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

Hamid Norouzi Nav; Mohammad Reza Jahed Motlagh; Ahmad Makui

doi:10.3934/jimo.2018002

Article Contents

2018, Volume 14, Issue 3: 1123-1141. Doi: 10.3934/jimo.2018002

This issue Previous Article Portfolio procurement policies for budget-constrained supply chains with option contracts and external financing Next Article A new proximal chebychev center cutting plane algorithm for nonsmooth optimization and its convergence

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

Received: November 30, 2015

Revised: July 31, 2017

Early access: January 2018

Published: June 2018

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Abstract

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.

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Mathematics Subject Classification: Primary: 35C20, 35P20; Secondary: 93D15.

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Figure 1. Schematic of the control system

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Figure 2. Supply chain network (SCN)

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Figure 3. Block diagram of entity operations

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Figure 4. Time series plot of DTI and FTI in the stable state

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Figure 5. Phase plot of DTI-FTI in the stable state

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Figure 6. Time series plot of DTI and FTI in the chaotic state

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Figure 7. Phase plot of DTI-FTI in the chaotic state

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Figure 8. Effect of the demand elasticity by the ES forecast method

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Figure 9. Effect of the inventory adjustment parameter by the ES forecast method

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Figure 10. Effect of the demand elasticity by the MA forecast method

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Figure 11. Effect of the inventory adjustment parameter by the MA forecast method

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Figure 12. Effect of the demand elasticity by the ordering policy based on the PD controller

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Figure 13. Effect of the inventory adjustment parameter by the ordering policy based on the PD controller

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Figure 14. Comparison of all scenarios with $\lambda _\max \prec 0$

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Figure 15. Comparison of all scenarios with 0 ≤ $\lambda _\max \prec 0.01$

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Table 1. Decision-making scenarios

Scenarios	MA forecast	ES forecast	PD controller-based ordering	Smooth ordering
1	*		*
2	*			*
3		*	*
4		*		*

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Table 2. 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 $	$0\mathrm{\le}\beta\mathrm{\le}2$

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Table 3. The number of Maximum LEs in different ranges

Scenario $ 0.02\le \lambda _{Max} $	$0.01\le \lambda _{Max} \prec 0.02$	$0\le \lambda _{Max} \prec 0.01$	$\lambda _{Max} \prec 0$
1	159	95	94	452
2	160	120	80	440
3	202	306	120	172
4	208	336	87	169

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