Guarantee Stability of Supply Chain with Dual Channels under Supply Failure of Goods

Abstract

The supply chain is an extremely important and complex system, which is widely related to the social system and the management system. The hierarchical scheduling and production planning of a dynamic supply chain can enable enterprises in the supply chain to obtain, and maintain, stable and lasting competitive advantages; thus, improving the overall competitiveness of the supply chain. Simultaneously, research on supply chains is also a hotspot in system engineering. At present, research on complex supply chain systems with dual channels is insufficient, and analysis of dynamic supply chains with supply failure of goods is rare. In the epidemic era, dynamic supply chain management was affected by the epidemic and became a very challenging topic. The control of the supply chain system under the impact of an epidemic situation is particularly important. This paper focused on ${\mathcal{H}}^{\infty }$ the fault-tolerant control of a complex supply chain system with dual channels. First, from the perspective of engineering practice and green production, we established a new green supply chain model. This supply chain model includes the dual-channel supply of goods and also considers the supply failure of goods. At the same time, production delay and supply delay are also considered in this model. Second, a supply scheduling scheme was designed through cybernetics to ensure the stability of a supply chain with dual channels under supply failure of goods, and we completed supply chain management. Finally, the effectiveness of our results was verified by specific Chinese enterprise data.

1. Introduction

A dynamic supply chain is the core issue in green manufacturing [1]. The control and management of dynamic supply chains occupy a core position in the sustainable development strategy of today’s society and is an internal requirement for the scientific concept of development and the concept of sustainable development [2]. We propose a high-quality dynamic supply chain model and present a high-quality supply chain scheduling plan, which is in line with the development requirements of the national economy [1,2]. The Earth is the only home we live in at present, and its resources are limited. Therefore, it is necessary to build a recycling channel to save resources to ensure the sustainable development of human society. The recycling channel is particularly important in the dynamic supply chain of the national economy [3]. If there are dual channels in complex supply chain systems, the supply of goods not only ensures the continuous supply of goods from downstream suppliers but also saves the inventory costs of upstream suppliers [4]. In the face of the increasingly developing international trade environment, various unstable factors and external disturbances endlessly emerge, and the stability of a dynamic supply chain is greatly challenged. For example, the recent COVID-19 epidemic had a huge impact on the national economy [5]. In consideration of the lack of trust, uncertain demand, and information asymmetry in enterprises in the supply chain network, the authors in Ref. [6] constructed a network dynamics model of trust representation, calculation, and dissemination. Fault-tolerant control [7] and ${\mathcal{H}}^{\infty }$ control [8] provide ideas to solve disturbance and actuator failures of complex systems [9,10]. Therefore, it is very important to research the supply chain system with dual channels under supply failure of goods based on fault-tolerant control and ${\mathcal{H}}^{\infty }$ control.

Stability analysis and control of supply chains have frontier hot topics [11]. Traditional retail channels can no longer meet the current research progress. With the rapid development of network technology, online channels need to be taken into account. Abbaspour et al. comprehensively considered the retail channel and online channel, and also discussed relevant important issues, such as the demand for goods, the lead time of goods schedule, the delivery time of customer service in each channel, inventory control strategy, market pricing strategy, etc [11]. The final question is how to maximize the profits of enterprises. Salami et al. considered coordination and recovery issues, which are issues the current industry is paying more and more attention to. These two issues have broad practical significance and can offer enterprises ways to further improve profits. At the same time, the authors also considered the manufacturing capacity and processing capacity of the enterprise [12]. It can also be seen that product recycling is very important. We need to consider establishing recycling channels in the supply chain. Zhang et al. proposed a new model, but it only involved a single retailer and a single manufacturer [13]. Xu et al. proposed a dual-channel supply chain model, which includes online channel suppliers and offline retailers, but did not consider the problem of supply failure and did not give an accurate scheduling scheme from the perspective of cybernetics [14]. The supply chain has become a key research object in the field of management science. Green supply chain management is an important part of sustainable development, and faces great challenges in all walks of life [15]. The control and optimization of a supply chain were investigated, and the optimal scheme was designed via cybernetics [16]. The manufacturer’s behavior concerning fairness was considered in the green closed-loop supply chain [17]. Soon et al. proposed a robust multi-objective mathematical model for a sustainable closed-loop supply chain network. The network model can meet different types of customer needs and has different quality levels [18]. Although many outstanding achievements have been made in hierarchical scheduling and production planning of complex supply chain systems, there are still some thorny problems to be solved. Ref. [19] does not consider the situation of a supply failure in each link of the supply chain. The situation considered in Ref. [20] is also relatively simple and does not consider the supply of goods through multiple channels, such as offline convenience stores, supermarkets, snack bars, and online Taobao, JD.com, Pinduoduoduo, and other channels. The single-channel supply chain is extremely fragile and unstable, which, facing an increasingly complex international trade environment, no longer meets the needs of social development. The situations of dual-channel supply of goods and supply failure were not considered [16]. Ref. [17] did not study the hierarchical scheduling and production planning of complex supply chain systems in the case of commodity supply failures. Ref. [16] focused on the supply chain network with a single channel and without supply failure, and we need to consider dual channels and supply failure.

