A relationship between the gravity model for trip distribution and the transportation problem in linear programming

Abstract

A gravity model for trip distribution describes the number of trips between two zones as a product of three factors; one is associated with the zone in which a trip begins, one with the zone in which it ends and the third with the separation between the zones. The separation or deterrence factor is usually a decreasing function of the generalized cost of travelling between the zones, where generalized cost is usually some combination of the time of travel, the distance travelled and the actual monetary costs. If the deterrence factor is of the exponential form exp (-α c) and if the total numbers of origins and destinations in each zone are known, then the resulting trip matrix depends solely on α. In this paper it is shown that as α tends to infinity, this 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. That is to say the limit is a cost-minimizing solution to the linear programming transportation problem having the same origin and destination totals. If this transportation problem has many cost-minimizing solutions then it is shown that the limit is one particular solution in which each non-zero flow from an origin i to a destination j is of the form r is j . A numerical example is given.