41 Theory. Journal of Political Economy, Vol. 74, Models. TRB, Transportation Research Record April 1966. 610, 1976, pp. 12-18 . 6. C. F. Manski. The Analysis of Qualitative Choice. 11. W. Pecknold and J. Suhrbier. Tests of Transfer­ Department of Economics, MIT, Cambridge, PhD ability and Validation of Disaggregate Behavioral dissertation, June 1973. Demand Models for Evaluating the Energy Conser­ 7. D. McFadden. Conditional Logit Analysis of Quali­ vation Potential of Alternative Transportation Poli­ tative Behavior. In Frontiers in Econometrics cies in Nine U.S. Cities. Cambridge Systematics (P. Zarembka, ed'J, Academic Press, New York, and Office of Energy Conservation Policy, Federal 1974. Energy Administration, April 1977. 8. M. E. Ben-Akiva and S. R. Lerman. Disaggregate 12. S. R. Lerman. A Disaggregate Behavioral Model Travel and Mobility Choice Models and Measures of Urban Mobility Decisions. Department of Civil of Accessibility. In Behavioral T r avel Model.ling Engineering, MIT, Cambridge, PhD dissertation, (D. A. Hensher and P. R. St opher, eds .), Croom­ June 1975. helm, London, in preparation. 13. Task Termination Report-Internal CBD Travel De­ 9. T. J. Atherton. Approaches for Transferring Dis­ mand Modeling. Barton-Aschman Associates and aggregate Travel Demand Models. Department of Cambridge Systematics, Aug. 1976. Civil Engineering, MIT, Cambridge, MS thesis, Feb. 1975. 10. T. J. Atherton and M. Ben-Akiva. Transferability Publication of this paper sponsored by Committee on Urban Activity and Updating of Disaggregate Travel Demand Systems. Optimizing Urban Mass Transit Systems: A General Model Alan Black, Graduate Program in Planning, University of Texas at Austin This paper describes a model for determining the general dimensions of hypothesize a circular city with a definite center and an optimal mass transit system for an idealized urban area. The model with dens ity declining uniformly from the center in all is based on a circular city with a definite center and with density declin­ directions. The transit system consists of routes that ing uniformly from the center in all directions according to the negative exponential function. The transit system consists of radial routes that emanate from the center and contain discrete stops. By emanate from the center and contain discrete stops. Only trips to or use of integral calculus, a model was derived that rep­ from the center are considered, and travel is assumed to occur only in resented the total community costs of building and using radial and circumferential directions. The model represents total com­ such a system. By use of differential calculus, a pro­ munity costs of the system, defined to include travel time, operating cedure was developed to optimize the principal design costs, equipment, and construction. A recursive procedure was devised variables in the system: the number of radial routes, to find a simultaneous minimum with respect to the spacing of routes, their length, and the number and spacing of stops on number and spacing of stops on each route, and average headway. Nu­ merical analyses were conducted for six hypothetical cities by using vary­ each route. Numerical analyses compared three com­ ing values for the parameters of the density function. In each case, three mon forms of conventional transit: buses on city streets, types of transit systems were compared: conventional bus service, buses buses on exclusive lanes, and rail rapid transit. on exclusive lanes, and rail rapid transit. The optimal system in the larg­ Such an abstract model cannot be mechanically ap­ est city examined was exclusive bus lanes; in the other five cases, the plied to the complex, irregular pattern of a real city. optimal system was conventional bus service. Other interesting relations Abstraction is an unavoidable compromise if a model is that appeared in the results are summarized. to be made mathematically tractable. Similar ap­ proaches have been followed in many previous studies of transit optimization. A few of these will be cited The United States has entered a new era of massive in­ here; a fuller review can be found elsewhere (1). vestment in urban mass transit, prompted by the willing­ Most previous studies can be divided into two geo­ ness of Congress to authorize billions of dollars in fed­ metrical approaches. One of these assumes a gridiron eral aid for local transit systems. However, there is network of transit routes laid on a homogeneous infinite yet no systematic procedure for allocating these re­ city, usually with a uniform density of tr ip ends . What sources and determining whether a transit proposal is may have been the first study of this type was done by worthwhile. Each proposal is evaluated on an ad hoc Creighton and others {2) and involved both highway and basis, and considerable weight is given to