Based on these analyses, this paper focuses on guaranteeing the stability in a supply chain with dual channels under supply failure of goods via ${\mathcal{H}}^{\infty }$ control and fault-tolerant control. The purpose of this paper was to provide hierarchical scheduling and production planning schemes for complex supply chain systems with dual channels in the presence of supply failures so that enterprises can ensure the stability of their interests and dynamic supply chains. The study objective of this paper is to the supply chain with dual channels under supply failure of goods. The main problem of this paper was in guaranteeing the stability of a supply chain with dual channels under the supply failure of goods. The main contributions here are as follows:

• A supply chain dynamic network model, including dual channels and supply failures, is established;
• Based on the fault-tolerant control principle and ${\mathcal{H}}^{\infty }$ control scheme, the stability of a closed supply chain system is guaranteed.

The article is organized as follows: The dynamic supply chain model established in this paper is given in Section 2. The hierarchical scheduling and production planning of a complex supply chain system with dual channels under supply failure is given in Section 3. A numerical analysis to verify the validity of the theoretical results in Section 3 is given in Section 4. Some in-depth analysis and discussion are given in Section 5. Section 6 provides the conclusion of this paper.

2. Problem Definition & Formulation

This section establishes a model of a supply chain system with dual channels [21] and supply failures [5]. We completed the work in four steps.

Through the investigation of Gree Electric Co., Ltd., and TCL Company, it was found that home appliance enterprises have complete information flows [22]. Manufacturers need raw material suppliers to supply materials according to certain demands. Manufacturers’ products need to flow to distributors. The number of distributors’ products fluctuates under the influence of market conditions. There are two transaction modes, online (such as Taobao, JD, Pinduoduo, etc.) and offline (such as supermarkets, brand stores, wholesale cities, etc.), between manufacturers and distributors, and between distributors and markets [23]. Consumers can also purchase products directly from manufacturers through e-commerce channels. There are two destinations for old and damaged waste products: one is to discard them directly as garbage, and the other is to send them to manufacturers for regeneration and processing after being recycled by recyclers for sale. Based on this principle, the topology of a supply chain with dual channels in production–sales—recycling operation of household appliances enterprises could be established, as shown in Figure 1.

To facilitate our subsequent discussion, we used some variables to replace the specific physical meanings. The details are as follows:

• $x_1\left(k\right)$ represents the actual inventory of the manufacturer at the time k,
• $x_2\left(k\right)$ represents the actual inventory of the distributor at the time k,
• $x_3\left(k\right)$ represents the customer’s actual inventory at the time k,
• $x_4\left(k\right)$ represents the actual inventory of the recycler at the time k.
• $u_1\left(k\right)$ means the product produced by the manufacturer at the time k,
• $u_2\left(k\right)$ denotes the order quantity ordered by the distributor through an e-commerce channel at the time k,
• $u_3\left(k\right)$ denotes the order quantity ordered by the distributor through the physical channel at the time k,
• $u_4\left(k\right)$ denotes the waste disposal amount of the recycler at the time k.
• $\tau$ indicates the lag time in the supply chain with dual channels.
• ${\omega }_1\left(k\right)$ represents the customer demand at k,
• ${\omega }_2\left(k\right)$ represents the amount of product recovered at the time k,

where ${\omega }_1\left(k\right)$ and ${\omega }_2\left(k\right)$ are external input uncertainty variables. $\alpha (0⩽\alpha ⩽1)$ represents the remanufacturing rate, ${\beta }_1(0⩽{\beta }_1⩽1)$ represents the scrap rate, $\alpha$ and ${\beta }_1$ are constant parameters. $a,b,c(a+b+c=1,0⩽a⩽1,0⩽b⩽1,0⩽c⩽1)$ represent customer preference indices for different channels, respectively [19]. Therefore, combined with the engineering practice in the operation of supply of raw materials, production of products, distribution of products, online channels and recycling, and re-manufacturing, market fluctuations and supply failures, when considering the uncertainty of production time-delay, online channel order time delay, fluctuation of customer demand, supply failure, and recovery external input, the following dynamic models of production inventory, distribution inventory, customer virtual inventory, supply failure, and recovery inventory can be established. This is the equation of state as shown below

$$\begin{array}{cc}\hfill x_1(k+1)=& x_1\left(k\right)+\alpha x_4\left(k\right)+u_1\left(k\right)+u_1(k-\tau )\hfill \\ \hfill & -u_2\left(k\right)-u_2(k-\tau )-u_3\left(k\right)-a{\omega }_1\left(k\right),\hfill \\ \hfill x_2(k+1)=& x_2\left(k\right)+u_2\left(k\right)+u_2(k-\tau )+u_3\left(k\right)-b{\omega }_1\left(k\right)-c{\omega }_1\left(k\right),\hfill \\ \hfill x_3(k+1)=& x_3\left(k\right)-{\beta }_1{x}_3\left(k\right)+{\omega }_1\left(k\right)-{\omega }_2\left(k\right),\hfill \\ \hfill x_4(k+1)=& x_4\left(k\right)-\alpha x_4\left(k\right)-u_4\left(k\right)+{\omega }_2\left(k\right).\hfill \end{array}$$

(1)

Equation (1) describes the dynamic process of the supply chain manufacture, re-manufacture, and distributor inventory, as well as customer virtual inventory and recycler inventory in the demand market. The system is described by deviation.

