# Analysis of the Complexity Entropy and Chaos Control of the Bullwhip Effect Considering Price of Evolutionary Game between Two Retailers

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## Abstract

**:**

## 1. Introduction

## 2. Price Game Model

#### 2.1. Model Description

#### 2.2. The Dynamics of the Price Game Model

**Proposition**

**Proof.**

- $\begin{array}{c}{f}_{1}=2{a}_{2}{b}_{1}+{a}_{1}{c}_{2}+2{b}_{1}{b}_{2}c+{b}_{1}c{c}_{2},\hfill \\ {f}_{2}=2{a}_{1}{b}_{2}+{a}_{2}{c}_{1}+2{b}_{1}{b}_{2}c+{b}_{2}c{c}_{1},\hfill \\ {f}_{3}=4{b}_{1}{b}_{2}-{c}_{1}{c}_{2}.\hfill \end{array}$

#### 2.3. Experimental Design, Numerical Result and Discussion

- (a)
- Stable state: $\alpha =1.5,\beta =1.5;$
- (b)
- Period doubling: $\alpha =2.3,\beta =1.5;$
- (c)
- Chaos: $\alpha =2.8,\beta =1.5.$

- (a)
- The lead-time of two retailers: $L=3;$
- (b)
- The safety stock factor: $z=0;$
- (c)
- The demand smoothing index: $\gamma =0.3;$
- (d)
- The time length of numerical experiments: $T=52.$

