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

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  • 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.

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


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  • 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 * *
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    Table 2.  The initial data and parameters

    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$
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