Modelling supply networks with partial differential equations

Authors:
C. D'Apice, R. Manzo and B. Piccoli

Journal:
Quart. Appl. Math. **67** (2009), 419-440

MSC (2000):
Primary 35L65, 90B30

Published electronically:
May 5, 2009

MathSciNet review:
2547634

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A continuum-discrete model for supply networks is introduced. The model consists of a system of conservation laws: a conservation law for the goods density and an evolution equation for the processing rate. The network is formed by subchains and nodes at which, motivated by real cases, two routing algorithms are considered: the first maximizes fluxes taking into account the goods' final destinations, while the second maximizes fluxes without constraints. We analyze waves produced at nodes and equilibria for both algorithms, relating the latter to production rates in real supply networks. In particular, we show how the model can reproduce the well-known Bullwhip effect.

**1.**D. Armbruster, C. de Beer, M. Freitag, T. Jagalski and C. Ringhofer:*Autonomous Control of Production Networks using a Pheromone Approach*. Physica A: Statistical Mechanics and its applications, Vol. 363, Issue 1, pp. 104-114, 2006.**2.**D. Armbruster, P. Degond, and C. Ringhofer,*A model for the dynamics of large queuing networks and supply chains*, SIAM J. Appl. Math.**66**(2006), no. 3, 896–920. MR**2216725**, 10.1137/040604625**3.**D. Armbruster, P. Degond, and C. Ringhofer,*Kinetic and fluid models for supply chains supporting policy attributes*, Bull. Inst. Math. Acad. Sin. (N.S.)**2**(2007), no. 2, 433–460. MR**2325723****4.**D. Armbruster, D. Marthaler, and C. Ringhofer,*Kinetic and fluid model hierarchies for supply chains*, Multiscale Model. Simul.**2**(2003), no. 1, 43–61. MR**2044956**, 10.1137/S1540345902419616**5.**Alberto Bressan,*Hyperbolic systems of conservation laws*, Oxford Lecture Series in Mathematics and its Applications, vol. 20, Oxford University Press, Oxford, 2000. The one-dimensional Cauchy problem. MR**1816648****6.**Gabriella Bretti, Ciro D’Apice, Rosanna Manzo, and B. Piccoli,*A continuum-discrete model for supply chains dynamics*, Netw. Heterog. Media**2**(2007), no. 4, 661–694. MR**2357763**, 10.3934/nhm.2007.2.661**7.**Constantine M. Dafermos,*Hyperbolic conservation laws in continuum physics*, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2005. MR**2169977****8.**Carlos F. Daganzo,*A theory of supply chains*, Lecture Notes in Economics and Mathematical Systems, vol. 526, Springer-Verlag, Berlin, 2003. MR**1968682****9.**Ciro D’Apice and Rosanna Manzo,*A fluid dynamic model for supply chains*, Netw. Heterog. Media**1**(2006), no. 3, 379–398. MR**2247783**, 10.3934/nhm.2006.1.379**10.**Ciro D’apice, Rosanna Manzo, and Benedetto Piccoli,*Packet flow on telecommunication networks*, SIAM J. Math. Anal.**38**(2006), no. 3, 717–740. MR**2262939**, 10.1137/050631628**11.**D. Helbing, S. Lämmer, T. Seidel, P. Seba, and T. Platkowski:*Physics, stability and dynamics of supply networks*. Physical Review E 70, 2004, 066116.**12.**D. Helbing and S. Lämmer:*Supply and production networks: From the bullwhip effect to business cycles*, in: D. Armbruster, A. S. Mikhailov, and K. Kaneko (eds.) Networks of Interacting Machines: Production Organization in Complex Industrial Systems and Biological Cells, World Scientific, Singapore, pp. 33-66, 2005.**13.**S. Göttlich, M. Herty, and A. Klar,*Network models for supply chains*, Commun. Math. Sci.**3**(2005), no. 4, 545–559. MR**2188683****14.**S. Göttlich, M. Herty, and A. Klar,*Modelling and optimization of supply chains on complex networks*, Commun. Math. Sci.**4**(2006), no. 2, 315–330. MR**2219354****15.**M. Herty, A. Klar, and B. Piccoli,*Existence of solutions for supply chain models based on partial differential equations*, SIAM J. Math. Anal.**39**(2007), no. 1, 160–173. MR**2318380**, 10.1137/060659478**16.**M. J. Lighthill and G. B. Whitham,*On kinematic waves. II. A theory of traffic flow on long crowded roads*, Proc. Roy. Soc. London. Ser. A.**229**(1955), 317–345. MR**0072606****17.**E. Mosekilde and E.R. Larsen:*System Dynamics Rev.*, Vol. 4, 1/2, pp. 131-147, 1988.**18.**Takashi Nagatani and Dirk Helbing,*Stability analysis and stabilization strategies for linear supply chains*, Phys. A**335**(2004), no. 3-4, 644–660. MR**2044162**, 10.1016/j.physa.2003.12.020**19.**Paul I. Richards,*Shock waves on the highway*, Operations Res.**4**(1956), 42–51. MR**0075522****20.**J.D. Sterman:*Business Dynamics*, McGraw-Hill, Boston, 2000.**21.**http://www.sancarlo.it/it/publishing2.asp?ArticleId=5.

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Additional Information

**C. D'Apice**

Affiliation:
Department of Information Engineering and Applied Mathematics, University of Salerno, Fisciano (SA), Italy

Email:
dapice@diima.unisa.it

**R. Manzo**

Affiliation:
Department of Information Engineering and Applied Mathematics, University of Salerno, Fisciano (SA), Italy

Email:
manzo@diima.unisa.it

**B. Piccoli**

Affiliation:
Istituto per le Applicazioni del Calcolo “Mauro Picone”, Consiglio Nazionale delle Ricerche, Roma, Italy

Email:
b.piccoli@iac.cnr.it

DOI:
http://dx.doi.org/10.1090/S0033-569X-09-01129-1

Keywords:
Conservation laws,
supply networks

Received by editor(s):
October 3, 2007

Published electronically:
May 5, 2009

Article copyright:
© Copyright 2009
Brown University