今天推出的是存在共享投入的两阶段博弈交叉DEA模型。
网络 DEA 博弈交叉效率模型不仅适用于基本两阶段网络结构，还适用于多种网络结构，本文进一步将模型拓展为存在共享投入的两阶段DEA 博弈交叉效率模型。 存在共享投入的两阶段网络结构中，假设有 n 个 DMU;在第一阶段，D M U j ( j = 1 , 2 , . . . , n ) DMU_j(j=1,2,...,n) D M U j ( j = 1 , 2 , ... , n ) α i j ( i = 1 , 2 , . . . , m ) \alpha_{ij}(i=1,2,...,m) α ij ( i = 1 , 2 , ... , m ) x i j ( i = 1 , 2 , . . . , m ) x_ij(i=1,2,...,m) x i j ( i = 1 , 2 , ... , m ) z p j ( p = 1 , 2 , . . . , q ) ; z_pj(p=1,2,...,q); z p j ( p = 1 , 2 , ... , q ) ; D M U j ( j = 1 , 2 , . . . , n ) DMU_{j}(j=1,2,...,n) D M U j ( j = 1 , 2 , ... , n ) ( 1 − α i i ) (1-\alpha_{i_{i}}) ( 1 − α i i ) x i j ( i = 1 , 2 , . . . . , m ) x_{ij}(i=1,2,....,m) x ij ( i = 1 , 2 , .... , m ) z p j ( p = 1 , 2 , . . . , q ) z_{pj}(p=1,2,...,q) z p j ( p = 1 , 2 , ... , q ) y r j ( r = 1 , 2 , . . . . , s ) y_{rj}(r=1,2,....,s) y r j ( r = 1 , 2 , .... , s ) α i i ( i = 1 , 2 , . . . , m ) \alpha_{i_i}(i=1,2,...,m) α i i ( i = 1 , 2 , ... , m ) L i j L_{ij} L ij U i j U_{ij} U ij
其模型最终的规划式如下：
max E d k ( g ) = ∑ r = 1 s μ r k d ( g ) y r k + ∑ p = 1 q ω p k d ( g ) z p k s.t. ∑ i = 1 m v i k d ( g ) x i k + ∑ p = 1 q ω p k d ( g ) z p k = 1 , ∑ p = 1 q ω p k d ( g ) z p j − ∑ i = 1 m β i j d ( g ) x i j ≤ 0 , j = 1 , … , n , ∑ r = 1 s μ r k d ( g ) y r j − ∑ i = 1 m ν i k d ( g ) x i j + ∑ i = 1 m β i j d ( g ) x i j − ∑ p = 1 q ω p k d ( g ) z p j ≤ 0 , j = 1 , … , n E d ( g − 1 ) ∑ i = 1 m v i k d ( g ) x i d + ( E d ( g − 1 ) − 1 ) ∑ p = 1 q ω p k d ( g ) z p d − ∑ r = 1 s μ r k d ( g ) y r d ≤ 0 , L i j ν i k d ( g ) ≤ β i j d ( g ) ≤ U i j ν i k d ( g ) , i = 1 , … , m ; j = 1 , … , n , v i k d ( g ) , ω p k d ( g ) , μ r k d ( g ) ≥ 0 , i = 1 , … , m ; p = 1 , … , q ; r = 1 , … , s . \begin{array}{ll}\max & E_{d k}^{(g)}=\sum_{r=1}^{s} \mu_{r k}^{d(g)} y_{r k}+\sum_{p=1}^{q} \omega_{p k}^{d(g)} z_{p k} \\\text { s.t. } & \sum_{i=1}^{m} v_{i k}^{d(g)} x_{i k}+\sum_{p=1}^{q} \omega_{p k}^{d(g)} z_{p k}=1, \\& \sum_{p=1}^{q} \omega_{p k}^{d(g)} z_{p j}-\sum_{i=1}^{m} \beta_{i j}^{d(g)} x_{i j} \leq 0, j=1, \ldots, n, \\& \sum_{r=1}^{s} \mu_{r k}^{d(g)} y_{r j}-\sum_{i=1}^{m} \nu_{i k}^{d(g)} x_{i j}+\sum_{i=1}^{m} \beta_{i j}^{d(g)} x_{i j}-\sum_{p=1}^{q} \omega_{p k}^{d(g)} z_{p j} \leq 0, j=1, \ldots, n \\& E_{d}^{(g-1)} \sum_{i=1}^{m} v_{i k}^{d(g)} x_{i d}+\left(E_{d}^{(g-1)}-1\right) \sum_{p=1}^{q} \omega_{p k}^{d(g)} z_{p d}-\sum_{r=1}^{s} \mu_{r k}^{d(g)} y_{r d} \leq 0, \\& L_{i j} \nu_{i k}^{d(g)} \leq \beta_{i j}^{d(g)} \leq U_{i j} \nu_{i k}^{d(g)}, i=1, \ldots, m ; j=1, \ldots, n, \\& v_{i k}^{d(g)}, \omega_{p k}^{d(g)}, \mu_{r k}^{d(g)} \geq 0, \quad i=1, \ldots, m ; p=1, \ldots, q ; r=1, \ldots, s .