Let $z\left(k\right)$ represent the total cost of a supply chain with dual channels and $c_h$, $c_o$, $c_r$, $c_n$, $c_m^1$, $c_m^2$, $c_u$ represent the determined inventory cost, scrap cost, re-manufacturing cost, production cost, ordering cost through e-commerce channels, ordering cost through field channels, and scrap disposal cost, respectively. The total operational cost of a complex supply chain system with dual channels is

$$\begin{array}{cc}\hfill z\left(k\right)=& c_h\left(x_1\left(k\right)+x_2\left(k\right)+x_4\left(k\right)\right)+c_o{\beta }_1{x}_3\left(k\right)+c_r\alpha x_4\left(k\right)\hfill \\ \hfill & +c_n{u}_1\left(k\right)+c_m^1{u}_2\left(k\right)+c_m^2{u}_3\left(k\right)+c_u{u}_4\left(k\right).\hfill \end{array}$$

(2)

Let $z\left(k\right)$ represent the output variable. $x\left(k\right)={\left[x_1\left(k\right),x_2\left(k\right),x_3\left(k\right),x_4\left(k\right)\right]}^{\top }$ represent a state variable, $u\left(k\right)={\left[u_1\left(k\right),u_2\left(k\right),u_3\left(k\right),u_4\left(k\right)\right]}^{\top }$ is iuput, which represents the control variable, $\omega \left(k\right)={\left[{\omega }_1\left(k\right),{\omega }_2\left(k\right)\right]}^{\top }$ represents an external input uncertainty variable.

Equations (1) and (2) can be expressed as the state space model shown below

$$\begin{array}{cc}\hfill x(k+1)=& Ax\left(k\right)+Bu\left(k\right)+B_{\tau }u(k-\tau )+F\omega \left(k\right),\hfill \\ \hfill z\left(k\right)=& Cx\left(k\right)+Du\left(k\right).\hfill \end{array}$$

(3)

where

$$\begin{array}{c}\hfill A=\left[\begin{array}{cccc}1& 0& 0& \alpha \\ 0& 1& 0& 0\\ 0& 0& 1-{\beta }_1& 0\\ 0& 0& 0& 1-\alpha \end{array}\right],B=\left[\begin{array}{cccc}1& -1& -1& 0\\ 0& 1& 1& 0\\ 0& 0& 0& 0\\ 0& 0& 0& -1\end{array}\right],B_{\tau }=\left[\begin{array}{cccc}1& -1& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right],\\ \hfill F=\left[\begin{array}{cc}-a& 0\\ -b-c& 0\\ 1& -1\\ 0& 1\end{array}\right],C=\left[\begin{array}{cccc}{c}_h\hfill & c_h\hfill & c_o{\beta }_1\hfill & c_h+c_r\alpha \hfill \end{array}\right],D=\left[\begin{array}{cccc}{c}_n\hfill & c_m^1\hfill & c_m^2\hfill & c_u\hfill \end{array}\right].\end{array}$$

If the whole supply chain dynamic network can supply normally, its control variable can be designed as $u\left(k\right)=Kx\left(k\right)$.

In recent years, COVID-19 has been rampant and brings great challenges to the stability analysis of dynamic supply chains. For example, it can lead to the obstruction of goods transportation and sales channels, which may involve the reduction, or even suspension, of production of manufacturers, the interruption of supply channels of distributors or the reduction of goods supply, and the inability of recyclers to recycle waste products [23]. We use the actuator failure model in cybernetics for reference to describe this kind of situation and introduce the switch matrix L, $L=\mathrm{diag}\{l_1,l_2,l_3,l_4\}$ to represent the manufacturer’s production reduction or even shut down. Here

$$l_i=\left\{\begin{array}{cc}1,\hfill & u_i\left(k\right)\mathrm{can}\mathrm{be}\mathrm{supplied}\mathrm{normally},\hfill \\ \overline{\alpha }(0⩽\overline{\alpha }⩽1),\hfill & u_i\left(k\right)\mathrm{supply}\mathrm{decline},\hfill \\ 0,\hfill & u_i\left(k\right)\mathrm{interruption}\mathrm{of}\mathrm{supply}.\hfill \end{array}\right.$$

(4)

When the goods transportation and sales channels are blocked, the supply volume of the dynamic network is as follows

$$u\left(k\right)=LKx\left(k\right).$$

(5)

Bringing the supply quantity (5) into the supply chain dynamic system (1) derives the closed-loop supply chain dynamic system

$$x(k+1)=Ax\left(k\right)+BLKx\left(k\right)+B_{\tau }LKx(k-\tau )+Fw\left(k\right).$$

(6)

System (6) describes a closed-loop supply chain with dual channels under supply failures, which includes product recycling and re-manufacturing.

The main purposes of this manuscript were to find a suitable supply plan for when a closed-loop supply chain (6) experiences supply failure and to ensure that the closed-loop system (6) meets the following conditions:

• stable market demand, that is, customer demand is stable. At this time, we need to find a reliable supply scheduling scheme to ensure the stability of the entire dynamic supply chain (6).
• When market demand is unstable, that is, when the customer demand is unstable. In this case, it is very difficult to ensure the stability of closed-loop supply. We can only try to suppress the impact of external disturbances on the entire dynamic network and ensure the smooth and healthy operation of the dynamic supply chain.

The main objective of this paper was to provide hierarchical scheduling and production planning schemes for complex dynamic supply chains with dual channels in the presence of supply failures. The hierarchical scheduling here is mainly reflected in regard to manufacturers, distributors, consumers, and recyclers. They are located in different links of the dynamic supply chain.

3. Robust Control of Complex Supply Chain System with Dual Channels

We consider the impact of market disruption on the supply chain separately. First, we give the results without market interference, and then we give the results with market interference.

Theorem 1. 

