Modelling supply networks with partial differential equations
Authors:
C. D'Apice, R. Manzo and B. Piccoli
Journal:
Quart. Appl. Math. 67 (2009), 419440
MSC (2000):
Primary 35L65, 90B30
Published electronically:
May 5, 2009
MathSciNet review:
2547634
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Abstract: A continuumdiscrete 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 wellknown Bullwhip effect.
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 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. 104114, 2006.
 2.
 D. Armbruster, P. Degond and C. Ringhofer: A model for the dynamics of large queuing networks and supply chains. SIAM Journal on Applied Mathematics, Vol. 66, Issue 3, pp. 896920, 2006. MR 2216725 (2006k:90015)
 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.), Vol. 2(2), pp. 433460, 2007. MR 2325723 (2008e:90031)
 4.
 D. Armbruster, D. Marthaler and C. Ringhofer: Kinetic and fluid model hierarchies for supply chains. SIAM J. on Multiscale Modeling, Vol. 2(1), pp. 4361, 2004. MR 2044956 (2004m:90061)
 5.
 A. Bressan: Hyperbolic Systems of Conservation Laws. The Onedimensional Cauchy Problem. Oxford Univ. Press, 2000. MR 1816648 (2002d:35002)
 6.
 G. Bretti, C. D'Apice, R. Manzo, and B. Piccoli: A continuumdiscrete model for supply chains dynamics. Networks and Heterogeneous Media, Vol. 2, No. 4, pp. 661694, 2007. MR 2357763
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 C. Dafermos: Hyperbolic Conservation Laws in Continuum Physics. SpringerVerlag, 1999. MR 2169977 (2006d:35159)
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 C. F. Daganzo: A Theory of Supply Chains. Springer Verlag, New York, Berlin, Heidelberg, 2003. MR 1968682 (2004b:90003)
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 C. D'Apice and R. Manzo: A fluid dynamic model for supply chains. Networks and Heterogeneous Media, Vol. 1, No. 3, pp. 379398, 2006. MR 2247783 (2007f:35184)
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 C. D'Apice, R. Manzo, and B. Piccoli: Packets Flow on Telecommunication Networks. SIAM J. on Math. Anal., Vol. 38 (3), pp. 717740, 2006. MR 2262939 (2007h:35221)
 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. 3366, 2005.
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 S. Gottlich, M. Herty and A. Klar: Network models for supply chains. Comm. Math. Sci., Vol. 3(4), pp. 545559, 2005. MR 2188683 (2006f:90004)
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 S. Gottlich, M. Herty and A. Klar: Modelling and Optimization of Supply Chains on Complex Networks. Comm. Math. Sci., Vol. 4 (2), pp. 315330, 2006. MR 2219354 (2006m:90073)
 15.
 M. Herty, A. Klar, and B. Piccoli: Existence of solutions for supply chain models based on partial differential equations. SIAM J. Math. Anal., Vol. 39, No. 1, pp. 160173, 2007. MR 2318380
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 M. J. Lighthill and G. B. Whitham: On kinetic waves. II. Theory of Traffic Flows on Long Crowded Roads. Proc. Roy. Soc. London Ser. A, 229, pp. 317345, 1955. MR 0072606 (17:310a)
 17.
 E. Mosekilde and E.R. Larsen: System Dynamics Rev., Vol. 4, 1/2, pp. 131147, 1988.
 18.
 T. Nagatani and D. Helbing: Stability analysis and stabilization strategies for linear supply chains, Physica A: Statistical and Theoretical Physics, Vol. 335, Issues 34, pp. 644660, 2004. MR 2044162
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 P. I. Richards: Shock Waves on the Highway. Oper. Res., 4, pp. 4251, 1956. MR 0075522 (17:761b)
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 J.D. Sterman: Business Dynamics, McGrawHill, Boston, 2000.
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 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/S0033569X09011291
PII:
S 0033569X(09)011291
Keywords:
Conservation laws,
supply networks
Received by editor(s):
October 3, 2007
Published electronically:
May 5, 2009
Article copyright:
© Copyright 2009 Brown University
