## 3 Methods and Materials
### Data Entvelopment Analysis (DEA)  
A Data Entvelopment Analysis (DEA) treats each evaluation object as an independent decision-making unit, calculating efficiency values based on the existing input-output data of the decision units without altering the original data [46]. A major advantage of a Data Entvelopment Analysis (DEA) is that it does not require a predefined functional relationship between the inputs and outputs, nor does it necessitate the a priori setting of parameters or the assumption of weights, thereby reducing the influence of subjective judgments [47]. Additionally, this method is insensitive to the measurement units of input-output indicators, not requiring the data to be converted into specific units or to conform to a particular functional form prior to analysis. Basic Principle: Assume there are \(n\) evaluation objects, each regarded as an independent decision-making unit. Thus, there are \(n\) decision units in total. For each decision unit, they involve \(m\) types of \"inputs\", representing various amounts of resource inputs. Additionally, each decision unit generates \(s\) types of \"outputs\", representing the results or outputs following the resource inputs. \[ \begin{pmatrix}V_{1}&\cdots&1\\ \vdots&\ddots&\vdots\\ V_{m}&\cdots&m\end{pmatrix} \rightarrow \begin{pmatrix}X_{11}&\cdots X\cdots&X_{1n }\\ \vdots&\ddots&\vdots\\ X_{m1}&\cdots X_{mj}&X_{mn}\end{pmatrix}  \tag{1}\] \[ \begin{pmatrix}Y_{11}&\cdots Y\cdots&Y_{1n}\\ \vdots&\ddots&\vdots\\ Y_{m1}&\cdots Y_{mj}&Y_{mm}\end{pmatrix} \rightarrow \begin{pmatrix}V_{1}&\cdots &u_{1}\\ \vdots&\ddots&\vdots\\ s&\cdots&u_{s}\end{pmatrix} \]  
Equation (1) represents the input-output matrix of the decision-making units. The input amount of the \(i\)th type corresponds to the \(j\)th decision-making unit (DMU) (X\({}_{ij}>0\)). The output amount of the \(r\)th type corresponds to the \(j\)th decision-making unit (DMU) (X\({}_{ij}>0\)). The specific weight of the \(i\)th input, and \(\mathrm{U}_{r}\) is the weight of the \(r\)th output. Where \(i=\)1, 2,..., \(m\). \(j=\) 1, 2,..., \(n\). \(r=\) 1, 2,..., \(s\). (X\({}_{u}\), Y\({}_{u}\)) represents the known inputs. According to academic consensus, \(V_{i}\) and \(\mathrm{U}_{r}\) are variables, representing a set of corresponding weight coefficients. \[v=(v_{1},\,v_{2},\,\cdots,\,v_{n})^{T};\,u=(u_{1},\,u_{2},\,\cdots,\,u_{r})^{T} \tag{2}\]  
Efficiency evaluation index (total inputs vs. total outputs):  
\[u^{1}=\frac{ \begin{pmatrix}L-I\\ \sum\limits_{i=\infty}^{L-I}\pi^{L}\lambda^{L}\end{pmatrix} }{ \begin{pmatrix}L -I\\ \sum\limits_{i=\infty}^{L-I}\Lambda^{j}X^{ij}\end{pmatrix} } \tag{3}\]  
where each decision-making unit has a corresponding efficiency evaluation index \(j=\) 1, 2,..., \(n\). Select the appropriate weight coefficients, \(v\) and \(u\), to satisfy \(h_{j}<1\), for \(j=\) 1, 2,..., \(n\). By introducing an input-oriented efficiency measure, we constructed a model that minimizes the input consumption while maintaining the output levels. Here, \(\theta\) represents the efficiency score of the decision-making unit (DMU), with values between 0 and 1, where 1 indicates full efficiency. Based on the above analysis, a dual linear programming model was constructed. The duality theory in linear programming is an effective analytical tool. By constructing a dual model, it is possible to explore and analyze the problem more deeply from both theoretical and economic perspectives. The dual programming model \((D_{c2r})\) is as follows:  
\[\left\{ \begin{array}{l}Min\theta\\ s.t.\sum\limits_{j=1}^{n}\left(X_{i}\lambda_{j}\right)\leq\theta X_{0}\\ \sum\limits_{j=1}^{n}\left(Y_{i}\lambda_{j}\right)\leq Y_{0}\\ \lambda_{j}\geq 0,\,\,j=1,\,2,\,\ldots,\,n\end{array} \right. \tag{4}\]  
Since the model uses inequalities, the efficiency determined might be a weak efficiency. Simultaneously introducing positive and negative deviation variables ensures the duality of the linear programming, aiming to determine whether there is input excess or output deficiency, as shown in Equation (5).