Abstract
In regard to the problem of determining minimal-cost routes in a region with variable cost density, it has been shown elsewhere that, for a radially symmetric cost density which is inversely proportional to the distance from a central point O, the minimal cost between two pointsP 1 andP 2 which are equidistant from O is attained along a circular arc. This is not true in general for an arbitrary, radially symmetric cost density. In the present paper, critical conditions for determining when a circular arc will be a relative minimal-cost path are derived. These criteria are then employed to examine a class of special cases in which the cost density is constant outside the city limits.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Zitron, N. R.,A Continuous Model of Optimal-Cost Routes in a Circular City, Journal of Optimization Theory and Applications, Vol. 14, pp. 291–303, 1974.
Angel, S., andHyman, G. M.,Urban Velocity Fields, Environment and Planning, Vol. 2, pp. 211–224, 1970.
Angel, S., andHyman, G. M.,Urban Spatial Interaction, Environment and Planning, Vol. 4, pp. 99–118, 1972.
Angel, S., andHyman, G. M.,Urban Fields, Pion Limited, London, England, 1976.
Von Stackelberg, H.,Das Brechnungsgesetz des Verkehrs, Jahrbucher für Nationalokonomie und Statistik, Vol. 148, pp. 680–694, 1938.
Lösch, A.,The Economics of Location. Yale University Press, New Haven, Connecticut, 1954.
Wardrop, J. G.,Minim-Cost Paths in Urban Areas, Beiträge zur Theorie des Verkehrsflusses, Edited by W. Leutzbach and P. Baron, Bundesminster fur Verkehr, Bonn, Germany, 1969.
Beckman, M. J.,A Continuous Model of Transportation, Econometrica, Vol. 20, pp. 643–646, 1952.
Luneberg, R. K.,Mathematical Theory of Optics, University of California Press, Berkeley, California, 1966.
Hardy, G. H., Littlewood, J. E., andPolya, G.,Inequalities, Cambridge University Press, Cambridge, England, 1964.
Author information
Authors and Affiliations
Additional information
Communicated by G. Leitmann
The author would like to express his appreciation to Professor J. B. Keller for calling his attention to the work of R. K. Luneberg and to Professor K. A. Brakke for some helpful advice.
Rights and permissions
About this article
Cite this article
Zitron, N.R. A critical condition for the cost density in the circular city model. J Optim Theory Appl 24, 507–512 (1978). https://doi.org/10.1007/BF00932893
Issue Date:
DOI: https://doi.org/10.1007/BF00932893