When market demand is stable, that is, when customer demand is stable, i.e., $w\left(k\right)=0$, if there is a positive constant ${\alpha }_1$, and positive definite symmetric constant matrix $X,Y$, and any constant matrix $Z$ makes inequalities

$$\left[\begin{array}{cc}-X+Y& X{A}^{\top }+Z^{\top }{L}^{\top }{B}^{\top }\\ AX+BLZ& -\frac{1}{1+{\alpha }_1}X\end{array}\right]<0,$$

(7)

$$\left[\begin{array}{cc}-Y& Z^{\top }{L}^{\top }{B}^{\top }\\ BLZ& -{\left(1+\frac{1}{{\alpha }_1}\right)}^{-1}X\end{array}\right]<0$$

(8)

hold, then, in the case of possible supply failure L, the closed-loop dynamic network (6) is stable by having the following supply volume

$$u\left(k\right)=LZ{X}^{-1}x\left(k\right).$$

(9)

Proof. 

Let us first construct a Lyapunov function, as shown below

$$V\left(k\right)=X^{\top }\left(k\right)Px\left(k\right)+\sum _{i=1}^h{x}^{\top }(k-i)Qx(k-i),$$

(10)

where $P,Q>0$. The difference of (10) along the closed-loop dynamic network (6) is

$$\begin{array}{cc}\hfill \Delta V\left(k\right)=& V(k+1)-V\left(k\right)\hfill \\ \hfill =& x^{\top }(k+1)Px(k+1)-x^{\top }\left(k\right)Px\left(k\right)\hfill \\ \hfill & +\sum _{i=1}^h\left[x^{\top }(k+1-i)Qx(k+1-i)-x^{\top }(k-i)Qx(k-i)\right]\hfill \\ \hfill =& \left\{x^{\top }\left(k\right){[A+BLK]}^{\top }+x^{\top }(k-h){\left[B_{\tau }LK\right]}^{\top }\right\}P·\left\{(A+BLK)x\left(k\right)\right.\hfill \\ \hfill & \left.+B_{\tau }LKx(k-h)\right\}-x^{\top }\left(k\right)Px\left(k\right)+x^{\top }\left(k\right)Qx\left(k\right)-x^{\top }(k-h)Qx(k-h)\hfill \\ \hfill =& x^{\top }\left(k\right){\left[A+BLK\right]}^{\top }P\left[A+BLK\right]x\left(k\right)+2x^{\top }\left(k\right){\left[A+BLK\right]}^{\top }P\left[B_{\tau }LK\right]x(k-h)\hfill \\ \hfill & +x^{\top }(k-h){\left(B_{\tau }LK\right)}^{\top }P\left(B_{\tau }LK\right)x(k-h)-x^{\top }\left(k\right)Px\left(k\right)+x^{\top }\left(k\right)Qx\left(k\right)\hfill \\ \hfill & -x^{\top }(k-h)Qx(k-h).\hfill \end{array}$$

(11)

It can be obtained by the Cauchy matrix inequality

$$\begin{array}{cc}\hfill 2x^{\top }\left(k\right){\left[A+BLK\right]}^{\top }P\left[B_{\tau }LK\right]x(k-h)⩽& {\alpha }_1{x}^{\top }\left(k\right){\left(A+BLK\right)}^{\top }P\left(A+BLK\right)x\left(k\right)\hfill \\ \hfill & +\frac{1}{{\alpha }_1}{x}^{\top }(k-h){\left[B_{\tau }LK\right]}^{\top }P\left[B_{\tau }LK\right]x(k-h).\hfill \end{array}$$

(12)

Bringing the inequality (12) into Equation (11), there is

$$\begin{array}{cc}\hfill \Delta V\left(k\right)⩽& x^{\top }\left(k\right){\left[A+BLK\right]}^{\top }P\left[A+BLK\right]x\left(k\right)\hfill \\ \hfill & +{\alpha }_1{x}^{\top }\left(k\right){\left(A+BLK\right)}^{\top }P\left(A+BLK\right)x\left(k\right)\hfill \\ \hfill & +\frac{1}{{\alpha }_1}{x}^{\top }(k-h){\left[B_{\tau }LK\right]}^{\top }P\left[B_{\tau }LK\right]x(k-h)\hfill \\ \hfill & +x^{\top }(k-h){\left[B_{\tau }LK\right]}^{\top }P\left[B_{\tau }LK\right]x(k-h)\hfill \\ \hfill & -x^{\top }\left(k\right)Px\left(k\right)+x^{\top }\left(k\right)Qx\left(k\right)-x^{\top }(k-h)Qx(k-h)\hfill \\ \hfill =& x^{\top }\left(k\right)\left\{(1+{\alpha }_1){\left[A+BLK\right]}^{\top }P\left[A+BLK\right]-P+Q\right\}x\left(k\right)\hfill \\ \hfill & +x^{\top }(k-h)\left\{\left(1+\frac{1}{{\alpha }_1}\right){\left[B_{\tau }LK\right]}^{\top }P\left[B_{\tau }LK\right]-Q\right\}x(k-h).\hfill \end{array}$$

(13)

Let $X=P^{-1},Y=P^{-1}Q{P}^{-1},Z=K{P}^{-1}$, then the inequality (7) can be rewritten as

$$\left[\begin{array}{cc}-P^{-1}+P^{-1}Q{P}^{-1}& P^{-1}{A}^{\top }+P^{-1}{K}^{\top }{L}^{\top }{B}^T\\ A{P}^{-1}+BLK{P}^{-1}& -\frac{1}{1+{\alpha }_1}{P}^{-1}\end{array}\right]<0.$$

(14)

