# Hierarchical Scheduling of Complex Dynamic Supply Chain
## 4 Numerical Verification  
Combined with the survey and statistics of the operation of a household appliance enterprise in China, the simulation calculation is carried out. The relevant cost matrix parameters are set as follows: The order price of the distributor through the field channel is 3. The ordering cost through e-commerce is 3. The possibility for customers to directly purchase household appliances from household appliance manufacturers through e-commerce channels is 0.4. The remanufacturing rate of recycled products is 0.6, the probability of waste products being directly discarded by customers is 0.1, and the inventory holding cost is \(c_{h}=0.2\). Discard processing cost \(c_{o}=0.1,c_{u}=0.1\), remanufacturing cost of recycledwaste \(c_{r}=1.5\), new material manufacturing cost \(c_{n}=1\) (unit: 1000 yuan/set). Here, Let \(h=2\) and \(l=1\). The system matrice of the complex dynamic supply chain (5) are set as follows:  
\[A=\left[\begin{array}{cccc}1&0&0&0.8\\ 0&1&0&0\\ 0&0&0.6&0\\ 0&0&0&0.2\\ \end{array}\right],B=\left[\begin{array}{cccc}1&-1&-1&0\\ 0&1&1&0\\ 0&0&0&0\\ 0&0&0&-1\\ \end{array}\right],B_{1}=\left[\begin{array}{cccc}1&-1&0&0\\ 0&1&0&0\\ 0&0&0&0\\ 0&0&0&0\\ \end{array}\right], \tag{30}\] \[A_{1}=\left[\begin{array}{cccc}1&0&0&0.7\\ 0&1&0&0\\ 0&0&0.9&0\\ 0&0&0&0.3\\ \end{array}\right],A_{2}=\left[\begin{array}{cccc}1&0&0&0.6\\ 0&1&0&0\\ 0&0&0.8&0\\ 0&0&0&0.4\\ \end{array}\right].\]  
In order to verify the effectiveness of algorithm in this paper, we use MATLAB software to calculate the specific parameters of supply, and use SIMULINK software to draw the dynamic response curve of inventory of the closed-loop supply chain (8). The Feasp solver in the LMI toolbox of Matlab software is used to solve Theorem 2. The calculation result of LMI (22) can be obtained through 29 iterations. Finally, the supply relationship is written as follows:  
\[u(k)=\left[\begin{array}{cccc}-1.9624&0.9494&-0.2395&1.4785\\ -0.2486&-1.7894&-0.4857&-0.8954\\ -0.6253&-0.4571&-1.9758&-0.5251\\ 0.1241&0.4525&0.4171&-1.4543\\ \end{array}\right]x(k). \tag{31}\]  
At this time, the inhibition rate of closed-loop supply chain to uncertain disturbances and factors is \(\gamma=1\).  
The potential demand of customers is subject to a normal distribution with a mean value of 10. The mean value is normal distribution of 2. According to the transportation situation of the household appliance enterprise, the delay is set to be 5 days. The control result of closed-loop supply chain under the action of robust \(\mathcal{H}^{\infty}\) control can be obtained. The simulation results are shown in the follows Figures 2-5.  
the production and distributor ordering of household appliances enterprises and the waste control of recyclers will become very stable. The inventory status of the closed-loop supply chain also tends to be stable, and the total operating cost of the closed-loop supply chain also eventually reaches stability. That is, the production, distribution and waste control \(u(k)\) based on the robust \(\mathcal{H}^{\infty}\) control strategy can suppress normal demand and recovery disturbances, sinusoidal demand and recovery disturbances, and uncertain parameters, make closed-loop supply chain operate stably. The experimental data in this paper come from Chinese household appliance enterprises, but for the requirement of data confidentiality, we extract part of the data in proportion to do numerical simulation, which shows the effectiveness and feasibility of this work.  
This paper uses the original data of household appliance enterprises, which is similar to Ref. [16]. The parameter matrice of dynamic supply chain are shown in formula (30). We give the optimal numerical results through LMI toolbox of MATLAB. The inventories of manufacturer and distributors at all levels are shown in Figures 2-5. From the simulation results, we can see that the inventories will stabilize at our expected level on the 8 th day. The numerical examples perfectly match the feasibility and effectiveness of our proposed algorithm.