添加指标取值范围

在工作中，我们把碳储量作为绿色发展的期望产出指标之一，而在一定的区域内，其所能达到的最高碳储量是有限的（最理想的状况是将全部建设用地面积都变成林地，从而实现区域内碳储量最大化，而这在现实中也是不可能的事情）。但是在原始的SBM模型中，所有投入和产出要素是没有上限的。这就导致在一些DMU中，SBM投射(SBM-projection)改进后的碳储量远远超出了该DMU理论上所能达到的碳储量。因此有必要对指标的取值范围添加更为严格的取值范围。

原始的非期望产出SBM模型：

$$ \begin{aligned} \min &\rho=\frac{1-\frac1m\sum_{i=1}^m\left({s^-_i}/{x_{i0}}\right)} {1+\frac1{q_1+q_2}\left[\sum_{r_1=1}^{q_1}\left({s^+_r}/{y_{r0}}\right) + \sum_{r=1}^{q_2}\left({s^{+’}_r}/{z_{r0}}\right)\right]}\\ s.t. & \vec x_0 = X\vec\lambda + \pmb s^- \\ & \vec y_0 = Y\vec\lambda - \pmb s^+\\ &\vec z_0=Z\vec\lambda+\pmb s^{+’}\\ & \vec\lambda\ge0,~\pmb s^-\ge0,\pmb s^+\ge 0 \end{aligned} $$

决策单元$DMU_0$的期望产出指标上限由向量$\pmb l_{Y0}\in R^{q_1\times 1}$所决定，该决策单元在经过SBM投射优化后的期望产出亦不能超过$\pmb l_{Y0}$，即 $$ \pmb y_0 +\pmb s^+ \le \pmb l_{Y0} $$ 由于期望产出在经过SBM投射优化之后是变大的，因此没有必要对其设置下边界。但是，投入和非期望产出指标在优化之后将更小，虽然SBM模型本身设置了大于零的下边界，但是这一边界可能还是不够精确，所以同理可以添加投入指标和非期望产出指标的下边界，满足 $$ \begin{array} \pmb x_0 - \pmb s^- \ge \pmb l_{X0}\\ \pmb z_0 - \pmb s^{+’} \ge \pmb l_{Z0} \end{array} $$ 综上，得到对松弛变量的约束条件： $$ \begin{aligned} \pmb s^- &\le \pmb x_0 - \pmb l_{X0}\\ \pmb s^+ &\le -\pmb y_0 + \pmb l_{Y0}\\ \pmb s^{+’} &\le \pmb z_0 - \pmb l_{Z0} \end{aligned} $$ 写进规划模型：

$$ \begin{aligned} \min &\rho=\frac{1-\frac1m\sum_{i=1}^m\left({s^-_i}/{x_{i0}}\right)} {1+\frac1{q_1+q_2}\left[\sum_{r_1=1}^{q_1}\left({s^+_r}/{y_{r0}}\right) + \sum_{r=1}^{q_2}\left({s^{+’}_r}/{z_{r0}}\right)\right]}\\ s.t. & \pmb x_0 = X\pmb\lambda + \pmb s^- \\ & \pmb y_0 = Y\pmb\lambda - \pmb s^+\\ & \pmb z_0=Z\pmb\lambda+\pmb s^{+’}\\ & 0\le \pmb s^- \le \pmb x_0 - \pmb l_{X0}\\ & 0\le \pmb s^+ \le -\pmb y_0 + \pmb l_{Y0}\\ & 0\le \pmb s^{+’} \le \pmb z_0 - \pmb l_{Z0}\\ & \pmb\lambda\ge0 \end{aligned} $$

