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July 2018, 14(3): 1123-1141. doi: 10.3934/jimo.2018002

Modeling and analyzing the chaotic behavior in supply chain networks: a control theoretic approach

1. 

Department of Electrical Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran

2. 

Department of Computer Engineering, Iran University of Science and Technology, Tehran, Iran

3. 

Department of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran

This research was partially supported by NSF.

Received  December 2015 Revised  August 2017 Published  January 2018

Supply chain network (SCN) is a complex nonlinear system and may have a chaotic behavior. This network involves multiple entities that cooperate to satisfy customers demand and control network inventory. The policy of each entity in demand forecast and inventory control, and constraints and uncertainties of demand and supply (or production) significantly affects the complexity of its behavior. In this paper, a supply chain network is investigated that has two ordering policies: smooth ordering policy and a new policy that is designed based on proportional-derivative controller. Two forecast methods are used in the network: moving average (MA) forecast and exponential smoothing (ES) forecast. The supply capacity of each entity is constrained. The effect of demand elasticity, which is the result of marketing activities, is involved in the SCN. The inventory adjustment parameter and demand elasticity are the most important decision parameters in the SCN. Overall, four scenarios are designed for modeling and analyzing the chaotic behavior of the network and in each scenario the maximum Lyapunov exponent is calculated and drawn. Finally, the best scenario for decision-making is obtained.

Citation: Hamid Norouzi Nav, Mohammad Reza Jahed Motlagh, Ahmad Makui. Modeling and analyzing the chaotic behavior in supply chain networks: a control theoretic approach. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1123-1141. doi: 10.3934/jimo.2018002
References:
[1]

T. Aslan, Simulated chaos in bullwhip effect, J. Manage. Marketing Logist, 2 (2015), 37-43. doi: 10.17261/Pressacademia.2015111603.

[2]

K. J. Astrom and T. Hagglund, PID Controller: Theory, Design and Tuning, Instrument Society of America, 1995.

[3]

L. Chong and L. Sifeng, A robust optimization approach to reduce the bullwhip effect of supply chains with vendor order placement lead time delays in an uncertain environment, Appl. Math. Model., 37 (2013), 707-718. doi: 10.1016/j.apm.2012.02.033.

[4]

C. F. Daganzo, A Theory of Supply Chains, Springer, Heidelberg, 2003.

[5]

C. F. Daganzo, On the stability of supply chain, Oper. Res., 52 (2004), 909-921.

[6]

J. DejonckheereS. M. DisneyM. R. Lambrecht and D. R. Towil, Transfer function analysis of forecasting induced bullwhip in supply chain, Int. Prod. Econ., 78 (2002), 133-144. doi: 10.1016/S0925-5273(01)00084-6.

[7]

J. DejonckheereS. M. DisneyM. R. Lambrecht and D. R. Towil, Measuring and avoiding the bullwhip effect: A control theoretic approach, Eur. J. Oper. Res., 147 (2003), 567-590.

[8] J. W. Forrester, Industrial Dynamics, MIT Press, Cambridge, 1961.
[9]

A. L. Fradkov and R. J. Evans, Control of chaos: Methods and applications in engineering, Annu. Rev. Contro, 29 (2005), 33-56. doi: 10.1016/j.arcontrol.2005.01.001.

[10]

G. F. Franklin, J. D. Powell and A. Emami-Naeini, Feedback Control of Dynamic Systems, Addison-Wesley, New York, 1986.

[11]

A. GoksuU. E. Kocamaz and Y. Uyaroglu, Synchronization and control of chaos in supply chain management, Comput. Ind. Eng., 86 (2015), 107-115. doi: 10.1016/j.cie.2014.09.025.

[12]

I. HeckmannT. Comes and S. Nickel, A critical review on supply chain risk-definition, measure and modeling, Omega, 52 (2015), 119-132. doi: 10.1016/j.omega.2014.10.004.

[13]

M. Hussain and P. R. Drake, Analysis of the bullwhip effect with order batching in multi-echelon supply chains, Inter. J. Phys. Distrib. Logist. Manage., 41 (2011), 972-990.

