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.

1.
Helbing
,
D.
,
Armbuster
,
D.
,
Mikhailov
,
A. S.
, and
Lefeber
,
E.
, 2006, “
Information and Material Flows in Complex Networks
,”
Physica A
,
363
, pp.
11
16
. 0378-4371
2.
Helbing
,
D.
, 2003, “
Modelling Supply Networks and Business Cycles as Unstable Transport Phenomena
,”
New J. Phys.
1367-2630,
5
, pp.
90.1
90.28
.
3.
Riddalls
,
C. E.
, and
Bennett
,
S.
, 2002, “
The Stability of Supply Chains
,”
Int. J. Prod. Res.
,
40
(
2
), pp.
459
475
. 0020-7543
4.
Riddalls
,
C. E.
, and
Bennett
,
S.
, 2002, “
Production-Inventory System Controller Design and Supply Chain Dynamics
,”
Int. J. Syst. Sci.
,
33
(
3
), pp.
181
195
. 0020-7721
5.
Sipahi
,
R.
,
Lammer
,
S.
,
Niculescu
,
S. -I.
, and
Helbing
,
D.
, 2006, “
On Stability Analysis and Parametric Design of Supply Networks Under the Presence of Transportation Delays
,”
ASME-IMECE Conference
, Chicago, IL, Paper No. 14782.
6.
Sterman
,
J.
, 2000,
Business Dynamics: Systems Thinking and Modeling for a Complex World
,
McGraw-Hill
,
Boston
.
7.
Daganzo
,
C.
, 2003,
A Theory of Supply Chains
,
Springer
,
New York
.
8.
Simchi-Levi
,
D.
,
Kaminsky
,
P.
,
Simchi-Levi
,
E.
, 2003,
Designing & Managing the Supply Chain
,
McGraw-Hill
,
New York
.
9.
Warburton
,
D. H. R.
, 2004, “
An Exact Analytical Solution to the Production Inventory Control Problem
,”
Int. J. Prod. Econ.
,
92
(
1
), pp.
81
96
. 0925-5273
10.
Siegele
,
L.
, 2002, “
Chain Reaction: Managing a Supply Chain is Becoming a Bit Like Rocket Science
,”
Economist
,
362
(8258), pp.
13
15
. 0013-063X
11.
Agrawal
,
M.
,
Kumaresh
,
T. V.
, and
Mercer
,
G. A.
, 2001, “
The False Promise of Mass Customization
,”
McKinsey Quarterly
0047-5394,
3
, pp.
62
71
.
12.
Ceroni
,
J. A.
, and
Nof
,
S. Y.
, 2005, “
Task Parallelism in Distributed Supply Organizations: A Case Study in the Shoe Industry
,”
Prod. Plan. Control
,
16
(
5
), pp.
500
513
. 0953-7287
13.
Helbing
,
D.
,
Lämmer
,
S.
,
Witt
,
U.
, and
Brenner
,
T.
, 2004, “
Network-Induced Oscillatory Behavior in Material Flow Networks and Irregular Business Cycles
,”
Phys. Rev. E
1063-651X,
70
, p.
056118
.
14.
Nagatani
,
T.
, and
Helbing
,
D.
, 2004, “
Stability Analysis and Stabilization Strategies for Linear Supply Chains
,”
Physica A
0378-4371,
335
, pp.
644
660
.
15.
Helbing
,
D.
,
Lämmer
,
S.
,
Seidel
,
T.
,
Šeba
,
P.
, and
Płatkowski
,
T.
, 2004, “
Physics, Stability and Dynamics of Supply Networks
,”
Phys. Rev. E
1063-651X,
70
, p.
066116
.
16.
Mosekilde
,
E.
, and
Larsen
,
E. R.
, 1988, “
Deterministic Chaos in the Beer Production-Distribution Model
,”
Syst. Dyn. Rev.
,
4
(
1–2
), pp.
131
147
. 0883-7066
17.
Chen
,
F.
,
Drezner
,
Z.
,
Ryan
,
J. K.
, and
Simchi-Levi
,
D.
, 2000, “
Quantifying the Bullwhip Effect in a Simple Supply Chain: The Impact of Forecasting, Lead Times, and Information
,”
Manage. Sci.
0025-1909,
46
(
3
), pp.
436
443
.
18.
Dejonckheere
,
J.
,
Disney
,
S. M.
,
Lambrecht
,
M. R.
, and
Towill
,
D. R.
, 2002, “
Transfer Function Analysis of Forecasting Induced Bullwhip in Supply Chain
,”
Int. J. Prod. Econ.
0925-5273,
78
, pp.
133
144
.
19.
Hoberg
,
K.
