. This is a principle that has been advocated by many people over the past 30 years—the question is how to achieve it practically. The main objective of headway reduction, at this point in the analysis, is guideway cost reduction, not reduced waiting time.
Other design choices affect the weight and cost of the guideway:
1. Standing vs. seated passengers. If the vehicles are designed to permit adult passengers to stand, they must be higher and wider than if they are designed for seated passengers. Several analyses, including my own, show that the requirement of standing passengers roughly doubles the weight per unit of length of the guideway. Standing-passenger vehicles require that the fore-and-aft and lateral accelerations be restricted to about one-eighth of gravity, whereas, if all adult passengers are seated, these values can be doubled. The result is reduction in curve radii of a factor of two, which is often critical in engineering the system into a city without tearing down buildings. A third factor favoring the smaller cross-section of seated-passenger vehicles is air drag reduction. This has not been a major factor in large trains that stop frequently, but becomes more important as the vehicle size reduces.
2. Optimum cross section. As shown in Anderson (1978), there is always an optimum cross section that minimizes the weight and cost of a guideway per unit of length. The optimum is a narrow, deep beam. To make use of this fact of structural theory requires, however, that the vehicle be designed to fit the requirements of the guideway, and not vice versa, as has so often been the case. The narrow beam also has the advantage of being the smallest in visual impact and the easiest to engineer into the city.
3. Hanging vs. supported vehicles. Some inventors who have appreciated the obvious advantages of narrow-beam guideways have assumed that the vehicle must hang from the guideway. The main cost increase if this is done comes from the requirement for a certain clearance for road traffic. If the vehicles hang from the guideway, it must be roughly eight feet higher from the ground than if the vehicles ride on top. Moreover, the hanging vehicle requires that the guideway support post must be cantilevered. Analysis shows that the lateral bending moment at the root of the support posts due to maximum cross winds and vehicle weight is roughly twice as large if a hanging-vehicle configuration is used.
4. Guideway heating. Pushkarev (1982) makes a particular point of the large increase in operating cost that result in climates in which ice and snow are present in guideway transit systems that require snow melting. Because of the use of electric power rails to provide power to the vehicles, it is impractical to remove snow by mechanical means without damaging these rails. Therefore, heaters have been embedded into the guideways to melt snow. Unfortunately, the amount of energy required to remove snow in this manner is greatly increased (80 calories per gram).
The answer is not in more clever methods of removing snow, but in the development of a configuration that requires very little if any snow removal. This is more difficult in supported-vehicle systems than in hanging-vehicle systems, but, because of the higher foundation costs of the latter, is worthy of serious attention in the configuration design.
A final factor in the fixed costs per lane-mile is the requirement for land. This is often neglected because public land is used, and taking the land is then not a direct cost. Yet, with light rail transit (LRT) systems operating on city streets, the required width for a two-way system is about 27 feet. The main purpose of LRT is to penetrate the downtown during the rush periods, when the automobile traffic is the heaviest. Fixed costs will then be increased by the need to construct multi-story parking structures to replace on-street parking. Minimization of land-condemnation requirements and costs is compatible with the use of minimum-width guideways.
This section can be summarized with the observation that the trend in designing minimum-cost-guideway transit systems should be toward the smallest practical vehicles, designed for seated passengers only, riding on top of optimum-cross-section guideways that require no heating. Use of very small vehicles requires automation, and we will show that the need for cost reduction in guideways is a more important reason for moving toward automation than the reduction of operating costs.
Stations
Station costs increase with the size and number of vehicles that must be accommodated at one time, and with the amount of space required for waiting passengers. Smaller vehicles operating at closer headways reduce the required platform length as well as the passenger wait time, thus reducing the space required for waiting passengers. Maximum station throughput is attained by using a platooning operating procedure, such as described by Dais and York (1973). As an example of the station platform length reduction possible, Anderson (1980) showed that the maximum station flow during the busiest hours at the busiest station of the Philadelphia Lindenwold rapid rail system could be handled by three-passenger vehicles in a platooning operation using nine station berths. The present system uses eight-car trains, requiring a platform about 560 feet long. A nine-berth station for three-passenger, automated vehicles would be 81 feet long, a ratio of 6.9:1. Moreover, the required line headway for the small vehicle system would be 1.14 seconds.
