# A New Supply Chain System and Its Impulsive Synchronization

Authors: Yang Peng; Jiang Wu; Shiping Wen; Yuming Feng; Zhengwen Tu; Limin Zou
Journal: Complexity
 (2020)

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## Abstract

The purpose of this paper is to discuss modelling and synchronization of nonlinear supply chain system. Firstly, we present a new supply chain system which is sensitive to various uncertainties along with exogenous disturbances. Synchronization is an important method to reduce the negative impact of uncertainties and disturbances on the supply chain. Since impulsive control can reduce control cost and the amount of transmitted information drastically, we discuss impulsive synchronization behavior of two supply chain systems with the same structure. Finally, simulation experiments are given to show the effectiveness of our analytical results.


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## Body

## 1. Introduction

Supply chain globalization is conducive to increasing the competitiveness of enterprises, but it also makes the supply chain face various uncertainties more frequently, such as demand uncertainty, supply uncertainty, and operational uncertainty. These uncertainties lead to the supply chain system showing complex dynamic characteristics, such as bullwhip effect [1, 2].In recent years, academia and industry are actively studying the complexity of the supply chain system. For example, Anne et al. [3] presented a three-echelon supply chain model which can be stated as follows:(1)x˙1=mx2−n+1x1,x˙2=x1r−x3−x2,x˙3=x1x2+b−1x3,where x1 is the quantity demanded by retailer in current period, x2 is the quantity distributors can supply in current period, and x3 is the quantity produced in the current period depend on the order, m is the coefficient of distributor delivery, n is rate of customer demand satisfaction at retailer, r is production threshold, and b is manufacturer’s safety stock factor. Supply chain system (1) has complex dynamic behavior. For example, if we choose m=10,n=3,r=18,andb=3/7, then system (1) is chaotic and Figure 1 shows the chaotic phenomenon of system (1) with the initial condition x1,x2,x3T=10,10,10T.Figure 1
The chaotic phenomenon of system (1) with the initial condition x1,x2,x3T=10,10,10T. (a) 3D phase portraits. (b) Time series of xi, i = 1, 2, 3.
(a)(b)Chaotic supply chain system (1) is very sensitive to disturbances. When a node in the supply chain changes, then the change will spread to the other two nodes quickly and cause the supply chain to deviate from the original state. For example, suppose that m=10,n=3,r=18,andb=3/7, if the state variables Xt0=10,10,10 becomes Yt0=10.1,10,10 at some point, that is to say, the retailer’s demand will increase by 0.1 unit, then the operation state of the supply chain will be changed, as shown in Figure 2.Figure 2
The influence of uncertain factors on system (1). (a) Time series with different initial values. (b) Errors of each node.
(a)(b)From Figure2, it can be seen that the small change of retailer’s demand has little impact on the operation state of the supply chain at the beginning, but with the passage of time, the operation state of the supply chain will change greatly, resulting in the operation of the supply chain deviating from the preset state. Therefore, the general idea is to restore the deviated operation state to the expected track through appropriate control strategies, so as to reduce the impact of uncertainty factors.Synchronization is an important method to reduce the negative impacts of uncertainties and disturbances on the supply chain. Göksu et al. [4] discussed the synchronization of two identical chaotic supply chain systems, and active and linear feedback controllers are applied for the system. Kocamaz et al. [5] presented the control of chaotic supply chain with Artificial Neural Network-based controllers and the synchronization of two identical chaotic supply chains. Tirandaz [6, 7] discussed adaptive modified projective synchronization and adaptive integral sliding mode control synchronization of supply chain system (1), respectively. Geng and Xiao [8] discussed the outer synchronization and parameter identification approach to the resilient recovery of supply network with uncertainty. Xu et al. [9] designed the adaptive super-twisting sliding mode control algorithm for chaos suppression and synchronization of the supply chain system.Considering the fact that the demand of a product does not increase monotonically with the increase of inventory, Mondal [10] proposed a new supply chain model as follows:(2)x˙1=mx21+ax2−n+1x1,x˙2=x1r−x3−px21+ax2,x˙3=x1x2+b−1x3,where a determines the saturation rate of demand with the increase of inventory and p denotes the effect of demand variation of the inventory. The author also discussed synchronization behavior of two coupled identical supply chain models under both unidirectional and bidirectional coupling.However, system (2) has a singularity when x2=−1/a and the second equation is unreasonable according to the analysis of Anne et al. [3]. Therefore, in this paper, following the idea of Mondal, in which demand has a saturation level and it does not increase monotonically with the inventory, we will propose a new supply chain system which does not have singularity. In addition, considering the feasibility of the control strategy, we use the impulsive control method to realize the synchronization of the new system. For more information on impulsive control method and its advantages, the reader can refer to [11–22] and the references therein.


