## 3 Materials and Methodologies
### Nonradial Super Efficiency Model (Super-SBM)  
[PERSON] [8] introduced the non-radial slacks-based measure (SBM) of efficiency in 2001. This model is based on a non-radial estimation method, which also considers the difference between input and output items (slacks), and the estimated efficiency value is between 0 and 1. When the efficiency score is equal to 1, it means that this decision-making unit is on the frontier and is without slacks in input or output. The SBM model deals with n DUMs by the input (\(m\)) and output (\(s\)) with matrices \(\mathrm{X}=\left(X_{ij}\right)\in R^{m\times n}\) and \(\mathrm{Y}=\left(Y_{ij}\right)\in R^{s\times n}\). First, let \(\lambda\) be a non-negative vector in \(R^{n}\). Second, the vectors \(s^{-}\in R^{m}\) and \(s^{+}\in R^{s}\) represent \begin{table}
\begin{tabular}{c c c} \hline \hline
**Author(s)** & **Input** & **Output** \\ \hline \multirow{3}{*}{[PERSON] [24]} & total assets; operating & \\  & expense; administrative & total revenue; net sales \\  & expenses; inventory & \\ \multirow{3}{*}{[PERSON] et al. [25]} & fixed assets; employees; R\&D \\  & expense; cost of sales & net sales; market value \\ \multirow{3}{*}{[PERSON] et al. [26]} & employees; R\&D expense; \\  & patents; R\&D employees; \\ \multirow{3}{*}{[PERSON] and [PERSON] [27]} & operating cost & capital stocks \\ \multirow{3}{*}{[PERSON] and [PERSON] [28]} & assets; employees; equity & revenue; profit; EPS \\  & fixed assets; cost of goods & net income; revenue; \\ \multirow{3}{*}{[PERSON] and [PERSON] [28]} & sold; R\&D expense; \\  & operating expense & retained earnings \\ \hline \hline \end{tabular}
\end{table} 
Table 2: List of input and output factors from previous studies. \begin{table}
\begin{tabular}{c c} \hline \hline
**DMUS** & **Name of Companies** \\ \hline DMU1 & ASE \\ DMU2 & Amkor \\ DMU3 & Powertech Technology \\ DMU4 & Tongfu Microelectronics \\ DMU5 & Tianshuihuatian Microelectronics \\ DMU6 & UTAC \\ DMU7 & King Yuan Electronics \\ DMU8 & Chipbond Technology \\ DMU9 & Chipmos Technologies \\ DMU10 & Orient IC Electronics \\ DMU11 & SFA Semicon \\ DMU12 & AOI Electronics \\ DMU13 & Greatek Elec \\ DMU14 & Unisem Berhad \\ DMU15 & Sigurd Microelectronics \\ DMU16 & Formosa Advanced Technologies \\ DMU17 & Hana Micron \\ DMU18 & Walton Advanced Engineering \\ DMU19 & Ardentec \\ DMU20 & Tong Hsing \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of decision making units (DMUs). the input excess and insufficient input and output factor, respectively. This model provides variable returns on the scale. The program equation (Equation (1)) and restriction equation (Equation (2)) are presented below:  
\[\text{min}\rho=\frac{1-\frac{1}{m}\sum_{i=1}^{m}s_{i}^{-}/x_{i0}}{1+\frac{1}{s} \sum_{r=1}^{s}s_{i}^{+}/y_{i0}} \tag{1}\]  
subject to  
\[ \begin{array}{l}x_{0}=X\lambda+s^{-},\\ y_{0}=Y\lambda-s^{+},\\ \sum_{j=1}^{m}\lambda_{j}=1\\ \lambda\geq 0,\ \ s^{-}\geq 0,\ \ s^{+}\geq 0\end{array}  \tag{2}\]  
The variables \(s^{+}\) and \(s^{-}\) measure the distance between the input \(X\lambda\) and output \(Y\lambda\) of the virtual unit and the evaluated unit. The numerator and denominator in the objective function measure the average distance between the input and output and the efficiency threshold. In order solve the problem of the SBM efficiency value of multiple decision-making units being the same as 1, [PERSON] [8] proposed a super-efficiency model for DMU efficiency ranking that is based on the SBM model. Let the optimal solution of SBM be (\(p^{*},\lambda^{n},s^{-*},s^{+*}\)). If \(p^{*}\), then DMU (\(X_{0},Y_{0}\)) is SBM-efficient. The condition is equivalent to \(s^{-*}=0\) and \(s^{+*}=0\) in any optimal solution, and there is no excessive input or output shortage. The program equation (Equation (3)) and restriction equation (Equation (4)) are presented below:  
\[\text{min}\delta=\frac{\frac{1}{m}\sum_{i=1}^{m}\overline{x}_{i}/x_{i0}}{\frac {1}{s}\sum_{r=1}^{s}\overline{y}_{r}/y_{r0}} \tag{3}\]  
subject to  
\[