A variational approach for continuous supply chain networks

TL;DR

This paper introduces a variational method for modeling continuous supply chain networks, offering a stable, efficient computational algorithm and formulating network flow optimization as mixed integer programs, improving upon existing models.

Contribution

It proposes a novel variational approach for supply chain network modeling, providing stability, error estimates, and computational efficiency, and compares it favorably with existing methods.

Findings

The variational method guarantees numerical stability.

The approach allows rigorous error estimates.

It demonstrates modeling and computational advantages over existing methods.

Abstract

We consider a continuous supply chain network consisting of buffering queues and processors first proposed by [D. Armbruster, P. Degond, and C. Ringhofer, SIAM J. Appl. Math., 66 (2006), pp. 896--920] and subsequently analyzed by [D. Armbruster, P. Degond, and C. Ringhofer, Bull. Inst. Math. Acad. Sin. (N.S.), 2 (2007), pp. 433--460] and [D. Armbruster, C. De Beer, M. Freitag, T. Jagalski, and C. Ringhofer, Phys. A, 363 (2006), pp. 104--114]. A model was proposed for such a network by [S. G\"ottlich, M. Herty, and A. Klar, Commun. Math. Sci., 3 (2005), pp. 545--559] using a system of coupling ordinary differential equations and partial differential equations. In this article, we propose an alternative approach based on a variational method to formulate the network dynamics. We also derive, based on the variational method, a computational algorithm that guarantees numerical stability, allows for rigorous error estimates, and facilitates efficient computations. A class of network flow optimization problems are formulated as mixed integer programs (MIPs). The proposed numerical algorithm and the corresponding MIP are compared theoretically and numerically with existing ones [A. F\"ugenschuh, S. G\"ottlich, M. Herty, A. Klar, and A. Martin, SIAM J. Sci. Comput., 30 (2008), pp. 1490--1507; S. G\"ottlich, M. Herty, and A. Klar, Commun. Math. Sci., 3 (2005), pp. 545--559], which demonstrates the modeling and computational advantages of the variational approach.