## 2. A New Supply Chain System and Its Impulsive Synchronization

Combining system (1) with (2), we propose a new supply chain system as follows:(3)x˙1=mx21+a2x22−n+1x1,x˙2=x1r−x3−x2,x˙3=x1x2+b−1x3.Puttinga=0 in system (3), we obtain system (1). The function fx2=x2/1+a2x22 is used to describe the saturation degree of retailer’s demand with respect to supply. It is easy to see that f is odd function and limx2⟶+∞fx2=1/a. The graph of function f is shown in Figure 3.Figure 3
The graph of functionf for different values of a.By decomposing the linear and nonlinear parts of system in (3), we can rewrite it as follows:(4)X˙=AX+ϕX,where(5)A=−n−100r−1000b−1,X=x1x2x3,ϕX=mx21+a2x22−x1x3x1x2.If we choosem=10,n=3,r=18,b=3/7,anda=0.3, then system (3) is chaotic and Figure 4 shows the chaotic phenomenon of system (3) with the initial condition x1,x2,x3T=10,10,10T.Figure 4
The chaotic phenomenon of system (3) with the initial condition x1,x2,x3T=10,10,10T. (a) 3D phase portraits. (b) Time series of xi, i = 1, 2, 3.
(a)(b)Next, we investigate impulsive synchronization of two supply chain systems with the same structure. System (3) is the driving system and the driven system is defined as(6)dYdt=AY+ϕY,t≠τk,ΔY=Be,t=τk,k=1,2,…,where Y=y1,y2,y3T and B is impulsive control gain matrix. The symbol e denotes system errors and e=e1,e2,e3T=y1−x1,y2−x2,y3−x3T. There will be a jump in system (6) when t=τk and t0=τ0<τ1<τ2<,…,limk⟶∞τk=∞. Through, system (6) minus system (3), we obtain(7)dedt=Ae+ψX,Y,t≠τk,Δe=Be,t=τk,k=1,2,…,where ψX,Y=ϕY−ϕX.Due to the boundedness of the chaotic system (3), we suppose that xi≤M,i=1,2,3. In the following theorem, we present a sufficient condition for the stability of error system (7).Theorem 1.
Letαandλ be the largest eigenvalues of AT+A and I+BTI+B, respectively. If(8)logλ+α+2M+3mτk−τk−1<0holds for all k=1,2,…, then the origin of impulsive synchronization error system (7) is asymptotically stable.Proof.
Let(9)Vet=eTtet.
Fort∈t0,τ1, we have(10)dVetdt=Ae+ψX,YTe+eTAe+ψX,Y=eTATe+ψX,YTe+eTAe+eTψX,Y=eTAT+Ae+2eTψX,Y.
By the arithmetic-geometric mean inequality, we have(11)eTψX,Y=y1−x1,y2−x2,y3−x3my21+a2y22−mx21+a2x22,−y1y3+x1x3,y1y2−x1x2T=e1my21+a2y22−mx21+a2y22+mx21+a2y22−mx21+a2x22+x1x3y2+x2y1y3−y3x1x2−x3y1y2=e1me21+a2y22+mx211+a2y22−11+a2x22+x2e1e3−x3e1e2=e1me21+a2y22+mx2a2x2−y2x2+y21+a2y221+a2x221+a2x22+1+a2y22+x2e1e3−x3e1e2=me1e21+a2y221−a2x21+a2x22x2+y21+a2x22+1+a2y22+x2e1e3−x3e1e2≤me1e21+a2y221+a2x21+a2x22x2+y21+a2x22+1+a2y22+x2e1e3+x3e1e2≤me1e21+a2y221+a21a2a+Me1e3+e1e2≤3me1e21+a2y22+Me1e3+e1e2≤32me12+e22+Me1e3+e1e2≤3m+2M2eTe.
It follows from (10) and (11) that(12)dVetdt≤α+2M+3mVet.
Let(13)gt=dVetdt−α+2M+3mVet.
Then, we obtain(14)Vet=eα+2M+3mt−t0∫t0te−α+2M+3mτgτdτ+Vet0.
Sincegt≤0, we have(15)Vet≤Vet0eα+2M+3mt−t0,for t∈t0,τ1.
Whent=τ1, we have(16)Veτ1=I+Beτ1−TI+Beτ1−=eTτ1−I+BTI+Beτ1−≤λeTτ1−eτ1−=λVeτ1−.
