Numerical simulation of optimal transport paths

Abstract

Abstract-Transport networks with branching structures are observable not only in nature as in trees, blood vessels, etc. but also in efficiently designed transport systems such as used in railway configurations and postage delivery networks. Mathematically, such a branching transport network is modeled by an optimal transport path between two probability measures (representing the source and the target). An essential feature of such a transport path is to favor group transportation in a large amount. This article provides numerical simulation of optimal transport paths. We first construct an initial transport path, and then modify the path as much as possible by using both local and global minimization algorithms.

Transport economies of scale and firm location

“…4 Clearly, G(m) represents the minimal transportation cost required for a given input bundle m. Unlike a general weighted Fermat problem, the three weights m α i , 3 For the sake of simplicity, we only consider concave power functions instead of more general concave ones. This power function is used in the ramified transportation literature to model branching transport structures as in Xia (2003Xia ( , 2010.…”

“…The first strand, known as ramified optimal transportation problems, studies how to find an optimal transport path or network used to move goods from given sources to targets. Examples include Gilbert (1967), Xia (2003Xia ( , 2010, Maddalena et al (2003), Bernot et al (2005Bernot et al ( , 2009, Xia and Xu (2013). The second strand, known as weighted Fermat problems, studies how to find a point in the plane such that the weighted sum of its distances to three given points is minimized.…”

The Geodesic Problem in Quasimetric Spaces

“…For the sake of visualization we provide some numerical simulations (see the forthcoming paper [15]) for different values of α. …”

Motivations, ideas and applications of ramified optimal transportation

“…The case of N = 1 is trivial. When N = 2, there exists an explicit formula in [30] for constructing the optimal transport path. Things become complicated when N ≥ 3, but still doable when N is small:…”

“…To calculate the transport cost d α (μ D , |D| δ O ), we used algorithms described in [30] to generate an approximated optimal transport path G P ∈ Path (μ D , |D| δ O ). This leads to Figure 8.…”