Multiply both sides of the inequality (14) by $\mathrm{diag}\{P,I\}$, so there is

$$\left[\begin{array}{cc}-P+Q\hfill & {\left(A+BLK\right)}^{\top }\hfill \\ A+BLK\hfill & -\frac{1}{1+{\alpha }_1}{P}^{-1}\hfill \end{array}\right]<0.$$

(15)

Through the Shure complement theorem, Equation (15) can be revised into

$$(1+{\alpha }_1){\left[A+BLK\right]}^{\top }P\left[A+BLK\right]-P+Q<0.$$

(16)

Let $X=P^{-1},Y=P^{-1}Q{P}^{-1},Z=K{P}^{-1}$, the above Equation (8) be converted into

$$\left[\begin{array}{cc}-P^{-1}Q{P}^{-1}& {\left(K{P}^{-1}\right)}^{\top }{L}^{\top }{B}_{\tau }^{\top }\\ B_{\tau }LK{P}^{-1}& -{\left(1+\frac{1}{{\alpha }_1}\right)}^{-1}{P}^{-1}\end{array}\right]<0.$$

(17)

Multiply both sides of the inequality (17) by $\mathrm{diag}\{P,I\}$, with

$$\left[\begin{array}{cc}-Q& K^{\top }{L}^{\top }{B}_{\tau }^{\top }\\ B_{\tau }LK& -{\left(1+\frac{1}{{\alpha }_1}\right)}^{-1}{P}^{-1}\end{array}\right]<0.$$

(18)

Through the Shure complement theorem, Equation (18) can be revised into

$$\left(1+\frac{1}{{\alpha }_1}\right){\left(B_{\tau }LK\right)}^{\top }P\left(B_{\tau }LK\right)-Q<0.$$

(19)

Bring (16) and (18) into (13), and there is

$$\begin{array}{cc}\hfill \Delta V\left(k\right)⩽& 0.\hfill \end{array}$$

(20)

Now we have completed the proof of the theorem. □

Theorem 2. 

When market demand is unstable, that is, when customer demand is unstable, i.e., $w\left(k\right)\ne 0$, if there are positive constants ${\alpha }_1,{\alpha }_2,{\alpha }_3$, and a positive definite symmetric constant matrix $X,Y$, and any constant matrix $Z$ makes inequalities

$$\left[\begin{array}{ccc}-X+Y& {\left(AX+BLZ\right)}^{\top }& {(CX+DLZ)}^{\top }\\ AX+BLZ& -\frac{1}{1+{\alpha }_1+{\alpha }_2}X& 0\\ CX+DLZ& 0& -I\end{array}\right]<0,$$

(21)

$$\left[\begin{array}{cc}-Y& Z^{\top }{L}^{\top }{B}_{\tau }^{\top }\\ B_{\tau }LZ& -{\left(\frac{1}{{\alpha }_2}+1+{\alpha }_3\right)}^{-1}X\end{array}\right]<0,$$

(22)

$$\left[\begin{array}{cc}-{\gamma }^{2}I& F^{\top }\\ F& -{\left(\frac{1}{{\alpha }_3}+\frac{1}{{\alpha }_4}+1\right)}^{-1}X\end{array}\right]<0$$

(23)

hold, then in the case of possible supply failure L, it can ensure that the chain dynamic network (6) is still gradually stable with the satisfaction performance index γ by using the following supply volume

$$u\left(k\right)=LZ{X}^{-1}x\left(k\right).$$

(24)

In this case, it is very difficult to ensure the stability of closed-loop supply. We can only try to suppress the impact of external disturbances on our entire dynamic network and ensure the smooth and healthy operation of the dynamic supply chain.

Proof. 

We still use the previous Lyapunov function (10), and the difference of (10) along the closed-loop dynamic network (6) is

$$\begin{array}{cc}\hfill \Delta V\left(k\right)=& V(k+1)-V\left(k\right)\hfill \\ \hfill =& x^{\top }(k+1)Px(k+1)-x^{\top }\left(k\right)Px\left(k\right)\hfill \\ \hfill & +\sum _{i=1}^h\left[x^{\top }(k+1-i)Q_{2}x(k+1-i)-x^{\top }(k-i)Q_{2}x(k-i)\right]\hfill \\ \hfill =& \left\{x^{\top }\left(k\right){[A+BLK]}^{\top }+x^{\top }(k-h){\left[B_{\tau }LK\right]}^{\top }+{\omega }^{\top }\left(k\right)F^{\top }\right\}P\hfill \\ \hfill & ·\left\{(A+BLK)x\left(k\right)+B_{\tau }LKx(k-h)+Fw\left(k\right)\right)\hfill \\ \hfill & -x^{\top }\left(k\right)Px\left(k\right)+x^{\top }\left(k\right)Qx\left(k\right)-x^{\top }(k-h)Qx(k-h)\hfill \\ \hfill =& x^{\top }\left(k\right){\left[A+BLK\right]}^{\top }P\left[A+BLK\right]x\left(k\right)\hfill \\ \hfill & +2x^{\top }\left(k\right){\left[A+BLK\right]}^{\top }P\left[B_{\tau }LK\right]x(k-h)\hfill \\ \hfill & +2x^{\top }\left(k\right){\left[A+BLK\right]}^{\top }PFw\left(k\right)\hfill \\ \hfill & +x^{\top }(k-h){\left(B_{\tau }LK\right)}^{\top }P\left(B_{\tau }LK\right)x(k-h)\hfill \\ \hfill & +2x^{\top }(k-h){\left[B_{\tau }LK\right]}^{\top }PFw\left(k\right)\hfill \\ \hfill & +w^{\top }\left(k\right)F^{\top }PFw\left(k\right)\hfill \\ \hfill & -x^{\top }\left(k\right)Px\left(k\right)+x^{\top }\left(k\right)Qx\left(k\right)-x^{\top }(k-h)Qx(k-h).\hfill \end{array}$$