Ift∈τ1,τ2, then repeating the process of deriving inequality (15), we obtain(17)Vet≤Veτ1eα+2M+3mt−τ1.
It follows from (16) and (17) that(18)Vet≤λVeτ1−eα+2M+3mt−τ1.
By inequality (18) and local order preserving of function limit, we obtain(19)Veτ1−≤Vet0eα+2M+3mτ1−t0.
Combining with inequalities (18) and (19), we have(20)Vet≤λVet0eα+2M+3mt−t0,for t∈τ1,τ2.
Whent=τ2, by the same method which is used to deduce inequality (16), we obtain(21)Veτ2≤λVeτ2−,and so(22)Vet≤Veτ2eα+2M+3mt−τ2≤λVeτ2−eα+2M+3mt−τ2≤λλVet0eα+2M+3mτ2−t0eα+2M+3mt−τ2=λ2Vet0eα+2M+3mt−t0,for t∈τ2,τ3.
Therefore, by the repeatability of the proof process and inequalities (15), (20), and (22), we know that(23)Vet≤λk−1Vet0eα+2M+3mt−t0=Vet0eα+2M+3mt−t0ek−1logλ=Vet0elogλ+α+2M+3mτ1−t0elogλ+α+2M+3mτ2−τ1⋯eα+2M+3mt−τk−1=λ−1Vet0elogλ+α+2M+3mτ1−t0elogλ+α+2M+3mτ2−τ1⋯elogλ+α+2M+3mt−τk−1,for t∈τk−1,τk.
So, ifα+2M+3m<0, then(24)Vet≤Vet0e∑i=1k−1logλ+α+2M+3mτi−τi−1.
On the other hand, ifα+2M+3m≥0, then(25)Vet≤λ−1Vet0e∑i=1klogλ+α+2M+3mτi−τi−1.
Hence, iflogλ+α+2M+3mτk−τk−1<0,k=1,2,…, then Vet⟶0 when k⟶+∞, that is, t⟶+∞. This completes the proof.
In generally, the intensity of each impulsive may not be the same, in view of this, we have the following result.Theorem 2.
Letαandλk be the largest eigenvalues of AT+A and I+BkTI+Bk, respectively. If(26)logλk+α+2M+3mτk−τk−1<0,holds for all k=1,2,…,, then the origin of impulsive synchronization error system (7) is asymptotically stable. The proof of this result is the same as Theorem 1.Recently, Hu et al. considered the effect of noise disturbances and provided some new methods [23–26]. Now, following the ideas of these authors, we consider system (3) with parameter uncertainty and state variables perturbation. The corresponding system can be described as follows:(27)dXdt=A+ΔAX+ϕX+fX,where fX=asinx1,bcosx2,csinx3T and(28)ΔA=δn00δr0000δb.Similarly, we can obtain the error system as follows:(29)dedt=A+ΔAe+ψX,Y+gX,Y,t≠τk,Δe=Be,t=τk,k=1,2,…,where gX,Y=fY−fX.Theorem 3.
Letαandλ be the largest eigenvalues of AT+A and I+BTI+B, respectively. Let γ=maxδ1,δ2 with δ1=2δb,δ2=δn+δn2+δr2, and ρ=2maxa,b,c. If(30)lnλ+α+γ+2M+3m+ρτi−τi−1<0holds for all k=1,2,…,, then the origin of impulsive synchronization error system (29) is asymptotically stable.Proof.
Note that(31)ΔA+ΔAT=2δnδr0δr00002δb.
Then, we have(32)xI−ΔA+ΔAT=x−2δnδr0δrx000x−2δb=x−2δbxx−2δn−δr2=x−2δbx2−2δnx−δr2.
So, the eigenvalues ofΔA+ΔAT are(33)δ1=2δb,δ2,3=δn±δn2+δr2.
Meanwhile, simple calculations show that(34)eTgX,Y=ae1sinY1−sinX1+be2cosY2−cosX2+ce3sinY3−sinX3≤ae1Y1−X1+be2Y2−X2+ce3Y3−X3=ae12+be22+ce32≤maxa,b,ceTe.
Let us construct the following Lyapunov function:(35)Vet=eTe.
Fort∈t0,τ1, we have(36)dVetdt=A+ΔAe+ψX,Y+gX,YTe+eTA+ΔAe+ψX,Y+gX,Ye=eTA+AT+ΔA+ΔATe+2eTψX,Y+2eTgX,Y.
It follows from (11) and (34) that(37)dVetdt≤eTA+AT+ΔA+ΔATe+2M+3meTe+ρeTe≤α+γeTe+2M+3meTe+ρeTe=α+γ+2M+3m+ρ.
The rest of proof is the same as Theorem1. This completes the proof.


