Existence of solutions to Cauchy problems for a mixed continuum-discrete model for supply chains and networks

D’Apice, Ciro; Manzo, Rosanna; Piccoli, Benedetto

doi:10.1016/j.jmaa.2009.07.058

Cited by 22 publications

(12 citation statements)

References 16 publications

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“…In contrast, macroscopic models describe the average behavior of the production system in terms of part density or flow. They naturally arise from microscopic limits and are usually based on ordinary differential equations (ODEs) [15], hyperbolic partial differential equations (PDEs) [1,5,6,8,11] or a mixture of both [4,7,14]. We refer to [3] and the references therein for a comprehensive overview of macroscopic production models.…”

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confidence: 99%

Semi-Markovian capacities in production network models

Göttlich¹,

Knapp²

2017

Discrete &Amp; Continuous Dynamical Systems - B

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In this paper, we focus on production network models based on ordinary and partial differential equations that are coupled to semi-Markovian failure rates for the processor capacities. This modeling approach allows for intermediate capacity states in the range of total breakdown to full capacity, where operating and down times might be arbitrarily distributed. The mathematical challenge is to combine the theory of semi-Markovian processes within the framework of conservation laws. We show the existence and uniqueness of such stochastic network solutions, present a suitable simulation method and explain the link to the common queueing theory. A variety of numerical examples emphasizes the characteristics of the proposed approach.

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confidence: 99%

Semi-Markovian capacities in production network models

Göttlich¹,

Knapp²

2017

Discrete &Amp; Continuous Dynamical Systems - B

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“…The well-posedness of the Riemann problem 7, 22- 24, as well as that of the corresponding Cauchy problem, is proved in [56], [87]. An extension of the model (22) has been proposed in [57], [60] to treat the case of supply chains with spatially and temporally depending capacities. Therein, j D j .t; x/ and equation 22is replaced by u j D .…”

Section: Product Flow In Supply Chainsmentioning

confidence: 99%

Flows on networks: recent results and perspectives

Bressan¹,

Čanić²,

Garavello³

et al. 2014

EMS Surv. Math. Sci.

156

151

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The broad research thematic of flows on networks was addressed in recent years by many researchers, in the area of applied mathematics, with new models based on partial differential equations. The latter brought a significant innovation in a field previously dominated by more classical techniques from discrete mathematics or methods based on ordinary differential equations. In particular, a number of results, mainly dealing with vehicular traffic, supply chains and data networks, were collected in two monographs: Traffic flow on networks, AIMSciences, Springfield, 2006, and Modeling, simulation, and optimization of supply chains, SIAM, Philadelphia, 2010. The field continues to flourish and a considerable number of papers devoted to the subject is published every year, also because of the wide and increasing range of applications: from blood flow to air traffic management. The aim of the present survey paper is to provide a view on a large number of themes, results and applications related to this broad research direction. The authors cover different expertise (modeling, analysis, numeric, optimization and other) so to provide an overview as extensive as possible. The focus is mainly on developments which appeared subsequently to the publication of the aforementioned books. 1 The author acknowledges partial support of 2013 GNAMPA project "Leggi di Conservazione: Teoria e Applicazioni". 2 The author acknowledges support by BMBF KinOpt, DFG Cluster of Excellence EXC128 and DAAD 54365630, 55866082. 3 The author acknowledges partial support of NSF Research Network in the Mathematical Sciences KI-Net "Kinetic description of emerging challenges in multiscale problems of natural sciences" Grant # : 1107444.

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“…Among the many examples where such systems arise are traffic flow [24,25,29,31], production networks [21,23,30], telecommunication networks [22], gas flow in pipe networks [3, 11-13, 15, 16] or water flow in canals [4,5,28,35]. Mathematically, flow problems on networks are boundary value problems where the boundary value is implicitly defined by a coupling condition.…”

Section: Introductionmentioning

confidence: 99%

Numerical Discretization of Coupling Conditions by High-Order Schemes

2016

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Abstract. We consider numerical schemes for 2 × 2 hyperbolic conservation laws on graphs. The hyperbolic equations are given on the spatially one-dimensional arcs and are coupled at a single point, the node, by a nonlinear coupling condition. We develop high-order finite volume discretizations for the coupled problem. The reconstruction of the fluxes at the node is obtained using derivatives of the parameterized algebraic conditions imposed at the nodal points in the network. Numerical results illustrate the expected theoretical behavior. 35R02, 35Q35, 35F30

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Existence of solutions to Cauchy problems for a mixed continuum-discrete model for supply chains and networks

Cited by 22 publications

References 16 publications

Semi-Markovian capacities in production network models

Semi-Markovian capacities in production network models

Flows on networks: recent results and perspectives

Numerical Discretization of Coupling Conditions by High-Order Schemes

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