# Hierarchical Scheduling of Complex Dynamic Supply Chain
## 2 Model of Complex Dynamic Supply Chain
Based on the above discussion and the analysis in Figure 1, we can see that considering lead time compression, the inventory state model of a complex dynamic supply chain composed of manufacturers, retailers and customers is as follows:  
\[x(k+1)= Ax(k)+\sum_{i=1}^{h}A_{i}x(k-i)+Bu(k)+\sum_{i=1}^{l}B_{i}u(k-i)+Dw(k), \tag{3}\]  
where \(\omega(k)\) represents an external input uncertainty variable, e.g., the market customer demand at \(k\), the amount of product recovered at the time of \(k\). Therefore, in order to minimize the total cost of complex dynamic supply chain, based on the comprehensive consideration of different inventory levels of manufacturers and retailers, as well as compression costs and out of stock costs, this paper formulates the following cost optimization strategies: when the inventory is less than the safety stock, if the lead time compression cost is greater than the out of stock cost, the lead time will not be compressed; otherwise, the lead time is compressed. Based on the above inventory model and the actual operation process of complex dynamic supply chain, a complex dynamic supply chain's total cost model considering inventory cost, production cost, compression cost, ordering cost and out of stock cost is constructed as follows:  
\[z(k)= Cx(k)+\sum_{i=1}^{h}C_{i}x(k-i)+D_{1}u(k)+\sum_{i=1}^{l}D_{1i}u(k-i)+D_{2 }w(k), \tag{4}\]  
where \(z(k)\) is the total operation cost of complex dynamic supply chain; \(C_{i}\) is the unit inventory cost of manufacturers and retailers in complex dynamic supply chain. \(D_{1}\), \(D_{1i}\), and \(D_{2}\) are the transportation cost of all levels and types in complex dynamic supply chain. The theme of this paper is about green manufacturing. This paper mainly studies the production of manufacturers and the supply of distributors, so this paper refers to production planning and scheduling. Since the COVID-I9 has had a great impact on the economy since 2019, in order to ensure the continuous supply of goods, we have arranged the order volume and inventory in advance through the model in Formulas (1)-(4), so as to ensure the dynamic stability of the closed-loop supply chain under the impact similar to the COVID-I9 pandemic. Because the above model is based on different inventory levels. In addition, manufacturers, the core enterprises in complex dynamic supply chain, will adopt corresponding production strategies according to their own different inventory levels. Therefore, different systems will be formed under different inventory levels, which are uniformly expressed in the form of matrix in this paper as follows:  
\[ \begin{split} x(k+1)=& Ax(k)+\sum_{i=1}^{h}A_{i}x(k-i )+Bu(k)+\sum_{i=1}^{l}B_{i}u(k-i)+Dw(k),\\ z(k)=& Cx(k)+\sum_{i=1}^{h}C_{i}x(k-i)+D_{1}u(k)+ \sum_{i=1}^{l}D_{1i}u(k-i)+D_{2}w(k),\end{split}  \tag{5}\]  
where \(x(k),u(k),z(k),w(k)\) are respectively the state vector, control input vector, controlled output vector and interference input vector with bounded energy of the complex dynamic supply chain. The correlation matrice are known well dimensional constant matrice. \(h\) and \(l\) are positive integers, representing the maximum delay steps of state and control inputs, respectively. Therefore, we establish a dynamic supply chain model with \((n-1)\)-level distributors/retail dealers and \(l\)-level lead time. **Remark 3**. : _When analyzing the quantitative relationship between the compression cost of lead-time and the compression amount, this paper adopts the calculation method of piecewise accumulation, that is, the lead-time is divided into 1 mutually independent parts, such as the order quantity when \(l=1\), the order processing quantity when \(l=2\), and so on. By calculating the compression cost of each independent part in turn and accumulating, we can establish the quantitative relationship model in this paper._  
Generally, the initial conditions for complex dynamic supply chain (5) are  
\[x(k)= \begin{cases}0,&k<0,\\ x_{0},&k=0.\end{cases}