## 2 Research Methods
### Two-Stage Dynamic Network Cross-Efficiency Model  
Traditional DEA models are both black-box in nature (assessing the efficiency of a decision unit through external input and output data without considering the specific mechanisms of its internal processes) and static self-assessment models (not being able to assess the efficiency of a decision unit dynamically and uniquely). In addition, in practice, many production processes often contain multiple stages, and different stages may share the same input indicators, which makes the traditional DEA models have some limitations in dealing with such problems. In order to overcome these limitations, this paper proposes an innovative approach, combining neutral cross-efficiency and two-stage dynamic network modeling, which can effectively deal with two-stage production structures with shared inputs. The specific model is as follows:  
\[ \begin{array}{ll}\max W=&\min\limits_{\begin{array}{c}a\in\{1,\ldots,A \}\\ r\in\{1,\ldots,R\}\\ m\in\{1,\ldots,M\}\\ s\in\{1,\ldots,S\}\end{array}}&\frac{\sum\limits_{t=1}^{T}\left(\varphi_{ao}z_{ao }^{t}+\phi_{ro}^{1}k_{ro}^{1t}+v_{mo}y_{mo}^{t}+\phi_{so}^{2}k_{so}^{2t}\right)} {\sum\limits_{t=1}^{T}\left(\sum\limits_{i=1}^{I}u_{io}x_{io}^{t}+\sum\limits_ {r=1}^{R}\phi_{rb}^{1}k_{ro}^{1(t-1)}+\sum\limits_{a=1}^{A}\varphi_{ao}z_{ao }^{t}+\sum\limits_{b=1}^{B}\eta_{bo}f_{bo}^{t}+\sum\limits_{s=1}^{S}\phi_{so}^{2 }k_{so}^{2(t-1)}\right)}\\ &m\in\{1,\ldots,M\}\\ &s\in\{1,\ldots,S\}\end{array}  \tag{9}\]s.t._  
\[\frac{\sum\limits_{t=1}^{T}\left(\sum\limits_{a=1}^{A}\theta_{ao}z_{ad}^{t}+ \sum\limits_{r=1}^{R}\phi_{ro}^{1}k_{rd}^{1t}+\sum\limits_{m=1}^{M}v_{mo}y_{md}^ {t}+\sum\limits_{s=1}^{S}\phi_{so}^{2}k_{sd}^{2t}\right)}{\sum\limits_{t=1}^{T} \left(\sum\limits_{i=1}^{I}u_{i\alpha}x_{id}^{t}+\sum\limits_{r=1}^{R}\phi_{ro}^ {1}k_{rd}^{1(t-1)}+\sum\limits_{a=1}^{A}\varphi_{ao}z_{ad}^{t}+\sum\limits_{b=1 }^{B}\eta_{bo}f_{bd}^{t}+\sum\limits_{s=1}^{S}\phi_{so}^{2}k_{sd}^{2(t-1)} \right)}=\theta_{dd}^{MT*} \tag{10}\] \[\frac{\sum\limits_{t=1}^{T}\left(\sum\limits_{i=1}^{A}\theta_{ao}z_ {aj}^{t}+\sum\limits_{r=1}^{R}\phi_{ro}^{1}k_{rj}^{1t}+\sum\limits_{a=1}^{A} \varphi_{no}y_{mj}^{t}+\sum\limits_{s=1}^{S}\phi_{so}^{2}k_{sj}^{2(t-1)}\right) }{\sum\limits_{t=1}^{T}\left(\sum\limits_{i=1}^{I}u_{i\alpha}x_{ij}^{t}+\sum \limits_{r=1}^{R}\phi_{ro}^{1}k_{rj}^{1(t-1)}+\sum\limits_{a=1}^{A}\varphi_{ao }z_{aj}^{t}+\sum\limits_{b=1}^{B}\eta_{bo}f_{bj}^{t}+\sum\limits_{s=1}^{S}\phi _{so}^{2}k_{sj}^{2(t-1)}\right)}\leqslant 1\] \[(j=1,\ldots,n;j\
eq 0)\] \[\frac{\sum\limits_{a=1}^{A}\varphi_{ao}z_{aj}^{t}+\sum\limits_{r=1 }^{R}\phi_{ro}^{1}k_{rj}^{1t}}{\sum\limits_{i=1}^{I}u_{i\alpha}t_{i}^{t}+\sum \limits_{r=1}^{R}\phi_{ro}^{1}k_{rj}^{1(t-1)}}\leqslant 1\ (j=1,\ldots,n;t=1,\ldots,T)\] \[\frac{\sum\limits_{m=1}^{M}v_{mo}y_{mj}^{t}+\sum\limits_{s=1}^{S} \phi_{so}^{2}k_{sj}^{2t}}{\sum\limits_{i=1}^{I}u_{io}(1-\alpha_{i}^{t})x_{ij}^ {t}+\sum\limits_{a=1}^{A}\varphi_{ao}z_{aj}^{t}+\sum\limits_{b=1}^{B}\eta_{bo} f_{bj}^{t}+\sum\limits_{s=1}^{S}\phi_{so}^{2}k_{sj}^{2(t-1)}}\leqslant 1\] \[(j=1,\ldots,n;t=1,\ldots,T)\] \[L_{i}^{t}u_{id}\leqslant x_{i}^{t}u_{id}\leqslant H_{i}^{t}u_{id }\ (i=1,\ldots,I;t=1,\ldots,T)\] \[u_{id}\geqslant 0,\phi_{ad}\geqslant 0,\phi_{rad}^{1}\geqslant 0,\phi_{ sd}^{2}\geqslant 0,\eta_{bd}\geqslant 0,v_{md}\geqslant 0\] \[u_{io}\geqslant 0,\phi_{ao}\geqslant 0,\phi_{ro}^{1}\geqslant 0, \phi_{so}^{2}\geqslant 0,\eta_{bo}\geqslant 0,v_{mo}\geqslant 0(o=1,\ldots,n\ and\ o\
eq d)\] \[(i=1,\ldots,I;a=1,\ldots,A;r=1,\ldots,R;s=1,\ldots,S;b=1,\ldots,B;m= 1,\ldots,M)\]  
This model is equivalently transformed into the following linear programming form via the Charnes-Cooper transform [26]. \[