# Hierarchical Scheduling of Complex Dynamic Supply Chain
## 2 Model of Complex Dynamic Supply Chain  
This section establishes the dynamic model of complex dynamic supply chain with multi-distributors or retail dealers, multi-delay inventory and multi-lead time. This paper establishes the dynamic model of complex dynamic supply chain based on the following information: (1) ensure information sharing, that is, the node enterprises in complex dynamic supply chain share product information, inventory information, production information, order status information, sales and forecast information through information technology; (2) ensure the manufacturer's core position, that is, the lead time considered is the manufacturer's production lead time; (3) shortage is allowed and only occurs within the lead time. Based on above principle, the network topology of complex dynamic supply chain, which is with multi-distributors/retail dealers, multi-delay inventory and multi-lead time, model composed of manufacturers, retailers and customers built in this paper is shown in Figure 1. Let \(x_{0}(k)\) represents the manufacturer's actual inventory at the time of \(k\), \(x_{i}(k)\) represents the \(i\)-th distributor's actual inventory at the time of \(k\). \(u_{0}(k)\) means the product produced by the manufacturer at the time of \(k\), \(u_{i}(k)\) represents the order quantity ordered by the \(i\)-th distributor at the time of \(k\), \(\omega_{i}(k)\) represents the market customer demand at \(k\). Let \(x(k)=[x_{0}(k),x_{1}(k),x_{2}(k),\cdots,x_{n-1}(k)]^{T}\) and \(u(k)=[u_{0}(k),u_{1}(k),u_{2}(k),\cdots,u_{n-1}(k)]^{T}\). We use \(x_{0}(k)\) to represent the manufacturer's actual inventory at the time of \(k\). Use \(x_{i}(k)\) to represent actual inventory of all levels distributors at the time of \(k\). The information flow of goods is shown by the arrows in the vertical direction in Figure 1.  
Various coefficients in complex dynamic supply chain are uniformly expressed as elements of the correlation matrix. By analyzing the quantitative relationship between the lead time compression cost and the compression amount, and adopting the calculation method of piecewise accumulation (that is, the lead time is divided into \(l\) mutually independent parts, such as order processing time, etc., and the compression cost of each independent part is calculated in turn, and the cumulative sum is performed), the quantitative relationship model is built as follows:  
\[\sum_{i=1}^{l}B_{i}u(k-i), \tag{1}\]  
Figure 1: The illustration of complex dynamic supply chain. where \(B_{i}\) is the coefficient matrix of lead time term; \(u(k-i)\) is the control input vector at the time of \(k-i\). The time-delay phenomenon in complex dynamic supply chain is the main reason for the uncertainty and instability of the system. Once the external and internal ordering lead time of complex dynamic supply chain is too long, each subsystem of complex dynamic supply chain will correspondingly increase the inventory to prevent the occurrence of stock outs, and thus can cause inventory time-delay. At the same time, the complex dynamic supply chain also has time delays in production, collaborative product supply and other logistics processes, which make complex dynamic supply chain less responsive to demand fluctuations. The quantitative relationship model of multi-level delay inventory of manufacturers, distributors and retail dealers is as follows:  
\[\sum_{i=1}^{h}A_{i}x(k-i), \tag{2}\]  
where \(h\) represents delay information; \(A_{i}\) is the coefficient matrix of time-delay term; \(x(k-i)\) is the state vector at the time of \(k-i\).