[14]

H. B. Hwarng and N. Xie, Understanding supply chain dynamics: A chaos perspective, Eur. J. Oper. Res., 184 (2008), 1163-1178. doi: 10.1016/j.ejor.2006.12.014.

[15]

D. Ivanov and B. Sokolov, Control and system-theoretic identification of the supply chain dynamics domain for planning, analysis and adaptation of performance under uncertainty, Eur. J. Oper. Res., 224 (2013), 313-323. doi: 10.1016/j.ejor.2012.08.021.

[16] W. E. Jarmain, Problems in Industrial Dynamics, MIT Press, Cambridge, 1963.
[17]

M. Jarsulic, A nonlinear model of the pure growth cycle, J. Econ. Behav. Organ., 22 (1993), 133-151. doi: 10.1016/0167-2681(93)90060-3.

[18]

Y. KristiantoP. HeloJ. Jiao and M. Sandhu, Adaptive fuzzy vendor managed inventory control for mitigating the Bullwhip effect in supply chains, Eur. J. Oper. Res., 216 (2012), 346-355. doi: 10.1016/j.ejor.2011.07.051.

[19]

E. R. LarsenJ. D. W. Morecroft and J. S. Thomsen, Complex behavior in a production-distribution model, Eur. J. Oper. Res., 119 (1999), 61-74.

[20]

H. L. LeeV. Padmanabhan and S. J. Whang, The bullwhip effect in supply chains, IEEE Engineering Management Review, 43 (2015), 108-117. doi: 10.1109/EMR.2015.7123235.

[21]

M. MarraW. Ho and J. S. Edwards, Supply chain knowledge management: A literature review, Expert Syst. Appl., 39 (2012), 6103-6110. doi: 10.1016/j.eswa.2011.11.035.

[22]

A. Matsumoto, Can inventory chaos be welfare improving, Int. J. Prod. Econ., 71 (2001), 31-43. doi: 10.1016/S0925-5273(00)00105-5.

[23]

E. Mosekilde and E. R. Larsen, Deterministic chaos in the beer production-distribution system, Syst. Dynam. Rev., 4 (1988), 131-147.

[24]

Y. Ouyang and X. Li, The bullwhip effect in supply chain networks, Eur. J. Oper. Res., 201 (2010), 799-810. doi: 10.1016/j.ejor.2009.03.051.

[25]

Q. QiangK. KeT. Anderson and J. Dong, The closed-loop supply chain network with competition, distribution channel investment, and uncertainties, Omega, 41 (2013), 186-194. doi: 10.1016/j.omega.2011.08.011.

[26]

C. A. G. SalcedoA. I. HernandezR. Vilanova and J. H. Cuartas, Inventory control of supply chain: Mitigating the bullwhip effect by centralized and decentralized internal model control approach, Eur. J. Oper. Res., 224 (2013), 261-272. doi: 10.1016/j.ejor.2012.07.029.

[27]

O. Sosnovtseva and E. Mosekilde, Torus destruction and chaos-chaos intermittency in a commodity distribution chain, Int. J. Bifurcat. Chaos, 7 (1997), 1225-1242. doi: 10.1142/S0218127497000996.

[28]

V. L. M. SpieglerM. M. NaimD. R. Towill and J. Wikner, A technique to develop simplified and linearised models of complex dynamic supply chain systems, Eur. J. Oper. Res., 251 (2016), 888-903. doi: 10.1016/j.ejor.2015.12.004.

[29] J. C. Sprott, Chaos and Time -Series Analysis, Oxford University Press, 2003.
[30]

J. D. Sterman, Modeling managerial behavior: Misperceptions of feedback in a dynamic decision making experiment, Manage. Sci., 35 (1989), 321-339. doi: 10.1287/mnsc.35.3.321.

[31]

M. J. TarokhN. DabiriA. H. Shokouhi and H. Shafiei, The effect of supply network configuration on occurring chaotic behavior in the retailer's inventory, J. Ind. Eng. Int., 7 (2011), 19-28.