(25)

Using Cauchy matrix inequality we obtain

$$\begin{array}{cc}\hfill 2x^{\top }\left(k\right){\left[A+BLK\right]}^{\top }PFw\left(k\right)⩽& {\alpha }_2{x}^{\top }\left(k\right){\left(A+BLK\right)}^{\top }P\left(A+BLK\right)x\left(k\right)\hfill \\ \hfill & +\frac{1}{{\alpha }_2}{w}^{\top }\left(k\right)F^{\top }PFw\left(k\right),\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill 2x^{\top }(k-h){\left[B_{\tau }LK\right]}^{\top }PFw\left(k\right)⩽& {\alpha }_3{x}^{\top }(k-h){\left[B_{\tau }LK\right]}^{\top }P\left[B_{\tau }LK\right]x(k-h)\hfill \\ \hfill & +\frac{1}{{\alpha }_3}{w}^{\top }\left(k\right)F^{\top }PFw\left(k\right).\hfill \end{array}$$

(27)

Taking the inequalities (12), (26), and (27) into (25), yields

$$\begin{array}{cc}\hfill \Delta V\left(k\right)⩽& x^{\top }\left(k\right){\left[A+BLK\right]}^{\top }P\left[A+BLK\right]x\left(k\right)\hfill \\ \hfill & +{\alpha }_1{x}^{\top }\left(k\right){\left(A+BLK\right)}^{\top }P\left(A+BLK\right)x\left(k\right)\hfill \\ \hfill & +\frac{1}{{\alpha }_1}{x}^{\top }(k-h){\left[B_{\tau }LK\right]}^{\top }P\left[B_{\tau }LK\right]x(k-h)\hfill \\ \hfill & +{\alpha }_2{x}^{\top }\left(k\right){\left(A+BLK\right)}^{\top }P\left(A+BLK\right)x\left(k\right)+\frac{1}{{\alpha }_2}{w}^{\top }\left(k\right)F^{\top }PFw\left(k\right)\hfill \\ \hfill & +x^{\top }(k-h){\left(B_{\tau }LK\right)}^{\top }P\left(B_{\tau }LK\right)x(k-h)\hfill \\ \hfill & +{\alpha }_3{x}^{\top }(k-h){\left[B_{\tau }LK\right]}^{\top }P\left[B_{\tau }LK\right]x(k-h)+\frac{1}{{\alpha }_3}{w}^{\top }\left(k\right)F^{\top }PFw\left(k\right)\hfill \\ \hfill & +w^{\top }\left(k\right)F^{\top }PFw\left(k\right)\hfill \\ \hfill & -x^{\top }\left(k\right)Px\left(k\right)+x^{\top }\left(k\right)Qx\left(k\right)-x^{\top }(k-h)Qx(k-h)\hfill \\ \hfill =& x^{\top }\left(k\right)\{(1+{\alpha }_1+{\alpha }_2){\left[A+BLK\right]}^{\top }P\left[A+BLK\right]-P+Q\}x\left(k\right)\hfill \\ \hfill & +x^{\top }(k-h)\left\{\left(\frac{1}{{\alpha }_2}+1+{\alpha }_3\right){\left[B_{\tau }LK\right]}^{\top }P\left[B_{\tau }LK\right]-Q\right\}x(k-h)\hfill \\ \hfill & +w^{\top }\left(k\right)\left(\frac{1}{{\alpha }_2}+\frac{1}{{\alpha }_3}+1\right)F^{\top }PFw\left(k\right).\hfill \end{array}$$

(28)

Through simple calculation, the output secondary item can be written as

$$\begin{array}{cc}\hfill z^{\top }\left(k\right)z\left(k\right)=& {(Cx\left(k\right)+DLKx\left(k\right))}^{\top }(Cx\left(k\right)+DLKx\left(k\right))\hfill \\ \hfill =& x^{\top }\left(k\right)\left\{C^{\top }C+C^{\top }DLK+{\left(DLK\right)}^{\top }C+{\left(DLK\right)}^{\top }\left(DLK\right)\right\}x\left(k\right).\hfill \end{array}$$

(29)

Next, introduce the performance index function under zero initial condition, as follows:

$$J=\sum _{k=0}^{\infty }\left[z{\left(k\right)}^{\mathrm{T}}z\left(k\right)-{\gamma }^{2}w{\left(k\right)}^{\mathrm{T}}w\left(k\right)\right].$$

(30)

Then, for any nonzero $w\left(k\right)\in l_2[0,\infty )$,

$$J⩽\sum _{k=0}^{\infty }\left[z{\left(k\right)}^{\mathrm{T}}z\left(k\right)-{\gamma }^{2}w{\left(k\right)}^{\mathrm{T}}w\left(k\right)+\Delta V\left(k\right)\right].$$

(31)