在后续进行Charnes-Cooper转换的时候，松弛变量的不等式约束只需要在两侧都乘上$t$即可： $$ \begin{aligned} \min &\rho=t-\frac1m\sum_{i=1}^m\frac{S_i^-}{x_{i0}}\\ s.t. & 1=t+\frac1{q_1+q_2}\left(\sum_{r=1}^{q_1}\frac{S_r^+}{y_{r0}}+\sum_{r=1}^{q_2}\frac{S_r^{+’}}{y_{r0}}\right)\\ & t\pmb x_0=\pmb X \pmb \Lambda + \pmb S^-\\ & t\pmb y_0=\pmb Y \pmb \Lambda - \pmb S^+\\ & t\pmb z_0=\pmb Z \pmb \Lambda + \pmb S^{+’}\\ &0\le \pmb S^- \le t(\pmb x_0 - \pmb l_{X0})\\ &0\le \pmb S^+ \le t(-\pmb y_0 + \pmb l_{Y0})\\ &0\le \pmb S^{+’} \le t(\pmb z_0 - \pmb l_{Z0})\\ & \pmb \Lambda\ge0, t\ge 0 \end{aligned} $$ 在基于Python scipy.optimize库的线性规划中，这一约束体现在添加不等式条件参数A_ub, b_ub。

添加指标权重

观察原始的目标函数： $$ \rho=\frac{1-\frac1m\sum_{i=1}^m\left({s^-_i}/{x_{i0}}\right)} {1+\frac1{q_1+q_2}\left[\sum_{r_1=1}^{q_1}\left({s^+_r}/{y_{r0}}\right) + \sum_{r=1}^{q_2}\left({s^{+’}_r}/{z_{r0}}\right)\right]} $$ 可以看出，最终在分子上影响目标函数的是所有投入指标松弛变量占比的平均值；在分母上影响目标函数的是期望产出和非期望产出指标松弛变量占比的平均值。当指标之间存在重要性差异时，为每个指标添加权重系数$\pmb w_x, \pmb w_y, \pmb w_z$，则目标函数改为

$$ \rho=\frac{1-\frac1{\sum_{i=1}^m w_{xi}}\sum_{i=1}^m (w_{xi}{s^-_i}/{x_{i0}) }} {1+\frac1{\sum_{r_1=1}^{q_1} w_{yr_1}+\sum_{r_2=1}^{q_2} w_{zr_2}} \left[\sum_{r_1=1}^{q_1}\left({w_{yr_1}s^+_r}/{y_{r0}}\right) + \sum_{r=1}^{q_2}\left({w_{zr_2}s^{+’}_r}/{z_{r0}}\right)\right]} $$