,
Thonemann
,
U. W.
, and
Bradley
,
J. R.
, 2003, “
Analyzing the Bullwhip Effect of Installation-Stock and Echelon-Stock Policies With Linear Control Theory
,”
Operations Research Proceedings 2003
,
D.
Ahr
,
R.
Fahrion
,
M.
Oswald
, and
G.
Reinelt
, eds.,
Springer
,
Heidelberg
, pp.
63
70
.
20.
Lee
,
H. L.
,
Padmanabhan
,
V.
, and
Whang
,
S.
, 1997, “
Information Distortion in a Supply Chain: The Bullwhip Effect
,”
Manage. Sci.
,
43
(
4
), pp.
546
558
. 0025-1909
21.
Toker
,
O.
, and
Ozbay
,
H.
, 1996, “
Complexity Issues in Robust Stability of Linear Delay-Differential Systems
,”
Math. Control, Signals, Syst.
,
9
, pp.
386
400
. 0932-4194
22.
Sipahi
,
R.
, and
Delice
,
I. I.
, 2009, “
Advanced Clustering With Frequency Sweeping Methodology for the Stability of Multiple Time Delay Systems
,”
IEEE Trans. Autom. Control
, submitted.
23.
Sipahi
,
R.
,
Atay
,
F. M.
, and
Niculescu
,
S. -I.
, 2008, “
Stability of Traffic Flow Behavior With Distributed Delays Modeling the Memory Effects of the Drivers
,”
SIAM J. Appl. Math.
0036-1399,
68
(
3
), pp.
738
759
.
24.
Sipahi
,
R.
, and
Niculescu
,
S. -I.
, 2006, “
Some Remarks on the Characterization of Delay Interactions in Deterministic Car Following Models
,”
17th International Symposium on Mathematical Theory of Networks and Systems (MTNS)
, Kyoto, Japan, July 24–28.
25.
Sterman
,
J.
, 1989, “
Modeling Managerial Behavior: Misperceptions of Feedback in a Dynamic Decision Making Experiment
,”
Manage. Sci.
,
35
(
3
), pp.
321
339
. 0025-1909
26.
Halanay
,
A.
, 1966,
Differential Equations: Stability, Oscillations, Time Lags
,
Academic
,
New York
.
27.
Hale
,
J. K.
, and
Verduyn Lunel
,
S. M.
, 1993,
An Introduction to Functional Differential Equations
,
Springer-Verlag
,
New York
.
28.
Niculescu
,
S. -I.
, 2001,
Delay Effects on Stability: A Robust Control Approach
, Vol.
269
,
Springer-Verlag
,
Heidelberg
.
29.
S. -I.
Niculescu
and
K.
Gu
, eds., 2004,
Advances in Time-Delay Systems
,
Springer-Verlag
,
Berlin
.
30.
Niculescu
,
S. -I.
,
Verriest
,
E. I.
,
Dugard
,
L.
, and
Dion
,
J. M.
, 1997, “
Stability and Robust Stability of Time-Delay System: A Guided Tour
,”
Stability and Control of Time Delay Systems
, Vol.
228
,
L.
Dugard
and
E. I.
Verriest
, eds.,
Springer
,
London
, pp.
1
71
.
31.
Chen
,
J.
, and
Latchman
,
H. A.
, 1995, “
Frequency Sweeping Tests for Stability Independent of Delay
,”
IEEE Trans. Autom. Control
0018-9286,
40
, pp.
1640
1645
.
32.
Gu
,
K.
,
Niculescu
,
S. -I.
, and
Chen
,
J.
, 2005, “
On Stability Crossing Curves for General Systems With Two Delays
,”
J. Math. Anal. Appl.
0022-247X,
311
, pp.
231
253
.
33.
Michiels
,
W.
,
Niculescu
,
S. -I.
, 2007, “
Stability and Stabilization of Time-Delay Systems: An Eigenvalue-Based Approach
,”
SIAM Advances in Design and Control
,
12
.
34.
Sipahi
,
R.
, and
Olgac
,
N.
, 2006, “
Stability Map of Systems With Three Independent Delays
,”
Proceedings of the American Control Conference
, Minneapolis, MN.
35.
Richard
,
J. P.
, 2003, “
Time-Delay Systems: An Overview of Some Recent Advances and Open Problems
,”
Automatica
0005-1098,
39
, pp.
1667
1694
.
36.
Engelborghs
,
K.
, 2000, “
DDE-BIFTOOL: A Matlab Package for Bifurcation Analysis of Delay Differential Equations
,” Department of Computer Science, Katholieke Universiteit Leuven, Leuven, Belgium.
37.
Michiels
,
W.
, 2002, “
Stability and Stabilization of Time-Delay Systems
,” Ph.D. thesis, Katholieke Universiteit Leuven, Leuven.