## 3. The Mitigation of the Bullwhip Effect by Chaos Control

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. The Proof of Proposition

## References

- Forrester, J.W. Industrial Dynamics; MIT Press: Cambridge, MA, USA, 1961. [Google Scholar]
- Sterman, J.D. Modeling managerial behavior: Misperceptions of feedback in a dynamic decision making experiment. Manag. Sci.
**1989**, 35, 321–339. [Google Scholar] [CrossRef] - Lee, H.L.; Padmanabhan, P.; Whang, S. Information distortion in a supply chain: The bullwhip effect. Manag. Sci.
**1997**, 43, 546–558. [Google Scholar] [CrossRef] - Chen, F.; Drezner, Z.; Ryan, J.K.; Simchi-Levi, D. Quantifying the bullwhip effect in a simple supply chain. Manag. Sci.
**2000**, 46, 436–443. [Google Scholar] [CrossRef] - Chen, F.; Drezner, Z.; Ryan, J.K.; Simchi-Levi, D. The impact of exponential smoothing forecasts on the bullwhip effect. Naval Res. Logist.
**2000**, 47, 269–286. [Google Scholar] [CrossRef] - Zhang, X. The impact of forecasting methods on the bullwhip effect. Int. J. Prod. Econ.
**2004**, 88, 15–27. [Google Scholar] [CrossRef] - Luong, H.T. Measure of bullwhip effect in supply chains with autoregressive demand process. Eur. J. Oper. Res.
**2007**, 180, 1086–1097. [Google Scholar] [CrossRef] - Agrawal, S.; Sengupta, R.N.; Shanker, K. Impact of information sharing and lead time on bullwhip effect and on-hand inventory. Eur. J. Oper. Res.
**2009**, 192, 576–593. [Google Scholar] [CrossRef] - Chatfield, D.C.; Kim, J.G.; Harrison, T.P.; Hayya, J.C. The bullwhip effect-impact of stochastic lead time, information quality, and information sharing: A simulation study. Prod. Oper. Manag.
**2004**, 13, 340–353. [Google Scholar] [CrossRef] - Cannella, S.; Ciancimino, E. On the bullwhip avoidance phase: Supply chain collaboration and order smoothing. Int. J. Prod. Res.
**2010**, 48, 6739–6776. [Google Scholar] [CrossRef] - Cannella, S.; Ciancimino, E.; Framinan, J.M. Inventory policies and information sharing in multi-echelon supply chains. Prod. Plan. Control
**2011**, 22, 649–659. [Google Scholar] [CrossRef] - Cannella, S.; Framinan, J.M.; Barbosa-Povoa, A. An IT-enabled supply chain model: A simulation study. Int. J. Syst. Sci.
**2014**, 45, 2327–2341. [Google Scholar] [CrossRef] - Cannella, S.; Bruccoleri, M.; Framinan, J.M. Closed-loop supply chains: What reverse logistics factors influence performance? Int. J. Prod. Econ.
**2016**, 175, 35–49. [Google Scholar] [CrossRef] - Cannella, S.; Barbosa-Povoa, A.P.; Framinan, J.M.; Relvas, S. Metrics for bullwhip effect analysis. J. Oper. Res. Soc.
**2013**, 64, 1–16. [Google Scholar] [CrossRef] - Trapero, J.R.; Kourentzes, N.; Fildes, R. Impact of information exchange on supplier forecasting performance. Omega
**2012**, 40, 738–747. [Google Scholar] [CrossRef] [Green Version] - Ciancimino, E.; Cannella, S.; Bruccoleri, M.; Framinan, J.M. On the bullwhip avoidance phase: The synchronised supply chain. Eur. J. Oper. Res.
**2012**, 221, 49–63. [Google Scholar] [CrossRef] [Green Version] - Dominguez, R.; Cannella, S.; Framinan, J.M. On bullwhip-limiting strategies in divergent supply chain networks. Comput. Ind. Eng.
**2014**, 73, 85–95. [Google Scholar] [CrossRef] - Dominguez, R.; Cannella, S.; Framinan, J.M. The impact of the supply chain structure on bullwhip effect. Appl. Math. Model.
**2015**, 39, 7309–7325. [Google Scholar] [CrossRef] - Duc, T.T.; Luong, H.T.; Kim, Y.D. A measure of bullwhip effect in supply chains with a mixed autoregressive-moving average demand process. Eur. J. Oper. Res.
**2008**, 187, 243–256. [Google Scholar] [CrossRef] - Ma, Y.; Wang, N.; Che, A.; Huang, Y.; Xu, J. The bullwhip effect under different information-sharing settings: A perspective on price-sensitive demand that incorporates price dynamics. Int. J. Prod. Res.
**2013**, 51, 3085–3116. [Google Scholar] [CrossRef] [Green Version] - Wang, N.; Lu, J.; Feng, G.; Ma, Y.; Liang, H. The bullwhip effect on inventory under different information sharing settings based on price-sensitive demand. Int. J. Prod. Res.
**2016**, 54, 4043–4064. [Google Scholar] [CrossRef] - Ma, Y.; Wang, N.; He, Z.; Lu, J.; Liang, H. Analysis of the bullwhip effect in two parallel supply chains with interacting price-sensitive demands. Eur. J. Oper. Res.
**2015**, 243, 815–825. [Google Scholar] [CrossRef] - Larsen, E.R.; Morecroft, J.D.W.; Thomsen, J.S. Complex behaviour in a production-distribution model. Eur. J. Oper. Res.
**1999**, 119, 61–74. [Google Scholar] [CrossRef] - Hwarng, H.B.; Xie, N. Understanding supply chain dynamics: A chaos perspective. Eur. J. Oper. Res.
**2008**, 184, 1163–1178. [Google Scholar] [CrossRef] - Hwarng, H.B.; Yuan, X. Interpreting supply chain dynamics: A quasi-chaos perspective. Eur. J. Oper. Res.
**2014**, 233, 566–579. [Google Scholar] [CrossRef] - Mosekilde, E.; Laugesen, J.L. Nonlinear dynamic phenomena in the beer model. Syst. Dyn. Rev.
**2007**, 23, 229–252. [Google Scholar] [CrossRef] - Wang, X.; Disney, S.M.; Wang, J. Stability analysis of constrained inventory systems with transportation delay. Eur. J. Oper. Res.
**2012**, 223, 86–95. [Google Scholar] [CrossRef] - Wang, X.; Disney, S.M.; Wang, J. Exploring the oscillatory dynamics of a forbidden returns inventory system. Int. J. Prod. Econ.
**2014**, 147, 3–12. [Google Scholar] [CrossRef] - Gao, B.; Wu, C.; Wu, Y.; Tang, Y. Expected Utility and Entropy-Based Decision-Making Model for Large Consumers in the Smart Grid. Entropy
**2015**, 17, 6560–6575. [Google Scholar] [CrossRef] - Zou, Y.; Yu, L.; He, K. Wavelet Entropy Based Analysis and Forecasting of Crude Oil Price Dynamics. Entropy
**2015**, 17, 7167–7184. [Google Scholar] [CrossRef] - Ma, J.; Si, F. Complex Dynamics of a Continuous Bertrand Duopoly Game Model with Two-Stage Delay. Entropy
**2016**, 18, 266. [Google Scholar] [CrossRef] - Disney, S.M.; Towill, D.R. The effect of vendor managed inventory (VMI) dynamics on the Bullwhip Effect in supply chains. Int. J. Prod. Econ.
**2003**, 85, 199–215. [Google Scholar] [CrossRef] - Costantino, F.; Di Gravio, G.; Shaban, A.; Tronci, M. The impact of information sharing on ordering policies to improve supply chain performances. Comput. Ind. Eng.
**2015**, 82, 127–142. [Google Scholar] [CrossRef] - Xie, L.; Ma, J. Study the complexity and control of the recycling-supply chain of China’s color TVs market based on the government subsidy. Commun. Nonlinear Sci. Numer. Simul.
**2016**, 38, 102–116. [Google Scholar] [CrossRef] - Zhang, J.; Ma, J. Research on the price game model for four oligarchs with different decision rules and its chaos control. Nonlinear Dyn.
**2012**, 70, 323–334. [Google Scholar] [CrossRef]

**Figure 10.**Bifurcation diagram of the system with $\beta =1.5$, $K=0.6$ and α varying from 0 to 4.8.

**Figure 12.**Bullwhip effect of the system without control and under control with α varying from 0 to 5.

**Figure 13.**Bifurcation diagram of the system with $\alpha =2.8$ and $\beta =1.5$, K varying from 0 to 1.

**Figure 14.**Bullwhip effect of the system with $\alpha =2.8$ and $\beta =1.5$, K varying from 0 to 1.

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**MDPI and ACS Style**

Ma, J.; Ma, X.; Lou, W.
Analysis of the Complexity Entropy and Chaos Control of the Bullwhip Effect Considering Price of Evolutionary Game between Two Retailers. *Entropy* **2016**, *18*, 416.
https://doi.org/10.3390/e18110416

**AMA Style**

Ma J, Ma X, Lou W.
Analysis of the Complexity Entropy and Chaos Control of the Bullwhip Effect Considering Price of Evolutionary Game between Two Retailers. *Entropy*. 2016; 18(11):416.
https://doi.org/10.3390/e18110416

**Chicago/Turabian Style**

Ma, Junhai, Xiaogang Ma, and Wandong Lou.
2016. "Analysis of the Complexity Entropy and Chaos Control of the Bullwhip Effect Considering Price of Evolutionary Game between Two Retailers" *Entropy* 18, no. 11: 416.
https://doi.org/10.3390/e18110416