通过Charnes-Cooper转换后得到线性规划模型： $$ \begin{aligned} \min &\rho=t-\frac1{\sum_{i=1}^m w_{xi}}\sum_{i=1}^m\frac{w_{xi}S_i^-}{x_{i0}}\\ s.t. & 1=t+\frac1{\sum_{r=1}^{q_1}w_{yr}+\sum_{r=1}^{q_2}w_{zr}}\left(\sum_{r=1}^{q_1}\frac{w_{yr}S_r^+}{y_{r0}}+\sum_{r=1}^{q_2}\frac{w_{zr}S_r^{+’}}{y_{r0}}\right)\\ & t\pmb x_0=\pmb X \pmb \Lambda + \pmb S^-\\ & t\pmb y_0=\pmb Y \pmb \Lambda - \pmb S^+\\ & t\pmb z_0=\pmb Z \pmb \Lambda + \pmb S^{+’}\\ & \pmb \Lambda\ge0, t\ge 0,\pmb S^-\ge0,\pmb S^+\ge0,\pmb S^{+’}\ge0 \end{aligned} $$ 写成矩阵形式， $$ \begin{aligned} \min & \rho=C\cdot \pmb X\\ s.t. & A\cdot\pmb X = b\\ & \pmb X\ge0 \end{aligned} $$ 其中自变量为 $$ \pmb X=\left[ {\color{red} \lambda_1,\lambda_2,\dots,\lambda_n,}t, {\color{blue} S_1^-,S_2^-,\dots,S_m^-,} {\color{green} S_1^+,S_2^+,\dots,S_{q_1}^+, S_1^{+’},S_2^{+’},\dots,S_{q_2}^{+’}} \right]^\rm T, $$ 目标函数系数矩阵为 $$ C=\left[ {\color{red} 0,0,\dots,0,}1,{\color{blue} -\frac{w_{x1}}{\left(\sum_{i=1}^m w_{xi}\right)x_{10}},-\frac{w_{x2}}{\left(\sum_{i=1}^m w_{xi}\right)x_{20}},\dots,-\frac{w_{xm}}{\left(\sum_{i=1}^m w_{xi}\right)x_{m0}}},{\color{green}0,\dots,0} \right], $$ 约束条件方程系数为 $$ \displaylines{ A=\left[ \begin{matrix} \color{red} 0 & \color{red}\cdots & \color{red}0 & 1 & \color{blue}0 & \color{blue}\cdots & \color{blue}0 & \color{green}\alpha_{1} & \color{green}\cdots & \color{green}\alpha_{q_1} & \color{green}\beta_{1} & \color{green}\dots & \color{green}\beta_{q_2}\\ \\ \\ \color{red}x_{11} & \color{red}\cdots & \color{red}x_{1n} & -x_{10} & \color{blue}1 & \color{blue}\cdots & \color{blue}0 & \color{green}0 & \color{green}\cdots & \color{green}0 & \color{green}0 & \color{green}\cdots & \color{green}0\\ & \color{red}\vdots &&&& \color{blue}\ddots &&& \color{green}\vdots &&& \color{green}\vdots\\ \color{red}x_{m1} & \color{red}\cdots & \color{red}x_{mn} & -x_{m0} & \color{blue}0 & \color{blue}\cdots & \color{blue}1 & \color{green}0 & \color{green}\cdots & \color{green}0 & \color{green}0 & \color{green}\cdots & \color{green}0\\ \\ \\ \color{red}y_{11} & \color{red}\cdots & \color{red}y_{1n} & -y_{10} & \color{blue}0 & \color{blue}\cdots & \color{blue}0 & \color{green}-1 & \color{green}\cdots & \color{green}0 & \color{green}0 & \color{green}\cdots & \color{green}0\\ & \color{red}\vdots &&&& \color{blue}\vdots &&& \color{green}\ddots &&& \color{green}\vdots\\ \color{red}y_{q_11} & \color{red}\cdots & \color{red}y_{q_1n} & -y_{q_10} & \color{blue}0 & \color{blue}\cdots & \color{blue}0 & \color{green}0 & \color{green}\cdots & \color{green}-1 & \color{green}0 & \color{green}\cdots & \color{green}0\\ \\ \\ \color{red}z_{11} & \color{red}\cdots & \color{red}z_{1n} & -z_{10} & \color{blue}0 & \color{blue}\cdots & \color{blue}0 & \color{green}0 & \color{green}\cdots & \color{green}0 & \color{green}1 & \color{green}\cdots & \color{green}0\\ & \color{red}\vdots &&&& \color{blue}\vdots &&& \color{green}\vdots &&& \color{green}\ddots\\ \color{red}z_{q_21} & \color{red}\cdots & \color{red}z_{q_2n} & -z_{q_20} & \color{blue}0 & \color{blue}\cdots & \color{blue}0 & \color{green}0 & \color{green}\cdots & \color{green}0 & \color{green}0 & \color{green}\cdots & \color{green}1\\ \end{matrix} \right],\\ \alpha_{r} = \frac{w_{yr}}{\left(\sum_{i=1}^{q_1}w_{yr}+\sum_{i=1}^{q_2}w_{zr}\right)y_{r0}}~\left( r=1,2,\dots,q_1\right),\\ \beta_{r} = \frac{w_{zr}}{\left(\sum_{i=1}^{q_1}w_{yr}+\sum_{i=1}^{q_2}w_{zr}\right)z_{r0}}~\left( r=1,2,\dots,q_2\right), } $$

约束条件常数项为 $$ b=\left[ 1,0,0,\dots,0 \right]^\rm T $$

添加指标取值范围#

添加指标权重#

添加指标取值范围

添加指标权重