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.
Highlights
The transportation planning process as it is usually carried out consists of a number of stages and at each stage except the first, use is made of the results of previous stages
If the deterrence factor is of the power form and if the total number of origins and destination in each zone is known, the resulting trip matrix depends solely on parameter, which is generally estimated from data
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
Summary
The transportation planning process as it is usually carried out consists of a number of stages and at each stage except the first, use is made of the results of previous stages. Tij = Sj Ri exp − α log cij = Bj ∀ j It is called the doubly constrained gravity model with cost function as a power function. If the Ai, Bj, and cij are given, as they usually are, all we need is a value for the parameter α to enable us to solve for the trip matrix (Tij). The value of α used in the model is generally estimated from data, for example, from observations of trips being made at present and the corresponding trip costs. We suppose that the data, which is available, consists of two matrices, a matrix of the observed number of trips per unit time between each origin and destination, and the matrix of costs applied when these trips were observed. A numerical example is provided in support of the existence of the problem
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