## 3. Simulation Experiments

In this section, we will use some numerical simulation experiments to illustrate how to use the strategies presented in Section2 to eliminate or suppress the negative impacts of uncertain factors on the supply chain system.Experiment 1.
This example simulates the impact of the change of retailer demand on the system operation in supply chain system (3) and restores the operation state of supply chain to the expected track through impulsive control strategy. As mentioned above, if we choose m=10,n=3,r=18,b=3/7,anda=0.3, then system (3) is chaotic.
Small calculations show thatα=13.2483andM=27.5293. When t=1, the retailer’s demand is reduced by one unit. We find that the operation of the supply chain is seriously deviated from the expected state soon afterwards. We hope that the supply chain system can return to its original state by itself after a period. Unfortunately, the two tracks have not come together. We start to implement impulsive control when t=5 and assume that the intensity of each regulation is the same. According to Theorem 1, by simple calculations, we know that if the intensity and time interval of impulsive control satisfy the following inequality(38)logλ+98.3069τk−τk−1<0,then, the disturbed operation state will return to the expected orbit after appropriate impulsive control. In order to satisfy inequality (38), we choose impulsive control gain matrix B=−0.7I, that is, the value of each node in the supply chain Yi+1 becomes 0.7Xi+0.3Yi. So, the time interval of regulation should meet τk−τk−1<0.0245. Suppose that τk−τk−1=0.0200, which means that the time interval of impulsive regulation is 20 unit time (the unit time is taken as 0.001 in the experiment). The numerical simulation results are shown in Figure 5.Figure 5
Impulsive synchronization of supply chain system (3). (a) Time series of xi, i = 1, 2, 3. (b) Errors of each node.
(a)(b)Experiment 2.
In this example, we simulate the impact of the change of the manufacturer’s output on the system operation in the supply chain system (27), and through impulsive control strategy, the supply chain operation state is restored to the expected track. Let m=5,n=1,r=20,b=0.3,a=0.1,δm=δn=0.1,δr=0.2,andδb=−0.1 and ft,X=0.01sinx1,0.02cosx2,0.03sinx3T be input disturbance. Suppose that Xt0=10,10,10, the phase diagram of the system and state trajectory of each node are shown in Figure 6.
Small calculations show thatα=17.0250,M=31.7329,γ=0.3236,andρ=0.06. When t=1, the produced quantity of the manufacturer is reduced, say, 0.7 times of the original for some reason. We find that the operation of the supply chain is seriously deviated from the expected state soon afterwards. We start to implement impulsive control when t=15 and assume that the intensity of each regulation is the same. According to Theorem 3, by simple calculations we know that if the intensity and time interval of impulsive control satisfy the following inequality(39)logλ+96.1744τk−τk−1<0,then, the disturbed operation state will return to the expected orbit after appropriate impulsive control. In order to satisfy inequality (39), we choose impulsive control gain matrix B=−0.9I, that is, the value of each node in the supply chain Yi+1 becomes 0.9Xi+0.1Yi. So, the time interval of regulation should meet τk−τk−1<0.0479 and suppose that τk−τk−1=0.0450 which means that the time interval of impulsive regulation is 45 unit time (the unit time is taken as 0.001 in the experiment). The numerical simulation results are shown in Figure 7.
It can be seen from Figure7 that the operation state of the system quickly returns to the expected orbit.Figure 6
The chaotic phenomenon of system (27) with the initial condition Xt0=10,10,10. (a) 3D phase portraits. (b) Time series of xi, i = 1, 2, 3.
(a)(b)Figure 7
Impulsive synchronization of supply chain system (27). (a) Time series of xi, i = 1, 2, 3. (b) Errors of each node.
(a)(b)


## 4. Conclusion

As pointed out in [3], in today’s global marketplace, the supply chain is dynamic and unstable. This dynamic and unstable properties produce a variety of uncertainties in the supply chain, such as demand uncertainty, supply uncertainty, delivery uncertainty, and forecast uncertainty. These uncertainties make the upstream and downstream transmission of the supply chain complex and nonlinear. In this paper, we propose a new supply chain system which is more in line with the actual situation. As stated earlier, synchronization is an important way to avoid uncertainty in supply chains, and impulsive control is an effective method to realize synchronization. So, we also discuss impulsive synchronization of supply chain system and some impulsive synchronization strategies are obtained. These results may be useful to the management of the supply chain.



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