Optimization model of transport currents

An optimization model of a one-dimensional transportation network is considered under the condition that the distribution of sources and sinks (for example, the population and workplaces in urban planning problems) is expressed in terms of finite Borel measures ϕ⁺ and ϕ⁻ of the same total masses, whereas the unknowns are one-dimensional Federer-Fleming currents T and S modeling transportation of people with or without the use of a public transportation network respectively. The choice of the network must provide the minimum for the cost of mass transfer from sources to sinks (for example, the transportation of people to workplaces) together with the cost of construction and exploitation of the network. Thus, the problem is to minimize the functional among pairs of flat chains of finite mass such that where A is the cost of transportation with the use of the network, B is the cost of transportation without the use of the network, H is a given monotone nondecreasing function characterizing the cost of construction and exploitation of the network, α ∈ (0, 1) is a given parameter, and is the δ-mass of the current. To prove the existence theorem, we develop a mathematical tool based on representations of one-dimensional currents in terms of measures on a space of Lipschitz paths. Bibliography: 18 titles. Illustration: 1 figure. __________ Translated from Problemy Matematicheskogo Analiza, No. 32, 2006, pp. 21–43.