[32]

J. S. ThmomsenE. Mosekilde and J. D. Sterman, Hyper chaotic phenomena in dynamic decision making, Syst. Anal. Model. Sim., 9 (1992), 137-156.

[33]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems, Springer, New York, 1990. doi: 10.1007/978-1-4757-4067-7.

[34]

G. P. Williams, Chaos Theory Tamed, Taylor & Francis, Landon, 1997.

[35]

Y. Wu and D. Z. Zhang, Demand fluctuation and chaotic behavior by interaction between customers and suppliers, Int. J. Prod. Econ., 107 (2007), 250-259.

[36]

Y. R. WuL. H. HuatucoG. Frizelle and J. Smart, A method for analyzing operational complexity in supply chains, J. Oper. Res. Soc., 64 (2013), 654-667.

show all references

References:
[1]

T. Aslan, Simulated chaos in bullwhip effect, J. Manage. Marketing Logist, 2 (2015), 37-43. doi: 10.17261/Pressacademia.2015111603.

[2]

K. J. Astrom and T. Hagglund, PID Controller: Theory, Design and Tuning, Instrument Society of America, 1995.

[3]

L. Chong and L. Sifeng, A robust optimization approach to reduce the bullwhip effect of supply chains with vendor order placement lead time delays in an uncertain environment, Appl. Math. Model., 37 (2013), 707-718. doi: 10.1016/j.apm.2012.02.033.

[4]

C. F. Daganzo, A Theory of Supply Chains, Springer, Heidelberg, 2003.

[5]

C. F. Daganzo, On the stability of supply chain, Oper. Res., 52 (2004), 909-921.

[6]

J. DejonckheereS. M. DisneyM. R. Lambrecht and D. R. Towil, Transfer function analysis of forecasting induced bullwhip in supply chain, Int. Prod. Econ., 78 (2002), 133-144. doi: 10.1016/S0925-5273(01)00084-6.

[7]

J. DejonckheereS. M. DisneyM. R. Lambrecht and D. R. Towil, Measuring and avoiding the bullwhip effect: A control theoretic approach, Eur. J. Oper. Res., 147 (2003), 567-590.

[8] J. W. Forrester, Industrial Dynamics, MIT Press, Cambridge, 1961.
[9]

A. L. Fradkov and R. J. Evans, Control of chaos: Methods and applications in engineering, Annu. Rev. Contro, 29 (2005), 33-56. doi: 10.1016/j.arcontrol.2005.01.001.

[10]

G. F. Franklin, J. D. Powell and A. Emami-Naeini, Feedback Control of Dynamic Systems, Addison-Wesley, New York, 1986.

[11]

A. GoksuU. E. Kocamaz and Y. Uyaroglu, Synchronization and control of chaos in supply chain management, Comput. Ind. Eng., 86 (2015), 107-115. doi: 10.1016/j.cie.2014.09.025.

[12]

I. HeckmannT. Comes and S. Nickel, A critical review on supply chain risk-definition, measure and modeling, Omega, 52 (2015), 119-132. doi: 10.1016/j.omega.2014.10.004.

[13]

M. Hussain and P. R. Drake, Analysis of the bullwhip effect with order batching in multi-echelon supply chains, Inter. J. Phys. Distrib. Logist. Manage., 41 (2011), 972-990.

[14]

H. B. Hwarng and N. Xie, Understanding supply chain dynamics: A chaos perspective, Eur. J. Oper. Res., 184 (2008), 1163-1178. doi: 10.1016/j.ejor.2006.12.014.

[15]

D. Ivanov and B. Sokolov, Control and system-theoretic identification of the supply chain dynamics domain for planning, analysis and adaptation of performance under uncertainty, Eur. J. Oper. Res., 224 (2013), 313-323. doi: 10.1016/j.ejor.2012.08.021.

