Congestion and Penalization in Optimal Transport

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.