Further substituting (28), (29) into (31), then

$$\begin{array}{cc}\hfill J⩽& \sum _{k=0}^{\infty }\left[z{\left(k\right)}^{\mathrm{T}}z\left(k\right)-{\gamma }^{2}w{\left(k\right)}^{\mathrm{T}}w\left(k\right)+\Delta V\left(k\right)\right]\hfill \\ \hfill =& \sum _{k=0}^{\infty }\{x^{\top }\left(k\right)\left\{C^{\top }C+C^{\top }DLK+{\left(DLK\right)}^{\top }C+{\left(DLK\right)}^{\top }\left(DLK\right)\right\}x\left(k\right)\hfill \\ \hfill & +x^{\top }\left(k\right)\left\{(1+{\alpha }_1+{\alpha }_2){\left[A+BLK\right]}^{\top }P\left[A+BLK\right]-P+Q\right\}x\left(k\right)\hfill \\ \hfill & +x^{\top }(k-h)\left\{\left(\frac{1}{{\alpha }_2}+1+{\alpha }_3\right){\left[B_{\tau }LK\right]}^{\top }P\left[B_{\tau }LK\right]-Q\right\}x(k-h)\hfill \\ \hfill & +w^{\top }\left(k\right)\left(\frac{1}{{\alpha }_2}+\frac{1}{{\alpha }_3}+1\right)F^{\top }PFw\left(k\right)-{\gamma }^2{w}^{\mathrm{T}}\left(k\right)w\left(k\right)\}.\hfill \end{array}$$

(32)

Let $X=P^{-1},Y=P^{-1}Q{P}^{-1},Z=K{P}^{-1}$, the inequality (21) be rewritten as

$$\left[\begin{array}{ccc}-P^{-1}+P^{-1}Q{P}^{-1}& {\left(A{P}^{-1}+BLK{P}^{-1}\right)}^{\top }& {(C{P}^{-1}+DLK{P}^{-1})}^{\top }\\ A{P}^{-1}+BLK{P}^{-1}& -\frac{1}{1+{\alpha }_1+{\alpha }_2}{P}^{-1}& 0\\ C{P}^{-1}+DLK{P}^{-1}& 0& -I\end{array}\right]<0.$$

(33)

Multiply both sides of the inequality (33) by $\mathrm{diag}\{P,I,I\}$, with

$$\left[\begin{array}{ccc}-P+Q& {\left(A+BLK\right)}^{\top }& {(C+DLK)}^{\top }\\ A+BLK& -\frac{1}{1+{\alpha }_1+{\alpha }_2}{P}^{-1}& 0\\ C+DLK& 0& -I\end{array}\right]<0.$$

(34)

Through simple calculation, the inequality (34) can be written as

$$\begin{array}{cc}\hfill (1+{\alpha }_1+{\alpha }_2)& {\left[A+BLK\right]}^{\top }P\left[A+BLK\right]-P+Q\hfill \\ \hfill +& C^{\top }C+C^{\top }DLK+{\left(DLK\right)}^{\top }C+{\left(DLK\right)}^{\top }\left(DLK\right)<0.\hfill \end{array}$$

(35)

Let $X=P^{-1},Y=P^{-1}Q{P}^{-1},Z=K{P}^{-1}$, The inequality (22) be rewritten as

$$\left[\begin{array}{cc}-P^{-1}Q{P}^{-1}& {\left(K{P}^{-1}\right)}^{\top }{L}^{\top }{B}_{\tau }^{\top }\\ B_{\tau }LK{P}^{-1}& -{\left(\frac{1}{{\alpha }_2}+1+{\alpha }_3\right)}^{-1}{P}^{-1}\end{array}\right]<0.$$

(36)

Multiply both sides of the inequality (36) by $\mathrm{diag}\{P,I\}$, with

$$\left[\begin{array}{cc}-Q& K^{\top }{L}^{\top }{B}_{\tau }^{\top }\\ B_{\tau }LK& -{\left(\frac{1}{{\alpha }_2}+1+{\alpha }_3\right)}^{-1}{P}^{-1}\end{array}\right]<0.$$

(37)

The above inequality (37) can be converted into

$$\left(\frac{1}{{\alpha }_2}+1+{\alpha }_3\right){\left(B_{\tau }LK\right)}^{\top }P\left(B_{\tau }LK\right)-Q<0.$$

(38)

Let $X=P^{-1}$, the inequality (23) be rewritten as follows

$$\left[\begin{array}{cc}-{\gamma }^{2}I& F^{\top }\\ F& -{\left(\frac{1}{{\alpha }_3}+\frac{1}{{\alpha }_4}+1\right)}^{-1}{P}^{-1}\end{array}\right]<0.$$

(39)

The above inequality (39) can be converted into

$$\left(\frac{1}{{\alpha }_2}+\frac{1}{{\alpha }_3}+1\right)F^{\top }PF-{\gamma }^{2}I<0.$$

(40)

If the above three formulae (35), (38), (40) are brought into (32), $J<0$ can be proved. □

4. Numerical Verification

We verify the validity of the results in Section 3 through numerical simulation in this Section. The whole process of numerical simulation is divided into four parts. First, establish a specific model of a complex supply chain system with dual-channel. Second, establish the failure model of supply. Third, calculate the specific supply through the theoretical results in Section 3. Finally, confirm the actual simulation results match the theoretical results perfectly by drawing.

First, based on the operational status of a Chinese power enterprise, we completed its simulation calculation through investigation and statistics. According to the data normalization, we assumed that the distributor’s order price through the on-site channel was 3. We still used the data in Ref. [16] for the conventional parameters in the model. These data are from our research on relevant enterprises and have engineering significance. We did not need to adjust the data. The specific complex supply chain model could be obtained by bringing the above parameters into Formulas (1)–(3).

Remark 1. 