[16] W. E. Jarmain, Problems in Industrial Dynamics, MIT Press, Cambridge, 1963.
[17]

M. Jarsulic, A nonlinear model of the pure growth cycle, J. Econ. Behav. Organ., 22 (1993), 133-151. doi: 10.1016/0167-2681(93)90060-3.

[18]

Y. KristiantoP. HeloJ. Jiao and M. Sandhu, Adaptive fuzzy vendor managed inventory control for mitigating the Bullwhip effect in supply chains, Eur. J. Oper. Res., 216 (2012), 346-355. doi: 10.1016/j.ejor.2011.07.051.

[19]

E. R. LarsenJ. D. W. Morecroft and J. S. Thomsen, Complex behavior in a production-distribution model, Eur. J. Oper. Res., 119 (1999), 61-74.

[20]

H. L. LeeV. Padmanabhan and S. J. Whang, The bullwhip effect in supply chains, IEEE Engineering Management Review, 43 (2015), 108-117. doi: 10.1109/EMR.2015.7123235.

[21]

M. MarraW. Ho and J. S. Edwards, Supply chain knowledge management: A literature review, Expert Syst. Appl., 39 (2012), 6103-6110. doi: 10.1016/j.eswa.2011.11.035.

[22]

A. Matsumoto, Can inventory chaos be welfare improving, Int. J. Prod. Econ., 71 (2001), 31-43. doi: 10.1016/S0925-5273(00)00105-5.

[23]

E. Mosekilde and E. R. Larsen, Deterministic chaos in the beer production-distribution system, Syst. Dynam. Rev., 4 (1988), 131-147.

[24]

Y. Ouyang and X. Li, The bullwhip effect in supply chain networks, Eur. J. Oper. Res., 201 (2010), 799-810. doi: 10.1016/j.ejor.2009.03.051.

[25]

Q. QiangK. KeT. Anderson and J. Dong, The closed-loop supply chain network with competition, distribution channel investment, and uncertainties, Omega, 41 (2013), 186-194. doi: 10.1016/j.omega.2011.08.011.

[26]

C. A. G. SalcedoA. I. HernandezR. Vilanova and J. H. Cuartas, Inventory control of supply chain: Mitigating the bullwhip effect by centralized and decentralized internal model control approach, Eur. J. Oper. Res., 224 (2013), 261-272. doi: 10.1016/j.ejor.2012.07.029.

[27]

O. Sosnovtseva and E. Mosekilde, Torus destruction and chaos-chaos intermittency in a commodity distribution chain, Int. J. Bifurcat. Chaos, 7 (1997), 1225-1242. doi: 10.1142/S0218127497000996.

[28]

V. L. M. SpieglerM. M. NaimD. R. Towill and J. Wikner, A technique to develop simplified and linearised models of complex dynamic supply chain systems, Eur. J. Oper. Res., 251 (2016), 888-903. doi: 10.1016/j.ejor.2015.12.004.

[29] J. C. Sprott, Chaos and Time -Series Analysis, Oxford University Press, 2003.
[30]

J. D. Sterman, Modeling managerial behavior: Misperceptions of feedback in a dynamic decision making experiment, Manage. Sci., 35 (1989), 321-339. doi: 10.1287/mnsc.35.3.321.

[31]

M. J. TarokhN. DabiriA. H. Shokouhi and H. Shafiei, The effect of supply network configuration on occurring chaotic behavior in the retailer's inventory, J. Ind. Eng. Int., 7 (2011), 19-28.

[32]

J. S. ThmomsenE. Mosekilde and J. D. Sterman, Hyper chaotic phenomena in dynamic decision making, Syst. Anal. Model. Sim., 9 (1992), 137-156.

[33]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems, Springer, New York, 1990. doi: 10.1007/978-1-4757-4067-7.

[34]

G. P. Williams, Chaos Theory Tamed, Taylor & Francis, Landon, 1997.

[35]

Y. Wu and D. Z. Zhang, Demand fluctuation and chaotic behavior by interaction between customers and suppliers, Int. J. Prod. Econ., 107 (2007), 250-259.