\end{array} max s.t. E d k ( g ) = ∑ r = 1 s μ r k d ( g ) y r k + ∑ p = 1 q ω p k d ( g ) z p k ∑ i = 1 m v ik d ( g ) x ik + ∑ p = 1 q ω p k d ( g ) z p k = 1 , ∑ p = 1 q ω p k d ( g ) z p j − ∑ i = 1 m β ij d ( g ) x ij ≤ 0 , j = 1 , … , n , ∑ r = 1 s μ r k d ( g ) y r j − ∑ i = 1 m ν ik d ( g ) x ij + ∑ i = 1 m β ij d ( g ) x ij − ∑ p = 1 q ω p k d ( g ) z p j ≤ 0 , j = 1 , … , n E d ( g − 1 ) ∑ i = 1 m v ik d ( g ) x i d + ( E d ( g − 1 ) − 1 ) ∑ p = 1 q ω p k d ( g ) z p d − ∑ r = 1 s μ r k d ( g ) y r d ≤ 0 , L ij ν ik d ( g ) ≤ β ij d ( g ) ≤ U ij ν ik d ( g ) , i = 1 , … , m ; j = 1 , … , n , v ik d ( g ) , ω p k d ( g ) , μ r k d ( g ) ≥ 0 , i = 1 , … , m ; p = 1 , … , q ; r = 1 , … , s . 两阶段博弈DEA模型，由于需要大量计算，如有N个DMU，那么需要进行G*N^2次的线性规划，为了加快运行速度，在计算式采用文献中提供的新的算法设计。其具体思路如下：
第一步：将 g=1 时每个 DMU 博弈前的平均整体交叉效率设置为 0.001,即令 g=1 时(式 3.7)中E d ( 0 ) = 0.001 E_d^{( \mathbf{0} ) }= 0. 001 E d ( 0 ) = 0.001 ( d = 1 , 2 , . . . , n ) ( d= 1, 2, . . . , n) ( d = 1 , 2 , ... , n ) D M U k ( k = 1 , 2 , . . . , n ) DMU_k(k=1,2,...,n) D M U k ( k = 1 , 2 , ... , n ) E k ( l ) E_k^{(\mathrm{l})} E k ( l ) E k l ( l ) E_k^{\mathrm{l}(\mathrm{l})} E k l ( l ) E k 2 ( l ) ; E_k^{2(\mathrm{l})}; E k 2 ( l ) ; D M U k ( k = 1 , 2 , . . . , n ) DMU_k(k=1,2,...,n) D M U k ( k = 1 , 2 , ... , n ) E k ( 2 ) E_k^{(2)} E k ( 2 ) E k 1 ( 2 ) E_k^{1(2)} E k 1 ( 2 ) E k 2 ( 2 ) E_k^{2( 2) } E k 2 ( 2 )
第三步：当 g=3 时，令E k ( 3 ) = E k ( 1 ) + E k ( 2 ) 2 ( k = 1 , 2 , . . . , n ) E_k^{(3)}=\frac{E_k^{(1)}+E_k^{(2)}}2(k=1,2,...,n) E k ( 3 ) = 2 E k ( 1 ) + E k ( 2 ) ( k = 1 , 2 , ... , n ) E k 1 ( 3 ) = E k 1 ( 1 ) + E k 1 ( 2 ) 2 E_k^{1(3)}=\frac{E_k^{1(1)}+E_k^{1(2)}}2 E k 1 ( 3 ) = 2 E k 1 ( 1 ) + E k 1 ( 2 )
( k = 1 , 2 , . . . , n ) (k=1,2,...,n) ( k = 1 , 2 , ... , n ) E k 2 ( 3 ) = E k 2 ( 1 ) + E k 2 ( 2 ) 2 ( k = 1 , 2 , . . . , n ) ; E_{k}^{2(3)}=\frac{E_{k}^{2(1)}+E_{k}^{2(2)}}{2}\left(k=1,2,...,n\right); E k 2 ( 3 ) = 2 E k 2 ( 1 ) + E k 2 ( 2 ) ( k = 1 , 2 , ... , n ) ;
第四步：当g ≥ 4 g\geq4 g ≥ 4 D M U k ( k = 1 , 2 , . . . , n ) DMU_k(k=1,2,...,n) D M U k ( k = 1 , 2 , ... , n ) E k ( g ) E_k^{(\mathrm{g})} E k ( g ) E k 1 ( s ) E_k^{1(\mathrm{s})} E k 1 ( s ) E k 2 ( g ) ; E_k^{2(g)}; E k 2 ( g ) ; D M U k ( k = 1 , 2 , . . . , n ) DMU_k(k=1,2,...,n) D M U k ( k = 1 , 2 , ... , n ) ∣ E k ( g + 1 ) − E k ( g ) ∣ ≤ ε , ∣ E k l ( g + 1 ) − E k l ( g ) ∣ ≤ ε , ∣ E k 2 ( g + 1 ) − E k 2 ( g ) ∣ ≤ ε ( k = 1 , 2 , . . . , n ) \left|E_k^{(\mathrm{g}+1)}-E_k^{(\mathrm{g})}\right|\leq\varepsilon,\left|E_k^{\mathrm{l}(\mathrm{g}+1)}-E_k^{\mathrm{l}(\mathrm{g})}\right|\leq\varepsilon,\left|E_k^{2(\mathrm{g}+1)}-E_k^{2(\mathrm{g})}\right|\leq\varepsilon(k=1,2,...,n) E k ( g + 1 ) − E k ( g ) ≤ ε , E k l ( g + 1 ) − E k l ( g ) ≤ ε , E k 2 ( g + 1 ) − E k 2 ( g ) ≤ ε ( k = 1 , 2 , ... , n )
关于该模型，使用matlab和julia分别完成了模型建模，均能得到和论文相同的结果（指总效率，分阶段效率有差异，但是排名与论文中的结果近似。这是由于最优解不唯一造成的。）
对比如下：
参考文献：《网络DEA中的交叉效率及其应用研究》
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