Our modeling is derived from engineering practice. The model parameters were determined according to the actual needs of the project, so we could not modify them at will. At the same time, our specific data came from China’s home appliance enterprises. As it relates to data confidentiality requirements, we do not make specific statements.

Second, considering the impact of COVID-19, and assuming the manufacturer reduces production and supplies are insufficient, consider $l_1=0.6$. Considering the closed management of the epidemic situation and the interruption of traffic, the distributor is unable to supply goods with all its strength, $l_2=0.8$, considering customers’ normal demand for goods, $l_3=1$. Consider that the waste recyclers cannot transport the recovered goods back to the manufacturer, due to traffic suspension under the epidemic situation, $l_4=0$. Of course, the fault design principle in cybernetics was actually used here.

Third, we used Matlab’s LMI tool to solve the main theorems in this paper, iterated 29 times under the condition that LMIs (21)–(23) were satisfied, and obtained the supply quantity relationship

$$u\left(k\right)=\left[\begin{array}{cccc}-0.3896& 0.1236& -0.0026& -0.0043\\ 0.04512& -0.0592& -0.0091& 0.0854\\ -0.0359& -0.0854& -0.0094& -0.4589\\ 0.1456& -0.2564& -0.5746& -0.4752\end{array}\right]x\left(k\right).$$

(41)

Here, we calculated the interference suppression rate as $\gamma =1$.

Finally, according to the robust ${\mathcal{H}}^{\infty }$ control result solved by the closed-loop supply chain model. Referring to the general model of household appliance enterprises, we assumed that the potential demand of the market followed a normal distribution with an average value of 10. We assumed that the waste collected by the recycler followed a normal distribution with an average value of 2. In addition, for the general transportation scheme of household appliance enterprises, and the actual situation of modern logistics, there would generally be a delay of about 5 days. The simulation results are shown in Figure 2, Figure 3, Figure 4 and Figure 5.

The numerical verification was completed through the above steps. The numerical simulation results perfectly matched the theoretical results in the previous Section 3. Next, we conduct an in-depth analysis and discussion.

5. Discussion

Sensitivity analysis: The concept of sensitivity analysis comes from operational research and cybernetics, and different disciplines have different concepts. In supply chain management, sensitivity analysis quantifies the response relationship between influence variables and response variables. It helps enterprises identify and capture many factors that may affect enterprise performance and then provide a basis for enterprise decision-making. Therefore, sensitivity analysis is a very important concept in operational research, cybernetics, and public management [24]. However, the research focus of this paper was “Hierarchical Scheduling and Production Planning of Complex Supply Chain System with Dual-Channel via Cybernetics”. The key method was cybernetics, and the mathematical tool was linear matrix inequality. The solution was an optimal value. Our modeling came from engineering practice. The model parameters were determined according to the actual needs of the project, so we could not modify them at will. The final scheduling scheme was calculated by using linear matrix inequality, so we could not adjust its specific value at will. Therefore, sensitivity analysis was no longer needed in this paper.

Calculation and solution: The solution of the results of this paper was based on the LMI toolbox of Matlab. No matter how the parameters changed, we could use the LMI toolbox to solve them, which ensured the universality and practicability of the theoretical results in this paper. When the LMI toolbox iteration stopped, we obtained a satisfactory scheduling scheme if the result indicated was feasible. At the same time, this greatly reduced the human computing burden.

Simulation results: It can be seen from the four pictures that the final inventories of manufacturers, distributors, consumers, and scrap recyclers maintained dynamic stability under the effect of the supply scheduling scheme designed in this paper. It can be seen that the numerical simulation results of this paper perfectly matched the theoretical analysis results.

Generality: Through the theoretical analysis and numerical analysis in this paper, we see that the dual-channel supply chain network proposed in this paper has strong robustness. The results of this paper are general. As long as relevant enterprises meet our theoretical design requirements, relevant enterprises can manage their businesses according to the principles and mechanisms presented in this paper. There is no need to restrict the scale of enterprises, the way of participation of enterprises, and the cooperation mode of enterprises. In the face of an increasingly complex international trade environment, the theoretical results proposed in this paper have great application.

Advantages: Medium-sized manufacturers or distributors, having limited financial and material resources, are extremely vulnerable compared with large enterprises in the whole supply chain process. The dual-channel supply chain model built in this paper can greatly enhance the robustness of the dynamic supply chain. Even in the face of turbulence in the market environment, medium-sized enterprises can also be guaranteed to have strong a self-repair ability.

Limitation: The limitation of this paper is that the model we built is linear. However, with increasingly diversified economic development, it is difficult for linear models to describe the dynamic and steady-state characteristics of the current dynamic supply chain. In the future, it is necessary to establish nonlinear models to meet the needs of social and economic development.

6. Conclusions

This paper establishes a dynamic supply chain network model with dual channels, including supply interruption; that is, distributors at all levels delivering goods, and direct delivery of goods through e-commerce channels. At the same time, this paper gives the management mechanism of this kind of complex supply chain. The methods used in this paper are fault-tolerant control and ${\mathcal{H}}^{\infty }$ control. The main innovation of this paper is to construct a complex supply chain system with dual channels, and hierarchical scheduling and production planning schemes when supply failure occurs in such a complex supply chain system with dual channels. The simulation results are satisfactory and show the reliability and effectiveness of the results in a supply chain system with dual channels. The limitation of the work in this paper is that the time delay is relatively singular. We will establish a new supply chain model to deal with the increasingly complex international trade environment in the future and provide its stability analysis method through advanced cybernetics.