[36]

Y. R. WuL. H. HuatucoG. Frizelle and J. Smart, A method for analyzing operational complexity in supply chains, J. Oper. Res. Soc., 64 (2013), 654-667.

Figure 1.  Schematic of the control system
Figure 2.  Supply chain network (SCN)
Figure 3.  Block diagram of entity operations
Figure 4.  Time series plot of DTI and FTI in the stable state
Figure 5.  Phase plot of DTI-FTI in the stable state
Figure 6.  Time series plot of DTI and FTI in the chaotic state
Figure 7.  Phase plot of DTI-FTI in the chaotic state
Figure 8.  Effect of the demand elasticity by the ES forecast method
Figure 9.  Effect of the inventory adjustment parameter by the ES forecast method
Figure 10.  Effect of the demand elasticity by the MA forecast method
Figure 11.  Effect of the inventory adjustment parameter by the MA forecast method
Figure 12.  Effect of the demand elasticity by the ordering policy based on the PD controller
Figure 13.  Effect of the inventory adjustment parameter by the ordering policy based on the PD controller
Figure 14.  Comparison of all scenarios with $\lambda _\max \prec 0$
Figure 15.  Comparison of all scenarios with 0 ≤ $\lambda _\max \prec 0.01$
Table 1.  Decision-making scenarios
Scenarios MA forecast ES forecast PD controller-based ordering Smooth ordering
1 * *
2 * *
3 * *
4 * *
Scenarios MA forecast ES forecast PD controller-based ordering Smooth ordering
1 * *
2 * *
3 * *
4 * *
Table 2.  The initial data and parameters
ItemValue
Total initial inventory (in each level)24
Total desired inventory (in factories)50
Total desired inventory (in distributers)40
Total desired inventory (in wholesalers)30
Total desired inventory (in retailers)20
Total initial supply line (in each level)12
Production capacity, $c_{p} $100
Basic demand, $d_{o} $12
Discount threshold, $x_{T} $40
Ratio of overstock and discount, $c$1.2
Maximum discount, $r_{\max } $0.7
Lead time, $\tau $5
Fixed updating parameter for expectations, $\theta _{i} $0.4
Number of periods used to compute the forecast, $T_{m} $4
Inventory adjustment parameter, $\alpha $$0\mathrm{\le}$ $\alpha$ $\mathrm{\le}1$
Derivative time, $\tau _{i}^{D} $$3\alpha$
Elasticity of demand, $\beta $$0\mathrm{\le}\beta\mathrm{\le}2$
ItemValue
Total initial inventory (in each level)24
Total desired inventory (in factories)50
Total desired inventory (in distributers)40
Total desired inventory (in wholesalers)30
Total desired inventory (in retailers)20
Total initial supply line (in each level)12
Production capacity, $c_{p} $100
Basic demand, $d_{o} $12
Discount threshold, $x_{T} $40
Ratio of overstock and discount, $c$1.2
Maximum discount, $r_{\max } $0.7
Lead time, $\tau $5
Fixed updating parameter for expectations, $\theta _{i} $0.4
Number of periods used to compute the forecast, $T_{m} $4
Inventory adjustment parameter, $\alpha $$0\mathrm{\le}$ $\alpha$ $\mathrm{\le}1$
Derivative time, $\tau _{i}^{D} $$3\alpha$
Elasticity of demand, $\beta $$0\mathrm{\le}\beta\mathrm{\le}2$
Table 3.  The number of Maximum LEs in different ranges
Scenario $ 0.02\le \lambda _{Max} $$0.01\le \lambda _{Max} \prec 0.02$$0\le \lambda _{Max} \prec 0.01$$\lambda _{Max} \prec 0$
11599594452
216012080440
3202306120172
420833687169
Scenario $ 0.02\le \lambda _{Max} $$0.01\le \lambda _{Max} \prec 0.02$$0\le \lambda _{Max} \prec 0.01$$\lambda _{Max} \prec 0$
11599594452
216012080440
3202306120172
420833687169
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