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Research Papers

On Stability Problems of Supply Networks Constrained With Transport Delay

[+] Author and Article Information
Rifat Sipahi

Mechanical and Industrial Engineering, Northeastern University, Boston, MA 02115rifat@coe.neu.edu

Stefan Lämmer

 Dresden University of Technology, Andreas-Schubert-Strasse 23, 01062 Dresden, Germanytraffic@stefanlaemmer.de

Dirk Helbing

 ETH Zurich, Universitätstrasse 41, 8092 Zurich, Switzerland; Collegium Budapest-Institute for Advanced Study, Szentháromság utca 2, H-1014 Budapest, Hungaryhelbing1@vwi.tu-dresden.de

Silviu-Iulian Niculescu

 Laboratoire de Signaux et Systèmes (L2S), Supélec, F-91192 Gif-sur-Yvette, Francesilviu.niculescu@lss.supelec.fr

In Sec. 3, we shall show other directions to relax these assumptions.

J. Dyn. Sys., Meas., Control 131(2), 021005 (Feb 04, 2009) (9 pages) doi:10.1115/1.3072144 History: Received July 31, 2007; Revised October 27, 2008; Published February 04, 2009

Transportation is one of the most crucial components in supply networks. In transportation lines, there exists a finite time between products leaving a point and arriving to another point in the supply network. This period of time is the delay, which accompanies all transportation lines present in the entire network. Delay is a well-known limitation, which is inevitable and pervasive in the network causing synchronization problems, fluctuating or excessive inventories, and lack of robustness of inventories against cyclic perturbations. The end results of such undesirable effects directly reflect to costs. This paper is motivated to reveal the mechanisms leading to these problems by analytically characterizing qualitative behavior of supply network dynamics modeled by continuous-time differential equations. The presence of delay forms the main challenge in the analysis and this is tackled by developing/utilizing the tools emerging from delay systems and control theory. While the backbone of the paper addresses the qualitative behavior in presence of a single delay representing delays in all transportation paths, it also reveals how to choose production rates and transportation delay without inducing any undesirable effects mentioned. Thorough cases studies with single and multiple delays are presented to demonstrate the effectiveness of the approaches proposed.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

A generic supply-demand flow from source to customers, inspired by Sterman (6)

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Figure 2

Pure (discrete) delay modeling and its effects between an input and an output

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Figure 3

Two possible scenarios for the geometric interaction between χ(λ) and the unit circle as per Lemma 1. Only subfigure on left is admissible as per Lemma 2.

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Figure 4

Interaction of the curve χ(λ) and unit circle when Jii=1

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Figure 5

Stability boundaries of the supply network for various A values, where B=0.1. The stability region is the area below the curves. Input rate: 0<Jii<1.

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Figure 6

Stability boundary of the supply network represented with iso-σ contours in the plane of Jii versus B for the nominal value Anom=0.2. Contour curves are labeled with their corresponding σ values. Input rate: 0<Jii<1.

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Figure 7

Contour plot of |μi0(α,σ∗)|=1 with respect to 0.01<Jii<0.99 and to A when B=0.1. Delay is σ∗=0.5. Bullwhip effects occur in regions R1, R1∪R2, and R1∪R2∪R3 for A=0.1, A=0.2, and A=0.3, respectively.

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Figure 8

Contour plot of |μi0(α,σ∗)|=1 with respect to 0.01<Jii<0.99 and to A when B=0.1. Delay is σ∗=3. Bullwhip effects occur in regions R1, R1∪R2, and R1∪R2∪R3 for A=0.1, A=0.2, and A=0.3, respectively.

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Figure 9

Stability map of the supply network with four delays. The stable regions are entrapped by the axes and the labeled boundaries.

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Figure 10

Simulations of variations in inventories and production rates with respect to scaled time for a given combination of four delays. Supply network is asymptotically stable with these delays, compare with Fig. 9.

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