# Switching Control of Closed-Loop Supply Chain Systems with Markov Jump Parameters
## 2 Closed-Loop Supply Chain Switching System Modeling  
This paper considers the control problem of a closed-loop supply chain system based on remanufacturing. The model assumes that a manufacturer produces a product, and at the same time, the manufacturer retrieves that product from the market for remanufacturing. This paper assumes that the quality standard of the remanufactured goods can meet the standard of the new products. This paper mainly considers the problem of inventory management. Manufactured and remanufactured goods are stored in the usable goods warehouse, and used goods recovered from the market are stored in the recycled goods warehouse. \(x_{1}(k)\) and \(x_{2}(k)\) represent the inventory levels of the available commodity warehouse and the recycled commodity warehouse and are the state vector of the system. \(u_{1}(k)\) and \(u_{2}(k)\) represent the manufacturing rate of the manufacturing equipment and the recycling rate of the used goods at the moment, respectively. For the system model, we have the following assumptions. Assumption 1: _Assume that all products are recyclable and that the manufacturer is the only determinant of the amount of product to be recycled, i.e., that there are sufficient products on the market to meet the demand for recycling and that the manufacturer only needs to recycle the amount of product it needs._  
Assumption 2: _The market demand \(d(k)\) is the sum of the constant \(\bar{d}\) and the perturbation \(\omega(k)\), i.e.,_  
\[d(k)=\bar{d}+\omega(k)\]  
**Note 1. ** For the sake of generality, it is assumed that the perturbations \(\omega(k)\) follow a normal distribution or are sinusoidal functions. Assumption 3: _Recycled products are disposed of in two ways, remanufacturing and disposal, so that \(\alpha\)\((0\leq\alpha\leq 1)\) represents the remanufacturing rate, \(\beta\)\((0\leq\beta\leq 1)\) the disposal rate, and \(\alpha\) and \(\beta\) are uncertain parameters; \(0<\alpha+\beta\leq 1\) is assumed in this paper._  
**Assumption 4**. : _The value of the products in the warehouse decreases over time. \(\rho_{1}\) and \(\rho_{2}\) represent the decay rates of the available and recovered warehouses, respectively._  
The closed-loop supply chain system considered in this paper have inventory levels as the state variable. First, for the available commodity warehouse, when \(0<x_{1}(k)<c_{\max}^{1}\), the system is given by the following equation  
\[x_{1}(k+1)=(1-\rho_{1})x_{1}(k)+\alpha x_{2}(k)+u_{1}(k)-d(k) \tag{1}\]  
where \(c_{\max}^{1}\) is the maximum capacity of the available warehouse. It should be noted that when \(x_{1}(k)\leq 0\) indicates that there are no items in the warehouse that can be used to meet the order demand, which also results in a stock-out phenomenon. At this point, the production-inventory model can be described as  
\[x_{1}(k+1)=x_{1}(k)+\alpha x_{2}(k)+u_{1}(k)-d(k) \tag{2}\]  
Similarly, for the recycled goods inventory, let \(c_{\max}^{2}\) be the maximum capacity of the recycled goods warehouse; when \(0<x_{2}(k)<c_{\max}^{2}\), the system can be represented as  
\[x_{2}(k+1)=(1-\rho_{2})x_{2}(k)-\alpha x_{2}(k)-\beta x_{2}(k)+u_{2}(k) \tag{3}\]  
When \(x_{2}(k)\leq 0\) is used, the system can be described as  
\[x_{2}(k+1)=x_{2}(k)+u_{2}(k) \tag{4}\]  
Let \(y(k)\) be the operating cost of the system, then  
\[y(k)=(C+\Delta C)x(k) \tag{5}\]  
where \(C=\left[ \begin{array}{cc}C_{h1}&0\\ 0&C_{h2}\end{array} \right]\), \(\Delta C=\left[ \begin{array}{cc}0&c_{r}\alpha+c_{0}\beta\end{array} \right]\), the parameters are as follows: \(c_{h1}\) is the cost of useful inventory, \(c_{h2}\) is the cost of remanufactured inventory, \(c_{r}\) is the cost of remanufactured product, and \(c_{0}\) is the cost of discard (all deterministic parameters). **Note 2**. : The total closed-loop supply chain cost is determined by the cost of useful inventory, the cost of remanufactured inventory, the cost of remanufactured products, and the cost of waste disposal.