the zeal of the transit grids; the object was to find the optimal com­ proponents, political pressures, and the current avail­ bination of investment in the two modes. Holroyd (3) ability of funds. Choice of technology has become a assumed a single grid of bus routes and derived a solu­ major issue in many areas, and the question of whether tion for the optimal spacing of routes and frequency of medium-sized cities should proceed with huge invest­ service. Two dissertations, one by Mattzie at Carnegie­ ments in fixed-guideway transit systems is particularly Mellon (4) and the other by Woodhull at Rensselaer controversial. Polytechnic (5), also dealt with grid systems of transit This paper summarizes a dissertation aimed at de­ routes. - termining the dimensions of an optimal tr ansit system The second approach is to examine a single transit for an idealized urban area (1). The approach was to line. Often one terminal is assumed to be in the central 42 business district (CBD), and only trips to or from this Objective terminal are considered. The object is usually to find The objective selected was to minimize total community the optimal spacing of stops. So~e studies have as­ costs, measured in dollars. Total costs were defined sumed uniCorm spacing, but the more interesting have to include capital investment for guideway and vehicles allowed for variable spacings. An eal'ly study of vari­ operating costs, and the door-to-door travel time of ' able spacing of stops was made by Schneider (6). Vuchic travelers. It would be desirable to include other types (J_) later did a fuller analys is of the problem. -Both re­ of community costs, such as externalities and intangi­ searchers assumed a constant density of boarding pas­ bles, but they were omitted because of the difficulty of sengers along the line and came to the conclusion that measuring them or converting them to dollar values. interstation spacings should increase as one approaches The approach to measuring time costs was to calcu­ the CBD. This is oppos ite to conditions normally found late the distance traveled from door to door and to esti­ on radial rail routes in which the spacings become mate average speed on the portions of the route tra­ smaller near the CBD. versed. There are several components in door-to-door A third geometrical approach was taken by Byrne (8), travel time. One of these is the time spent in the transit who assumed a circula1· city of given radius and for ffie vehicle, which must be divided into two parts. The first case in which population density varies only with dis­ is the time that would accrue if the vehicle moved at its tance from the center, derived a solution for the optimal cruising speed from the rider's point of embarkation to number of radial routes. Byrne presented results for the point of debarkation. The second component con­ four density functions including' the negative exponential. sists of the additional time penalties incurred when the The model described here has a similar geometry and vehicle accelerates, decelerates, and waits at stops to also uses the 11 egative exponential fuuction but it in­ load and discharge other passengers. volves simultaneously solving for the optil~al number of Two other components of door-to-door travel time radials and optimal length of radials as well as other were included. One is walking time-the time it takes variables. a traveler to walk from origin to boarding stop or from where he or she gets off to his or her final destination. DESCRIPTION OF THE MODEL The other is waiting time, which depends on the sched­ uling of vehicles. It was assumed that average waiting time is half the scheduled headway (the time between suc­ The hypothetical city is a complete circle, uniform cessive buses or trains). throughout its 360 degrees, uninterrupted by barriers Operating costs depend on a number of factors, but or irregularities, and extending to infinity. The city here they were based solely on vehicle kilometers. This has a center that is taken to represent the CBD. The appears to be the most significant relation and also the transit network consists of an unknown number of radial simplest. lines that emanate from the center and extend an unknown The 1·e are two major types of capital costs : fixed distance. Each line has discrete stops and access is facilities (sltCh as i·oaclbed, structures, and stations) and possible only at these fixed points, whi~h must be de­ l'Unni11g equipment (bus es a11d trains). In analyzing a termined. Because of the assumed symmetry of the city specific proposal, detailed estimates of capital costs are the radial lines are equally spaced. Each has the same ' based on engineering drawings. This cannot be done for number of stops, spaced in the same way, and is of the a hypothetical city; hence, the cost of fixed facilities was same length.