## 2 Research Methods
### Two-Stage Dynamic Network Model  
Suppose that the network structure is a classical two-stage network structure with T periods and n homogeneous decision units (\(DMU_{d}(d=1,\dots,n)\)). \(DMU_{d}^{Nt}\) is the Nth stage of \(DMU_{d}\) in period t; \(x_{id}^{t}\) is the ith shared input of \(DMU_{d}\) in period t, where \(a_{i}^{t}\) is the proportion of sub-matches in stage one; \(z_{ad}^{t}\) is the ath connecting variable between stage one and stage two of \(DMU_{d}\) in period t; \(k_{rd}^{1t}\) is the rth carry-over variable in stage one of \(DMU_{d}\) in period t, and \(k_{sd}^{2t}\) is the sth carry-over variable of \(DMU_{d}\) in period t, stage two; \(f_{bd}^{t}\) is the bth input of \(DMU_{d}\) in period t, stage two; and \(y_{md}^{t}\) is the mth output of \(DMU_{d}\) in period t, stage two, where \(d=1,\dots,n\), \(t=1,2,\dots,T\), \(N\in\{1,2\}\), \(i=1,\dots,I\), \(a=1,\dots,A\), \(r=1,\dots,R\), \(s=1,\dots,S\), \(b=1,\dots,B\), and \(m=1,\dots,M\). There is a mathematical structure of the model, as shown in Figure 1.  
Based on the assumption of constant returns to scale, the linear programming form of the two-stage dynamic network DEA model is as follows:  
Figure 1: Mathematical structure of two-stage dynamic network model.  
\[\begin{array}{ll}\max&\theta_{dd}^{NT}=\sum\limits_{t=1}^{T}\left(\sum\limits_{a= 1}^{A}\delta_{ad}z_{ad}^{t}+\sum\limits_{r=1}^{R}\psi_{rd}^{1}k_{rd}^{1t}+\sum \limits_{m=1}^{M}\xi_{md}y_{md}^{t}+\sum\limits_{s=1}^{S}\psi_{sd}^{2}k_{sd}^{2t }\right)\\ s.t.&\sum\limits_{t=1}^{T}\left(\sum\limits_{i=1}^{I}\mu_{id}x_{id}^{t}+ \sum\limits_{r=1}^{R}\psi_{rd}^{1}k_{rd}^{1(t-1)}+\sum\limits_{a=1}^{A}\delta_ {ad}z_{ad}^{t}+\sum\limits_{b=1}^{B}\xi_{bd}f_{bd}^{t}+\sum\limits_{s=1}^{S} \psi_{sd}^{2}k_{sd}^{2(t-1)}\right)=1\\ &\sum\limits_{t=1}^{T}\left(\sum\limits_{\begin{subarray}{c}a=1\\ \sum\limits_{a=1}^{A}\delta_{ad}z_{aj}^{t}+\sum\limits_{r=1}^{R}\psi_{rd}^{1}k _{rf}^{1t}+\\ \sum\limits_{m=1}^{M}\xi_{md}y_{mj}^{t}+\sum\limits_{s=1}^{S}\psi_{sd}^{2}k_{ sj}^{2t}\end{subarray}\right)\\ &-\sum\limits_{t=1}^{T}\left(\sum\limits_{i=1}^{I}\mu_{id}x_{ij}^{t}+ \sum\limits_{r=1}^{R}\psi_{rd}^{1}k_{rj}^{1(t-1)}+\right.\\ &\left.\sum\limits_{a=1}^{A}\delta_{ad}z_{aj}^{t}+\sum\limits_{b=1}^{B}\xi_{ bd}f_{bj}^{t}+\sum\limits_{s=1}^{S}\psi_{sd}^{2}k_{sj}^{2(t-1)}\right)\leqslant 0 \ (j=1,\ldots,n)\\ &\sum\limits_{a=1}^{A}\delta_{ad}z_{aj}^{t}+\sum\limits_{r=1}^{R}\psi_{rd}^{1} k_{rf}^{1t}-\left(\sum\limits_{i=1}^{I}\tau_{id}^{t}x_{ij}^{t}+\sum\limits_{r=1}^{R} \psi_{rd}^{1}k_{rf}^{1(t-1)}\right)\leqslant 0\\ &(j=1,\ldots,n;t=1,\ldots,T)\\ &\left.\sum\limits_{m=1}^{M}\xi_{md}y_{mj}^{t}+\sum\limits_{s=1}^{S}\psi_{sd}^ {2}k_{sj}^{2t}-\left(\begin{array}{c}\sum\limits_{i=1}^{I}(\mu_{id}-\tau_{ id}^{t})x_{ij}^{t}+\sum\limits_{a=1}^{A}\delta_{ad}z_{aj}^{t}\\ +\sum\limits_{b=1}^{B}\xi_{bd}f_{bj}^{t}+\sum\limits_{s=1}^{S}\psi_{sd}^{2}k_{ sj}^{2(t-1)}\end{array}\right)\leqslant 0\\ &(j=1,\ldots,n;t=1,\ldots,T)\\ &L_{t}^{t}\mu_{id}\leqslant\tau_{id}^{t}\leqslant H_{t}^{t}\mu_{id}\ (i=1, \ldots,I;t=1,\ldots,T)\\ &\mu_{id}\geqslant 0,\delta_{ad}\geqslant 0,\psi_{rd}^{1}\geqslant 0,\psi_{sd}^{2} \geqslant 0,\xi_{bd}\geqslant 0,\xi_{md}\geqslant 0\\ &(i=1,\ldots,I;a=1,\ldots,A;r=1,\ldots,R;s=1,\ldots,S;b=1,\ldots,B;m=1,\ldots,M)\\ \end{array} \tag{8}\]