Existence of Solutions for Supply Chain Models Based on Partial Differential Equations

Herty, Michaël; Klar, Axel; Piccoli, Benedetto

doi:10.1137/060659478

Cited by 65 publications

(68 citation statements)

References 13 publications

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“…For more detailed discussions, see e.g. [3,4,21,23]. In many aspects these models are quite similar to those of traffic flows [7] and pedestrian flows [8,9,15].…”

Section: Introduction and Main Resultsmentioning

confidence: 95%

Analysis and control of a scalar conservation law modeling a highly re-entrant manufacturing system

Shang

Wang

2011

Journal of Differential Equations

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International audienceIn this paper, we study a scalar conservation law that models a highly re-entrant manufacturing system as encountered in semi-conductor production. As a generalization of Coron et al. (2010) [14], the velocity function possesses both the local and nonlocal character. We prove the existence and uniqueness of the weak solution to the Cauchy problem with initial and boundary data in L∞L∞. We also obtain the stability (continuous dependence) of both the solution and the out-flux with respect to the initial and boundary data. Finally, we prove the existence of an optimal control that minimizes, in the LpLp-sense with 1⩽p⩽∞1⩽p⩽∞, the difference between the actual out-flux and a forecast demand over a fixed time period

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“…For more detailed discussions, see e.g. [3,4,21,23]. In many aspects these models are quite similar to those of traffic flows [7] and pedestrian flows [8,9,15].…”

Section: Introduction and Main Resultsmentioning

confidence: 95%

Analysis and control of a scalar conservation law modeling a highly re-entrant manufacturing system

Shang

Wang

2011

Journal of Differential Equations

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“…Following Herty et al (2007), we introduce a generalized tangent vector (η s , ξ i ρ , ξ i,p µ ) for the triplet (q s , ρ i , µ p i ) where η s ∈ R is a scalar shift of the queue q s (·), i.e. the shifted queue is q s (·) + η s , while the tangent vectors of ρ i and µ p i are defined in the same way as in Appendix A.2.…”

Section: 3mentioning

confidence: 99%

Continuity of the path delay operator for dynamic network loading with spillback

Han

Piccoli

Friesz

2016

Transportation Research Part B: Methodological

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This paper establishes the continuity of the path delay operators for dynamic network loading (DNL) problems based on the Lighthill-Whitham-Richards model, which explicitly capture vehicle spillback. The DNL describes and predicts the spatial-temporal evolution of traffic flow and congestion on a network that is consistent with established route and departure time choices of travelers. The LWR-based DNL model is first formulated as a system of partial differential algebraic equations (PDAEs). We then investigate the continuous dependence of merge and diverge junction models with respect to their initial/boundary conditions, which leads to the continuity of the path delay operator through the wave-front tracking methodology and the generalized tangent vector technique. As part of our analysis leading up to the main continuity result, we also provide an estimation of the minimum network supply without resort to any numerical computation. In particular, it is shown that gridlock can never occur in a finite time horizon in the DNL model.

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“…Existence and uniqueness for the non relaxed case were shown in [16], by directly using wave front tracking. Summarizing the above equations, we are left with the following problem: …”

Section: Supply Chainsmentioning

confidence: 99%

On the coupling of systems of hyperbolic conservation laws with ordinary differential equations

2010

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Motivated by applications to the piston problem, to a manhole model, to blood flow and to supply chain dynamics, this paper deals with a system of conservation laws coupled with a system of ordinary differential equations. The former is defined on a domain with boundary and the coupling is provided by the boundary condition. For each of the examples considered, numerical integrations are provided.

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Existence of Solutions for Supply Chain Models Based on Partial Differential Equations

Cited by 65 publications

References 13 publications

Analysis and control of a scalar conservation law modeling a highly re-entrant manufacturing system

Analysis and control of a scalar conservation law modeling a highly re-entrant manufacturing system

Continuity of the path delay operator for dynamic network loading with spillback

On the coupling of systems of hyperbolic conservation laws with ordinary differential equations

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