A Mass Transportation Model for the Optimal Planning of an Urban Region

Abstract

Abstract. We propose a model to describe the optimal distributions of residents and services in a prescribed urban area. The cost functional takes into account the transportation costs (according to a Monge-Kantorovich-type criterion) and two additional terms which penalize concentration of residents and dispersion of services. The tools we use are the Monge-Kantorovich mass transportation theory and the theory of nonconvex functionals defined on measures.Key words. urban planning, mass transportation, nonconvex functionals over measures AMS subject classifications. 49J45, 49K99, 90B06 DOI. 10.1137/S00361410034383131. Introduction. The efficient planning of a city is a tremendously complicated problem, both for the high number of parameters which are involved as well as for the several relations which intervene among them (price of the land, kind of industries working in the area, quality of the life, prices of transportations, geographical obstacles, etc.). Perhaps a careful description of the real situations could be only obtained through evolution models which take into account the dynamical behavior of the different parameters involved.Among the several ingredients in the description of a city which are considered by urban planners, two of the most important are the distribution of residents and the distribution of services (working places, stores, offices, etc.). These two densities have to be treated in a different way due to their features (see facts ii) and iii) below). Several interesting mathematical models for the description of the equilibrium structure of these two elements of a city have been studied in the spatial economical literature (see for instance the classical text by Fujita [11] and the more recent paper by Lucas and Rossi-Hansberg [12])). The first time where the Monge-Kantorovich theory of optimal transportation appears and plays an important role, is, to the best of our knowledge, in Carlier and Ekeland [6,7].All these papers are mostly equilibrium-oriented and this is the main difference with the present work, which wants to be focused on the following aspects: distribution of residents and services, optimization of a global criterion, optimal transport theory.We consider a geographical area as given, and we represent it through a subset Ω of R n (n = 2 in the applications to concrete urban planning problems). We want to study the optimal location in Ω of a mass of inhabitants, which we denote by µ, as well as of a mass of services, which we denote by ν. We assume that µ and ν are probability measures on Ω. This means that the total amounts of population and production are fixed as problem data, and this is a difference from the model in [6]. The measures µ and ν represent the unknowns of our problem that have to be found in such a way that a suitable total cost functional F(µ, ν) is minimized. The definition of this total cost functional takes into account some criteria we want the two densities µ and ν to satisfy: (i) there is a transportation cost for moving from the residential ar...

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