Optimal transport-based mathematical models for resource flow regulation in networked engineering systems

Abstract

In this paper, develop a new mathematical formulation using the concept of optimal transport to regulate resource flows in interconnected engineering networks. Specifically, this formulation represents supply and demand distributions defined on the connected nodes and computes a feasible transport plan based on the network structure, the cost of using the shortest path, flow conservation, and capacity constraints. Kantorovich duality theory is used to obtain the optimal regulation cost and interpret the dual variables as decision-making signals at the network level. A Wasserstein distance-based approach is used to analyze the stability of the transport cost with respect to the demand perturbation. An entropic regularization method and Sinkhorn algorithm are considered to reduce the computational complexity of the optimal transport problem for large and sparse networks. In a numerical experiment conducted on a 100-node radial distribution network with 30 supply nodes and 70 demand nodes, the proposed Sinkhorn algorithm can achieve a transport cost of 0.40% deviation from the exact optimal solution computed by linear programming solver in 74 iterations and 0.18 s of computing time. For the same problem, the exact network simplex method takes 2.90 s of computing time which corresponds to an approximately 16-times improvement of the computational time compared with the exact optimal solution. Under randomized ±10% demand perturbations in 500 runs, the theoretical stability bound stays within a factor of 1.4 from the worst cost change of all demand perturbations. It is further shown that the optimal gap decreases from 0.92% to 0.15% when decreasing the regularization coefficient from 0.10 to 0.02, but the iteration number grows from 51 to 146.