38.
Sipahi
,
R.
, and
Olgac
,
N.
, 2005, “
Complete Stability Robustness of Third-Order LTI Multiple Time-Delay Systems
,”
Automatica
0005-1098,
41
, pp.
1413
1422
.
39.
Sipahi
,
R.
, and
Olgac
,
N.
, 2003, “
Active Vibration Suppression With Time Delayed Feedback
,”
ASME J. Vibr. Acoust.
0739-3717,
125
, pp.
384
388
.
40.
Datko
,
R.
, 1978, “
A Procedure for Determination of the Exponential Stability of Certain Differential-Difference Equations
,”
Q. Appl. Math.
,
36
, pp.
279
292
. 0033-569X
41.
El’sgol’ts
,
L. E.
, and
Norkin
,
S. B.
, 1973,
Introduction to the Theory and Application of Differential Equations With Deviating Arguments
,
Academic
,
New York
.
42.
Hsu
,
C. S.
, 1970, “
Application of the Tau-Decomposition Method to Dynamical Systems Subjected to Retarded Follower Forces
,”
ASME J. Appl. Mech.
,
37
, pp.
258
266
. 0021-8936
43.
Olgac
,
N.
, and
Holm-Hansen
,
B.
, 1994, “
A Novel Active Vibration Absorption Technique: Delayed Resonator
,”
J. Sound Vib.
0022-460X,
176
, pp.
93
104
.
44.
Bellman
,
R. E.
, and
Cooke
,
K. L.
, 1963,
Differential-Difference Equations
,
Academic
,
New York
.
45.
Cooke
,
K. L.
, and
van den Driessche
,
P.
, 1986, “
On Zeros of Some Transcendental Equations
,”
Funkc. Ekvac.
0532-8721,
29
, pp.
77
90
.
46.
Oguztoreli
,
M. N.
, 1966,
Time-Lag Control Systems
,
Academic
,
New York
.
47.
Stepan
,
G.
, 1989,
Retarded Dynamical Systems: Stability and Characteristic Functions
,
Longman Scientific & Technical
,
New York
/
Wiley
,
New York
.
48.
Volterra
,
V.
, 1931,
Théorie mathématique de la lutte pour la vie
,
Gauthier-Villars
,
Paris
, in French.
49.
Ergenc
,
A. F.
,
Olgac
,
N.
, and
Fazelinia
,
H.
, 2007, “
Extended Kronecker Summation for Cluster Treatment of LTI Systems With Multiple Delays
,”
SIAM J. Control Optim.
0363-0129,
46
(
1
), pp.
143
155
.
50.
Fazelinia
,
H.
,
Sipahi
,
R.
, and
Olgac
,
N.
, 2007, “
Stability Analysis of Multiple Time Delayed Systems Using ‘Building Block’ Concept
,”
IEEE Trans. Autom. Control
,
52
(
5
), pp.
799
810
. 0018-9286
51.
Jarlebring
,
E.
, 2008, “
Critical Delays and Polynomial Eigenvalue Problems
,”
J. Comput. Appl. Math.
0377-0427,
224
(
1
), pp.
296
306
.
52.
Sipahi
,
R.
, 2005, “
Cluster Treatment of Characteristic Roots, CTCR, A Unique Methodology for the Complete Stability Robustness Analysis of Linear Time Invariant Multiple Time Delay Systems Against Delay Uncertainties
,” Ph.D. thesis, University of Connecticut, Storrs.
53.
Breda
,
D.
,
Maset
,
S.
, and
Vermiglio
,
R.
, 2004, “
Computing the Characteristic Roots for Delay Differential Equations
,”
IMA J. Numer. Anal.
0272-4979,
24
(
1
), pp.
1
19
.
54.
Vyhlidal
,
T.
, and
Zitek
,
P.
, 2003, “
Quasipolynomial Mapping Based Rootfinder for Analysis of Time Delay Systems
,”
IFAC Workshop on Time-Delay Systems, TDS’03
, Rocquencourt.
55.
Ogata
,
K.
, 2002,
Modern Control Engineering
,
4th ed.
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
56.
Fait
,
B.
,
Arnold
,
D.
, and
Furmans
,
K.
, 2003, “
The Impact of the Exchange of Market and Stock Information on the Bullwhip Effect in Supply Chains
,”
Operations Research Proceedings 2003
,
D.
Ahr
,
R.
Fahrion
,
M.
Oswald
, and
G.
Reinelt
, eds.,
Springer
,
Heidelberg
, pp.
55
32
.
57.
Forrester
,
J. W.
, 1961,
Industrial Dynamics
,
MIT
,
Cambridge, MA
.
You do not currently have access to this content.