The constrained gravity model with power function as a cost function

摘要

A gravity model for trip distribution describes the number oftrips between two zones, as a product of three factors, one of thefactors is separation or deterrence factor. The deterrence factoris usually a decreasing function of the generalized cost oftraveling between the zones, where generalized cost is usuallysome combination of the travel, the distance traveled, and theactual monetary costs. If the deterrence factor is of the powerform and if the total number of origins and destination in eachzone is known, then the resulting trip matrix depends solely onparameter, which is generally estimated from data. In this paper,it is shown that as parameter tends to infinity, the trip matrixtends to a limit in which the total cost of trips is the leastpossible allowed by the given origin and destination totals. Ifthe transportation problem has many cost-minimizing solutions,then it is shown that the limit is one particular solution inwhich each nonzero flow from an origin to a destination is aproduct of two strictly positive factors, one associated with theorigin and other with the destination. A numerical example isgiven to illustrate the problem.