Congestion and Penalization in Optimal Transport

TL;DR

This paper introduces a new optimal transport model that incorporates congestion costs and penalization, providing more realistic solutions for demand-supply imbalances and heterogeneous costs, along with an efficient algorithm and analytical solutions.

Contribution

The paper presents a novel optimal transport model with congestion penalization, analytical interior solutions, and an efficient computational algorithm, improving realism in applications.

Findings

The model captures demand-supply imbalances more accurately.

An $O((N+L)N^2 L^2)$ algorithm for computing solutions.

Examples show solutions differ from classical approaches, improving accuracy.

Abstract

We introduce a novel model based on the discrete optimal transport problem that incorporates congestion costs and replaces traditional constraints with weighted penalization terms. This approach better captures real-world scenarios characterized by demand-supply imbalances and heterogeneous congestion costs. We develop an analytical method for computing interior solutions, which proves particularly useful under specific conditions. Additionally, we propose an $O((N+L)N^2 L^2)$ algorithm to compute the optimal interior solution. For certain cases, we derive a closed-form solution and conduct a comparative statics analysis. Finally, we present examples demonstrating how our model yields solutions distinct from classical approaches, leading to more accurate outcomes in specific contexts, such as Peru's health and education sectors.