Multicommodity routing optimization for engineering networks

Abstract

Optimizing passengers routes is crucial to design efficient transportation networks. Recent results show that optimal transport provides an efficient alternative to standard optimization methods. However, it is not yet clear if this formalism has empirical validity on engineering networks. We address this issue by considering different response functions-quantities determining the interaction between passengers-in the dynamics implementing the optimal transport formulation. Particularly, we compare a theoretically-grounded response function with one that is intuitive for settings involving transportation of passengers, albeit lacking theoretical justifications. We investigate these two modeling choices on the Paris metro and analyze how they reflect on passengers' fluxes. We measure the extent of traffic bottlenecks and infrastructure resilience to node removal, showing that the two settings are equivalent in the congested transport regime, but different in the branched one. In the latter, the two formulations differ on how fluxes are distributed, with one function favoring routes consolidation, thus potentially being prone to generate traffic overload. Additionally, we compare our method to Dijkstra's algorithm to show its capacity to efficiently recover shortest-path-like graphs. Finally, we observe that optimal transport networks lie in the Pareto front drawn by the energy dissipated by passengers, and the cost to build the infrastructure.

Sustainable optimal transport in multilayer networks

“…We briefly describe the model of [23] to find optimal paths in multilayer networks using optimal transport theory. It considers two main quantities on network edges: fluxes F e of passengers traveling through edge e and conductivities µ e determining F e passing through an edge e. To keep track of the various routes that passengers have, they consider a multi-commodity approach [22,24], by distinguishing passengers based on their entry station i ∈ S. With this approach, the flux F e is an Mdimensional vector, where entries F i e denote the flux of passengers of type i traveling on edge e. We assume the fluxes are determined by pressure potentials p i u and p i v defined on nodes as follow:…”

Infrastructure adaptation and emergence of loops in network routing with time-dependent loads

“…This result is complementary to the hierarchical formation of trees since loops provide alternative routes to accommodate fluctuations or guarantee robustness against broken links. Recently, loops formation has also been observed in multicommodity setups [27,28], where the loads are deterministic inputs of the problem. In this case, loops generation is a consequence of having different types of mass-commodities-interacting in a unique shared infrastructure.…”

“…Typically, optimal transport of mass in networks is set as a minimization problem where resources moving through the edges have to satisfy a set of constraints, e.g., conservation of mass, while minimizing a suitable transportation cost [1,3,19,[23][24][25][26][27][28][29][30]. Several efficient methods have been proposed to solve this problem.…”

Immiscible Color Flows in Optimal Transport Networks for Image Classification

“…Optimal Transport (OT) is a powerful method for computing the distance between two data distributions. This problem has a cross-disciplinary domain of applications, ranging from logistic and route optimization [1][2][3], to biology [4,5] and computer vision [6][7][8][9], among others. Within this broad variety of problems, OT is largely utilized in machine learning [10], and deployed for solving classification tasks, where the goal is to optimally match discrete distributions that are typically learned from data.…”

“…Notice that for M = 1 and Γ = 1, we recover the standard unicommodity OT setup. Our choice of using the 2-norm over a in J Γ is motivated by recent works [2,28,29] where it is shown that Eq. ( 1) corresponds to Joule's law, while transport paths follow Kirchhoff's law enforcing conservation of mass.…”