## 2 Problem Description and Modeling
### Mathematical Model  
Taking unit \(u_{1}\) as an example, when rotating from \(u_{1}\) to other units, the rotation number is \(c-1\), and the rotation process is shown in Figure 1.  
The decision matrix is divided into two parts. Let us introduce the 0-1 variable \(x_{i,j}(i=1,2,\ldots,c-1,j=S_{1,1},S_{1,2},\ldots,S_{1,N_{1}},c\geq 2)\), where \(S_{1}\) is the equipment number sequence of \(u_{1}\), \(N_{1}\) is the equipment quantity of \(u_{1}\). By analogy, the 0-1 variable is also introduced for other units to build a rotation decision matrix. The selected equipment is moved to the corresponding unit, and the equipment numbers of the current unit and the corresponding unit are updated. \(S^{\prime}\) is the updated equipment number sequence of each unit and \(N^{\prime}\) is the updated equipment quantity sequence of each unit. \[x_{i,j}=\left\{ \begin{array}{c}0\ S_{i}\ is\ not\ selected\ in\ j_{th}\ rotation\\ 1\ S_{i}\ is\ selected\ in\ j_{th}\ rotation\end{array} \right.\]  
Let us introduce the 0-1 variable \(y_{i,j}=(i=1,2,\ldots,a*U_{1};j=1,2,\ldots,n)\), where \(U_{1}\) is the task quantity of \(u_{1}\). By analogy, the 0-1 variable is also used for other units to construct the task decision matrix and update the working life of the selected equipment. \[y_{i,j}=\left\{ \begin{array}{c}0\ S_{i}^{\prime}\ is\ not\ selected\ in\ j_{th}\ task\\ 1\ S_{i}^{\prime}\ is\ selected\ in\ j_{th}\ task\end{array} \right.\]  
Next, let us establish the rotation decision matrix, \(X\), and task decision matrix, \(Y\), of \(u_{1}\), as shown in Tables 1 and 2. \[X=\left( \begin{array}{ccc}x_{1,1}&\ldots&x_{1,N_{1}}\\ \vdots&\ddots&\vdots\\ x_{c-1,1}&\ldots&x_{c-1,N_{1}}\end{array} \right)\]  
\[Y=\left( \begin{array}{ccc}y_{1,1}&\ldots&y_{1,N_{1}^{\prime}}\\ \vdots&\ddots&\vdots\\ y_{as\ t_{1},1}&\ldots&y_{as\ t_{1},N_{1}^{\prime}}\end{array} \right)\]  
Figure 1: Rotation process. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Rotation Time**Equipment Number** & \(S_{\mathbf{1,1}}\) & \(S_{\mathbf{1,2}}\) & \(\cdots\) & \(S_{\mathbf{1,N_{1}-1}}\) & \(S_{\mathbf{1,N_{1}}}\) \\ \hline
1 & \(x_{1,1}\) & \(x_{1,2}\) & \(\cdots\) & \(x_{1,N_{1}-1}\) & \(x_{1,N_{1}}\) \\
2 & \(x_{2,1}\) & \(x_{2,2}\) & \(\cdots\) & \(x_{2,N_{1}-1}\) & \(x_{2,N_{1}}\) \\ \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\cdots\) & \(\vdots\) & \(\vdots\) \\ \(c-2\) & \(x_{c-2,1}\) & \(x_{c-2,2}\) & \(\cdots\) & \(x_{c-2,N_{1}-1}\) & \(x_{c-2,N_{1}}\) \\ \(c-1\) & \(x_{c-1,1}\) & \(x_{c-1,2}\) & \(\cdots\) & \(x_{c-1,N_{1}-1}\) & \(x_{c-1,N_{1}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Construction rotation decision matrix \(X\) of \(u_{1}\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Task**Equipment Number** & \(S_{\mathbf{u1}}^{\prime}\) & \(S_{\mathbf{1,2}}^{\prime}\) & \(\cdots\) & \(S_{\mathbf{1,N_{1}-1}}^{\prime}\) & \(S_{\mathbf{1,N_{1}}}^{\prime}\) \\ \hline
1 & \(y_{1,1}\) & \(y_{1,2}\) & \(\cdots\) & \(y_{1,N_{1}^{\prime}-1}\) & \(y_{1,N_{1}^{\prime}}\) \\
2 & \(y_{2,1}\) & \(y_{2,2}\) & \(\cdots\) & \(y_{2,N_{1}^{\prime}-1}\) & \(y_{2,N_{1}^{\prime}}\) \\ \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\cdots\) & \(\vdots\) & \(\vdots\) \\ \(a*U_{1}-1\) & \(y_{a*U_{1}-1,1}\) & \(y_{a*U_{1}-1,2}\) & \(\cdots\) & \(y_{a*U_{1}-1,N_{1}^{\prime}-1}\) & \(y_{a*U_{1}-1,N_{1}^{\prime}}\) \\ \(a*U_{1}\) & \(y_{a*U_{1},1}\) & \(y_{a*U_{1},2}\) & \(\cdots\) & \(y_{a*U_{1},N_{1}^{\prime}-1}\) & \(y_{a*U_{1},N_{1}^{\prime}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Construction task decision matrix \(Y\) of \(u_{1}\). 2. The average life-matching function, \(F_{2}\). The life-matching function, \(Z_{1}\), of \(u_{1}\) is shown in Formula (2) \[Z_{1}= \begin{pmatrix}1-\frac{\sum\limits_{i=1}^{N^{\prime}_{1}}H_{1,i}}{N^{ \prime}_{1}+V_{1}}\\ \\ \end{pmatrix} \] (2) \[H_{1,i}=\left\{