A flexible gravity-opportunities model for trip distribution

This paper develops a flexible gravity-opportunities model for trip distribution in which standard forms of the gravity and opportunities models are obtained as special cases of a general opportunities (GO) model. Hence the question of choice between gravity or opportunities approaches is decided empirically and statistically by restrictions on parameters which control the global functional form of the trip distribution mechanism. The test for the gravity model is shown to be equivalent to a test of the IIA axiom where alternatives are destinations. The notational dichotomy between the two approaches is resolved by employing ordered trip matrices and transformations to permit row and column sum constraints to be applied. These constraints, often interpreted in various ways, are treated as normalisation terms and are therefore not strictly part of the form of the model. Doubly constrained, singly constrained and unconstrained versions of both models are developed throughout. A key step in the integration is the specification of an opportunity function which has as arguments destination-attribute variables such population, income or some other measure of opportunities and generalized cost/impedance-type variables relating origin and destination. This device obviates the mutual exclusiveness ordinarily required of these two sets of variables. The opportunity function is incorporated into a general proportionality factor which is defined by the difference in functions of cumulative opportunities; the latter are subjected to a convex combination of direct and inverse Box-Cox transformations. Different values of the parameters controlling these transformations generate contrasting families of models, notably the exponential and logarithmic intervening opportunities models and the gravity model. All models are shown to be embedded in a transformed triangular region over which likelihood function, response surface or simultaneous confidence interval contours may be plotted. These generalised gravity-opportunity concepts are applied to two well-known models: direct demand multimodal travel demand models, and the estimation of the OD matrix from link volumes. The second case is estimated empirically and here it is shown that a significant improvement is obtained over the gravity model, which is rejected, along with the logarithmic intervening opportunity model, in favour of a more general direct opportunities version.