The costs of the guideways of a transit system can be considered the economic drag on the system, whereas the stations are the economic generators. The more stops that are added, the greater is the accessibility to the system and hence the greater the potential patronage. The problem with conventional transit systems is that adding stations slows the average speed because it increases the number of stops per trip. This is the fundamental dilemma of on-line-station transit systems: high speed is attained with wide station spacing and hence poor access; yet good access requires close station spacing, which results in low average speed. People do not ride the system either because it is too slow or because they can't get to it conveniently. The answer is a new configuration using off-line stations. This is not new, but has been considered for over 30 years. The reason for off-line stations primarily is not capacity, as widely assumed, but speed. Adequate capacity can be obtained by the above-mentioned platooning operational procedure, but adequate speed requires that the trip bypass intermediate stations. Use of nonstop trips permits saving of the kinetic energy that is lost each time the vehicle stops and provides remarkable improvements in service.
Freed of the speed limitation of station spacing, we are able to consider the economics of adding stations without loss of patronage due to speed reduction. The problem is solved mathematically in Appendix B, in which a criterion is derived to show whether or not adding a station increases or decreases the total cost per passenger-mile. In the example shown, substantially smaller station spacing would be economical if adding stations does not reduce the average trip speed.
Vehicle Cost Characteristics
The variable cost term in equation (A-3), VC, is composed of two terms, given in equation (A-5). The first term is related to vehicle capital costs, and the second to operating and maintenance costs. In this section, I consider the numerator of the first term. The factors involved are the manufacturing cost of a vehicle per place, the associated support-service capital costs, the lifetime of the equipment, and the size of the maintenance float.
On page 98 of Anderson (1978), I plotted data from the Lea Transit Compendium showing that the quoted capital cost of transit vehicles per unit of capacity or per place is not dependent on vehicle capacity. In particular, there is no economy of scale in going to larger vehicles. In these considerations, the capacity is not the number of seats, but the manufacturer's quoted design capacity including standees. To update this information, I give some more recent information taken from Thompson (1982) in Table 1. By comparison, in the data shown in Anderson (1978) the Morgantown system gave the highest cost per place, and was about three times the average of the 28 other systems shown. The Fairlane system is clearly very much out of line in Table 1. According to industrial sources, a 180-passenger LRT vehicle currently costs about $1,000,000 or about $5600 per place.
Table 1. Economic Data on Automated Guideway Transit Systems
System / Total Cost per Lane-Mile / Vehicle Capacity, Places / Vehicle Cost per Place / O&M Cost per Place-Mile, ¢ / Average Speed, mphAirtrans / $7,240,000 / 40 / $9,300 / 4.3 / 10
Atlanta / 29,900,000 / 80 / 10,200 / 4.4 / 13
Busch Gardens / 5,300,000 / 96 / 6,700 / 8.8 / 11
Disney World / 21,420,000 / 20 / 8,300 / 1.5 / 5
Fairlane / 15,300,000 / 24 / 23,700 / 45.6 / 10
Houston / 16,100,000 / 36 / 5,900 / 26.1 / 6
King’s Dominion / 4,100,000 / 96 / 6,100 / 0.9 / 6
Miami / 29,000,000 / 99 / 2,900 / 1.5 / 17
Minnesota Zoo / 7,000,000 / 94 / 10,400 / 32.1 / 7
Morgantown / 18,300,000 / 21 / 14,000 / 8.7 / 17
Orlando / 19,200,000 / 100 / 7,050 / 2.2 / 21
Seattle-Tacoma / 30,500,000 / 102 / 6,300 / 1.6 / 9
Tampa / 16,100,000 / 100 / 5,240 / 2.6 / 9
Source:Thompson (1982)