Journal of Applied Mathematics

  • J. Appl. Math.
  • Volume 2011, Special Issue (2011), Article ID 325690, 14 pages.

Modeling and Analysis of Material Flows in Re-Entrant Supply Chain Networks Using Modified Partial Differential Equations

Fenglan He, Ming Dong, and Xiaofeng Shao

Full-text: Open access


The basic partial differential equation (PDE) models for supply chain networks with re-entrant nodes and their macroscopic are proposed. However, through numerical examples, the basic continuum models do not perform well for multiple re-entrant systems. Then, a new state equation considering the re-entrant degree of the products is introduced to improve the effectiveness of the basic continuum models. The applicability of the modified continuum models for re-entrant supply chains is illustrated through numerical examples. Finally, based on the modified continuum model, numerical examples of different re-entrant degrees are given, meanwhile, the changes in the WIP and outflux are analyzed in details for multiple re-entrant supply chain systems.

Article information

J. Appl. Math., Volume 2011, Special Issue (2011), Article ID 325690, 14 pages.

First available in Project Euclid: 29 August 2011

Permanent link to this document

Digital Object Identifier

Zentralblatt MATH identifier


He, Fenglan; Dong, Ming; Shao, Xiaofeng. Modeling and Analysis of Material Flows in Re-Entrant Supply Chain Networks Using Modified Partial Differential Equations. J. Appl. Math. 2011, Special Issue (2011), Article ID 325690, 14 pages. doi:10.1155/2011/325690.

Export citation


  • Z. Wang and Q. Wu, “Object-oriented hybrid PN model of semiconductor manufacturing line,” in Proceedings of the 4th World Congress on Intelligent Control and Automation, pp. 1354–1358, June 2002.
  • C. Lin, M. Xu, D. C. Marinescu, F. Ren, and Z. Shan, “A sufficient condition for instability of buffer priority policies in re-entrant lines,” IEEE Transactions on Automatic Control, vol. 48, no. 7, pp. 1235–1238, 2003.
  • M. Dong and F. F. Chen, “Process modeling and analysis of manufacturing supply chain networks using object-oriented Petri nets,” Robotics and Computer-Integrated Manufacturing, vol. 17, no. 1-2, pp. 121–129, 2001.
  • M. Dong, “Inventory planning of supply chains by linking production authorization strategy to queueing models,” Production Planning and Control, vol. 14, no. 6, pp. 533–541, 2003.
  • M. Li, “Fractal time series–-a tutorial review,” Mathematical Problems in Engineering, vol. 2010, Article ID 157264, 26 pages, 2010.
  • M. Li, W. Zhao, and S. Y. Chen, “mBm-based scalings of traffic propagated in internet,” Mathematical Problems in Engineering, vol. 2011, Article ID 389803, 21 pages, 2011.
  • M. Dong and F. F. Chen, “Performance modeling and analysis of integrated logistic chains: an analytic framework,” European Journal of Operational Research, vol. 162, no. 1, pp. 83–98, 2005.
  • S. Kumar and P. R. Kumar, “Queueing network models in the design and analysis of semiconductor wafer fabs,” IEEE Transactions on Robotics and Automation, vol. 17, no. 5, pp. 548–561, 2001.
  • G. F. Newell, “Scheduling, location, transportation, and continuum mechanics: some simple approximations to optimization problems,” SIAM Journal on Applied Mathematics, vol. 25, no. 3, pp. 346–360, 1973.
  • D. Armbruster, D. E. Marthaler, C. Ringhofer, K. Kempf, and T. C. Jo, “A continuum model for a re-entrant factory,” Operations Research, vol. 54, no. 5, pp. 933–950, 2006.
  • J. G. Dai and G. Weiss, “Stability and instability of fluid models for reentrant lines,” Mathematics of Operations Research, vol. 21, no. 1, pp. 115–134, 1996.
  • S. Göttlich, M. Herty, and A. Klar, “Network models for supply chains,” Communications in Mathematical Sciences, vol. 3, no. 4, pp. 545–559, 2005.
  • J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series in Operations Research and financial Engineering, Springer, New York, NY, USA, 2006.
  • E. J. Anderson, “A new continuous model for job shop scheduling,” International Journal of Systems Science, vol. 12, no. 12, pp. 1469–1475, 1981.
  • Y. H. Lee, M. K. Cho, S. J. Kim, and Y. B. Kim, “Supply chain simulation with discrete-continuous combined modeling,” Computers and Industrial Engineering, vol. 43, no. 1-2, pp. 375–392, 2002.
  • D. Armbruster, C. Ringhofer, and T. C. Jo, “Continuous models for production flows,” in Proceedings of the American Control Conference (AAC '04), pp. 4589–4594, July 2004.
  • R. van den Berg, E. Lefeber, and K. Rooda, “Modeling and control of a manufacturing flow line using partial differential equations,” IEEE Transactions on Control Systems Technology, vol. 16, no. 1, pp. 130–136, 2008.
  • A. Unver, C. Ringhofer, and D. Armbruster, “A hyperbolic relaxation model for product flow in complex production networks,” Discrete and Continuous Dynamical Systems, supplement 2009, pp. 790–799, 2009.
  • R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 2007.
  • H. Vogt, “FabSim Mini-fab example,” 2007,
  • T. B. Qin and Y. F. Wang, Application Oriented Simulation Modeling and Analysis with ExtendSim, Tsinghua University Press, Beijing, China, 2009.
  • E. Lefeber and D. Armbruster, Aggregate modeling of manufacturing systems, Systems Engineering Group, 2007.
  • S. Sun and M. Dong, “Continuum modeling of supply chain networks using discontinuous Galerkin methods,” Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 13–16, pp. 1204–1218, 2008.