Suppose that we are given a hypothetical origin-destination table of trips (but no network); we know the costs (per mile) of building various facilities and the cost (per mile) of travel per trip as a function of the flow and the facility type. For any assignment procedure for assigning trips to routes, we wish to select a network of various facilities which will accommodate the o-d flows and minimize the sum of construction cost plus travel cost. If the assignment principle is to minimize total travel cost (for any given network), the choice of the optimal network can be formulated as a programming problem abstractly of the same form as the usual assignment problem, except that the objective function is not convex, in fact it is (in some sense) approximately concave. As a result of this concavity, which is due to an economy of scale in construction, one finds that most idealized problems with high degrees of symmetry in the o-d table lead to optimal networks that do not display the symmetries of the o-d table. In particular, for an o-d table invariant to 90 degrees rotations, a square grid of roads or transit lines is, generally, the most expensive network as compared with other rectangular grids. (a) for the covering abstract of the symposium, please see irrd abstract no. 224453.
Samenvatting