## 2 Research Methods  
### Cross-Efficiency Model  
In order to solve the problems of the self-assessment mode of the traditional DEA evaluation model and the inability to fully order the decision units, [PERSON] et al. proposed a cross-efficiency evaluation method that combines self-assessment and mutual assessment [23], which first utilizes the traditional DEA model to obtain the decision unit's own efficiency and the optimal weight and then combines them with a set of the optimal weights of all the decision units to evaluate the cross-efficiency of each decision unit. The cross-efficiency based on the CCR model is taken as an example here. Suppose that there are n homogeneous decision-making units \(DMU_{d}(d=1,\ldots,n)\), each with I inputs and M outputs, \(x_{id}\) denotes the ith input of the \(DMU_{d}\), and \(y_{md}\) denotes the mth output of the \(DMU_{d}\): \(u_{id}\) is the weight of the ith input, and \(v_{md}\) is the weight of the mth output. Assuming that the self-assessed efficiency value of the \(DMU_{d}\) is \(\theta_{dd}\), the following formulation is available:  
\[ \begin{array}{ll}\text{max}&\theta_{dd}=\frac{\sum\limits_{m=1}^{M}v_{md}y_ {md}}{\sum\limits_{i=1}^{I}u_{id}x_{id}}\\ s.t.&\sum\limits_{m=1}^{M}v_{md}y_{mj}\\ &\sum\limits_{i=1}^{I}u_{id}x_{ij}\\ &u_{id}\geq 0,&i=1,\ldots,I\\ &v_{md}\geq 0,&m=1,\ldots,M\end{array}  \tag{1}\]  
Assuming that the optimal weights of the above model are \(u_{id}^{*}\) and \(v_{md}^{*}\), the cross-efficiency (\(\theta_{dj}\)) of decision unit \(DMU_{j}\) with respect to \(DMU_{d}\) is  
\[\theta_{dj}=\frac{\sum\limits_{m=1}^{M}v_{md}^{*}y_{mj}}{\sum\limits_{i=1}^{I} \frac{u_{id}^{*}x_{ij}}{\sum\limits_{i=1}^{I}u_{id}^{*}x_{ij}}},\ \ \ d,j=1,2,\ldots,n \tag{2}\]  
After each \(DMU_{d}\) obtains a set of optimal weights through the traditional DEA model, all the weights are used to evaluate the efficiency of the decision unit (\(DMU_{j}\)) to obtain the cross-efficiency matrix (\(\theta\)). \[\theta= \begin{bmatrix}\theta_{11}&\theta_{12}&\ldots&\theta_{1n}\\ \theta_{21}&\theta_{22}&\ldots&\theta_{2n}\\ \ldots&\ldots&\ldots&\ldots\\ \theta_{n1}&\theta_{n2}&\ldots&\theta_{m}\end{bmatrix}  \tag{3}\]  
In the matrix, the dth row (\(DMU_{d}\)) evaluates the efficiency of all the decision units separately; the jth column represents the efficiency evaluation of the \(DMU_{j}\) by all the decision units, so the cross-efficiency value (\(\theta_{j}\)) of the \(DMU_{j}\) can be taken as the arithmetic mean of all the efficiencies in the jth column. \[\theta_{j}=\frac{1}{n}\sum\limits_{d=1}^{n}\theta_{dj} \tag{4}\]  
Since most weights obtained from the solution using the traditional DEA model may not be unique, this may lead to non-unique cross-efficiency values. To solve this problem, [PERSON] et al. introduced secondary objectives and proposed two strategic secondary objective models: adversarial type and benevolent type [24]. The linear programming form of the adversarial cross-efficiency model is as follows:  
\[ \begin{array}{ll}\min&\sum\limits_{m=1}^{M}\left(\xi_{md}\sum\limits_{j=1, j\
eq d}^{n}y_{mj}\right)\\ s.t.&\sum\limits_{i=1}^{I}\left(\mu_{id}\sum\limits_{j=1,j\
eq d}^{n}x_{ij} \right)=1,\\ &\sum\limits_{m=1}^{M}\xi_{md}y_{md}-\theta_{id}^{*}\sum\limits_{i=1}^{I}\mu_ {id}x_{id}=0,\\ &\sum\limits_{m=1}^{M}\xi_{md}y_{mj}-\sum\limits_{i=1}^{I}\mu_{id}x_{ij}\leq 0,\qquad j=1,\ldots,n;j\
eq d;\\ &\xi_{md}y_{md}-\delta\geq 0,\qquad\qquad\qquad\qquad\quad m=1,\ldots,M\\ &\mu_{id}\geq 0,\qquad\qquad\qquad\qquad\quad\quad\quad\quad\quad\quad i=1, \ldots,I\\ &\delta\geq 0\end{array}  \tag{5}\]  
The linear programming form of the benevolent cross-efficiency model is as follows:  
\[