Modeling, Simulation, and Optimization of Supply Chains

D’Apice, Ciro; Göttlich, Simone; Herty, Michaël; Piccoli, Benedetto

doi:10.1137/1.9780898717600

Cited by 72 publications

(113 citation statements)

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“…We also assume that the typical time scale is t 0 = τ0 . Then, e s (m, x s ) = e0t0 τ0L e(m, x) and this term is of order O (1).…”

Section: Scaling and Dimensionless Formulationmentioning

confidence: 99%

“…where we have already re-scaled the component −εf 1 of the solution f , belonging to the orthogonal complement of the null space of C, to order O(ε). Further, note that with the previous splitting f 1 dq = 0.…”

Section: Formal Asymptoticmentioning

confidence: 99%

“…The investigation of flows on networks is motivated by production, social or traffic flow networks where on a given graph dynamics on arcs and at vertices are prescribed. Flows on structured media have been studied widely in the literature and we refer to the review paper 6 and to the textbooks on traffic flow 14,19,26,28 and production processes, [1][2][3]18,27 respectively. They have also been studied in the context of structured media in Refs.…”

Section: Introductionmentioning

confidence: 99%

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Large time behavior of averaged kinetic models on networks

Herty

Ringhofer

2015

Math. Models Methods Appl. Sci.

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Communicated by P. DegondWe are interested in flows on general networks and derive a kinetic equation describing general production, social or transportation networks. Corresponding macroscopic transport equations for large time and homogenized behavior are obtained and studied numerically. This work continues a recent discussion [Averaged kinetic models for flows on unstructured networks, Kinetic Related Models 4 (2011) 1081-1096] and provides additionally explicit equilibrium solutions, second-order macroscopic approximations as well as numerical simulations on a large scale homogenized network.Possible applications are production networks. Here, a good is flowing from a raw material supplier through a certain number of layers (nodes in the networks) of intermediate producers to a final consumer. Due to possible machine breakdowns and service distributions travel and waiting times of goods may be described using probability distributions. Production networks of this type have been introduced originally in Refs. 5,8, 9 and 11, and optimized in Refs. 15 and 16. Another application might be air traffic where we consider passengers arriving and leaving airports. The links are the possible flight routes and waiting times reflect delays at airports due to, for example, stochastic weather conditions. A description of effects of possible network structures in application examples for traffic and social networks has been studied in Ref. 7. Therein, the so-called small world network structure is discussed. Those networks are constructed by a process described in Ref. 4 and the mean connection between nodes grows at most logarithmic with the number of nodes. 30 Within this paper we want to explore the connection between a simple dynamics with possible applications in production and air traffic and the structure of the network more closely. To this end we consider a general mathematical model for transport on graphs described as a multi-agent model. We apply asymptotic techniques (borrowed from classical kinetic theory) to derive a simplified model for flows on large networks on large time scales. This reduces the computational complexity of the study of long time phenomena in such flows. We develop a kinetic model for flows on arbitrary graphs, originally proposed in Ref. 20, under some simplifying assumptions, which make the large time scale model practically applicable. The macroscopic model consists of a convection-diffusion equation for the agent density, posed on a continuum in space, representing the graph of the network. In order for this model to be "reasonably smooth", i.e. not to involve measure-valued transport coefficients, it is necessary to locate the nodes of the graph in a certain way, i.e. to "draw the graph in R 2 " in a special way. Reorganizing the network this way results in the solution of an optimization problem for the coordinates of the network nodes. This represents the generalization of an idea, originally proposed in Ref. 7 to higher dimensions. The final macroscopic model is a convection-d...

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“…We also assume that the typical time scale is t 0 = τ0 . Then, e s (m, x s ) = e0t0 τ0L e(m, x) and this term is of order O (1).…”

Section: Scaling and Dimensionless Formulationmentioning

confidence: 99%

Section: Formal Asymptoticmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Large time behavior of averaged kinetic models on networks

Herty

Ringhofer

2015

Math. Models Methods Appl. Sci.

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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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“…Armbruster and Uzsoy's review [9] is closest to our approach focussing on the interplay between discrete event simulations and aggregate models based on clearing functions and partial differential equations. Mathematical approaches to control at the aggregate level are discussed in [10,11]. All the previous approaches dealing with aggregate models were typically controlling the order release or starts policies.…”

Section: Introductionmentioning

confidence: 99%

Feedback control for priority rules in re-entrant semiconductor manufacturing

Armbruster

Herty

et al. 2015

Applied Mathematical Modelling

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a b s t r a c tTypical semiconductor production is re-entrant and hence requires priority decisions when parts compete for production capacity at the same machine. A standard way to run such a factory is to start to plan and to finish according to demand. Often this results in a push policy where early production steps have priority over later production steps at the beginning of the production line and a pull policy where later steps have priority at the end of the production line. The point where the policies switch is called the push-pull-point (PPP). We develop a control scheme based on moving the PPP in a continuum model of the production flow. We show that this control scheme significantly reduces the mismatch between demand and production output. The success of the control scheme as a function of the frequency of control action is analyzed and optimal times between control actions are determined.

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Modeling, Simulation, and Optimization of Supply Chains

Cited by 72 publications

References 0 publications

Large time behavior of averaged kinetic models on networks

Large time behavior of averaged kinetic models on networks

Numerical Discretization of Coupling Conditions by High-Order Schemes

Feedback control for priority rules in re-entrant semiconductor manufacturing

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