# Change Control Design in Product Supply Chain System Based on Radial Basis Function Neural Network
## 2 Product and Supply Chain System Modeling  
Consider a multi-level product supply chain system. A schematic of this system is shown in Figure 2, where each factory can produce one product, which can be considered a node. The product from the preceding factory serves as the material for the next one, while the product from the final factory is directly placed on the market. For the last node, it is also be directly affected by market demand. For the purpose of expressing the product supply chain system dynamics, the system parameters are defined as  
\[x(t) = \left[x_{1}(t),x_{2}(t),\cdots,x_{n}(t)\right]^{T}\] \[u(t) = \left[u_{1}(t),u_{2}(t),\cdots,u_{n}(t)\right]^{T}\]  
where \(x\) indicates the inventory level of each link of the supply chain system; \(u\) represents the productivity of the facility. In order to model the product supply chain inventory system more accurately and reflect the actual situation of the supply chain inventory system more realistically, this paper puts forward the following assumptions:  
**Assumption 1**. : _In a single-chain product supply chain system, all links pick up goods only upstream and deliver them only to a specific downstream, and there is no crosstalk._  
**Assumption 2**. : _The uncertain market demand is first-order-derivable._  
**Assumption 3**. : _The product under consideration is a perishable product, and inventory self-loss occurs at each stage of the production process due to either the product itself or the storage conditions, and it is assumed that the rate of self-loss for each of these production processes in a given time period is Etc._  
**Assumption 4**. : _Due to the quality requirements in the production process, not all products produced can be used. Some products that do not meet the requirements are discarded, and there is a certain production yield rate at each stage of the production process. It is assumed that the product yield rate at each production stage is denoted as\(\rho_{1},\rho_{2}\), etc._  
According to the schematic diagram of the product supply chain inventory system in Figure 2, the mathematical model of its linear system is obtained as follows:  
\[ \begin{cases}\dot{x}_{1}(t)=\rho_{1}x_{1}(t)+b_{1}u_{1}(t)-u_{2}(t)\\ \dot{x}_{2}(t)=\rho_{2}x_{2}(t)+b_{2}u_{2}(t)-u_{3}(t)\\ \cdots\\ \dot{x}_{n}(t)=\rho_{n}x_{n}(t)+b_{n}u_{n}(t)-d(t)\end{cases}  \tag{1}\]  
Figure 2: Schematic diagram of direct sale supply chain system. A simplified linear mathematical model is derived for Equation (1) as follows:  
\[ \begin{cases}\dot{x}(t)=Ax(t)+Bu(t)+Dd(t)\\ y(t)=Cx(t)+v(t)\end{cases}  \tag{2}\]  
where \(y\) is the output of the system; \(C\) is a matrix of known constants that represent the supply chain system's operation parameters, which can reflect the actual operation of the supply chain system. The observation noise \(v(t)\) has a mean of zero and a covariance matrix \(R\), and it follows a normal distribution \(v\sim N(0,R)\), \(D=[D_{1},D_{2},\cdots,D_{n}]^{T}\) is the disturbance coefficient matrix, and  
\[x(t)= \begin{bmatrix}x_{1}(t)\\ x_{2}(t)\\ \vdots\\ x_{n}(t)\end{bmatrix} \text{, }u(t)= \begin{bmatrix}u_{1}(t)\\ u_{2}(t)\\ \vdots\\ u_{n}(t)\end{bmatrix} \text{, }\]  
\[A= \begin{bmatrix}\rho_{1}&0&\cdots&0\\ 0&\rho_{2}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&\rho_{n}\end{bmatrix} \text{, }B= \begin{bmatrix}b_{1}&-1&0&\cdots&0\\ 0&b_{2}&-1&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\cdots&b_{n}\end{bmatrix} \]  
When situations such as complex environmental changes occur in the product supply chain system, they affect the product supply chain system's parameters, specifically parameters \(\triangle A\) and \(\triangle B\). At the same time, the uncertain user demand in the supply chain introduces volatility due to unexpected impacts such as new coronavirus outbreaks, floods, and trade wars, so the dynamics of the supply chain after the change are represented as  
\[ \begin{cases}\dot{x}(t)=(A+\triangle A)x(t)+(B+\triangle B)u(t)+D\bar{d}(t)\\ y(t)=Cx(t)+v(t)\end{cases}