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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:

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##### References:
Schematic of the control system
Supply chain network (SCN)
Block diagram of entity operations
Time series plot of DTI and FTI in the stable state
Phase plot of DTI-FTI in the stable state
Time series plot of DTI and FTI in the chaotic state
Phase plot of DTI-FTI in the chaotic state
Effect of the demand elasticity by the ES forecast method
Effect of the inventory adjustment parameter by the ES forecast method
Effect of the demand elasticity by the MA forecast method
Effect of the inventory adjustment parameter by the MA forecast method
Effect of the demand elasticity by the ordering policy based on the PD controller
Effect of the inventory adjustment parameter by the ordering policy based on the PD controller
Comparison of all scenarios with $\lambda _\max \prec 0$
Comparison of all scenarios with 0 ≤ $\lambda _\max \prec 0.01$
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 * *
The initial data and parameters
 Item Value 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, $\beta$ $0\mathrm{\le}\beta\mathrm{\le}2$
 Item Value 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, $\beta$ $0\mathrm{\le}\beta\mathrm{\le}2$
The number of Maximum LEs in different ranges
 Scenario $0.02\le \lambda _{Max}$ $0.01\le \lambda _{Max} \prec 0.02$ $0\le \lambda _{Max} \prec 0.01$ $\lambda _{Max} \prec 0$ 1 159 95 94 452 2 160 120 80 440 3 202 306 120 172 4 208 336 87 169
 Scenario $0.02\le \lambda _{Max}$ $0.01\le \lambda _{Max} \prec 0.02$ $0\le \lambda _{Max} \prec 0.01$ $\lambda _{Max} \prec 0$ 1 159 95 94 452 2 160 120 80 440 3 202 306 120 172 4 208 336 87 169
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