Journal of Applied Mathematics and Decision Sciences
Volume 2006 (2006), Article ID 48632, 13 pages
doi:10.1155/JAMDS/2006/48632
The constrained gravity model with power function as a cost function
1Department of Engineering Science, Haldia Institute of Technology, Haldia, Midnapore (East) 721657, West Bengal, India
2Department of Mathematics, Bengal Engineering and Science University, Howrah 711103, West Bengal, India
Received 25 August 2005; Revised 5 June 2006; Accepted 5 June 2006
Copyright © 2006 Bablu Samanta and Sanat Kumar Mazumder. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
A gravity model for trip distribution describes the number of
trips between two zones, as a product of three factors, one of the
factors is separation or deterrence factor. The deterrence factor
is usually a decreasing function of the generalized cost of
traveling between the zones, where generalized cost is usually
some combination of the travel, the distance traveled, and the
actual monetary costs. If the deterrence factor is of the power
form and if the total number of origins and destination in each
zone is known, then the resulting trip matrix depends solely on
parameter, which is generally estimated from data. In this paper,
it is shown that as parameter tends to infinity, the trip matrix
tends to a limit in which the total cost of trips is the least
possible allowed by the given origin and destination totals. If
the transportation problem has many cost-minimizing solutions,
then it is shown that the limit is one particular solution in
which each nonzero flow from an origin to a destination is a
product of two strictly positive factors, one associated with the
origin and other with the destination. A numerical example is
given to illustrate the problem.