数据包络分析CCR、BCC模型

前言：本文给出数据包络分析(Data envelopment analysis, DEA)中的CCR、BCC模型的原规划、对偶规划以及matlab代码实现

用以评价决策单元(decision making unit, DMUs)的效率高低，如：180家热电企业那一家的生产效率更高？我国所有银行中那些银行的管理水平更先进？哪些企业中员工积极性更高？拓展探究：评价技术进步率、环境效率、预测等

CCR模型

线性规划

$$\begin{array}{rl}& \max \mu Y_0\\ & \text{ s.t. }\{\begin{array}{l}\omega X_j-\mu Y_j\ge 0,j=1,2,\cdots ,n\\ \omega X_0=1\\ \omega ,\mu \ge 0\end{array}\end{array}$$

形式展开
$$\begin{array}{rl}& \max {\mu }_1{y}_{01}+{\mu }_2{y}_{02}+\dots +{\mu }_t{y}_{0t}\\ & \{\begin{array}{l}{\omega }_1{x}_{11}+{\omega }_2{x}_{12}+\dots +{\omega }_s{x}_{1s}-{\mu }_1{y}_{11}-{\mu }_2{y}_{12}-\dots -{\mu }_t{y}_{1t}\ge 0\\ {\omega }_1{x}_{21}+{\omega }_2{x}_{22}+\dots +{\omega }_s{x}_{2s}-{\mu }_1{y}_{21}-{\mu }_2{y}_{22}-\dots -{\mu }_t{y}_{2t}\ge 0\\ \dots \dots \dots \dots \dots \dots .\dots \dots \\ {\omega }_1{x}_{n1}+{\omega }_2{x}_{n2}+\dots +{\omega }_s{x}_{ns}-{\mu }_1{y}_{n1}-{\mu }_2{y}_{n2}-\dots -{\mu }_t{y}_{nt}\ge 0\\ {\omega }_1{x}_{01}+{\omega }_2{x}_{02}+\dots +{\omega }_s{x}_{0s}=1\end{array}\end{array}$$

对偶规划
$$\begin{array}{rl}& \min \theta \\ & \text{ s.t. }\{\begin{array}{l}{\sum }_{i=1}^n{\lambda }_j{x}_{ij}\le \theta x_{0j},j=1,2,\cdots ,\text{ n }\\ {\sum }_{i=1}^n{\lambda }_j{y}_{ij}\ge y_{0j},j=1,2,\cdots ,\text{ n }\\ {\lambda }_j\ge 0,j=1,2,\cdots ,n\\ \theta 无约束\end{array}\end{array}$$

代码实现

clear
clc
x=[]; y=[];
n=size(x,1); m=size(x',1); s=size(y',1);
A=[-x y]; b=zeros(n,1); LB=zeros(m+s,1);  UB=[];
for i=1:n;
    f=[zeros(1,m) -y(i,:)];
    Aeq=[x(i,:) zeros(1,s)];    beq=1;
    w(:,i)=linprog(f,A,b,Aeq,beq,LB,UB);
    E(i,i)=y(i,:)*w(m+1:m+s,i);
    theta(i)=E(i,i);
end
w         %权重向量
theta'    %输出相对效率值

---

BCC模型

线性规划

$$\begin{array}{rl}& \min f\cdot w\\ & \text{ s.t. }\{\begin{array}{l}A\cdot w⩽b\\ \mathrm{A}\mathrm{e}\mathrm{q}\cdot w=1\\ w⩾0\end{array}\end{array}$$

$$\begin{array}{rl}& f=\left[\begin{array}{llllllllllll}0& 0& \cdots & 0& -y_{01}& -{\mathrm{y}}_{02}& \cdots & -y_{0s}& n{b}_{01}& n{b}_{01}& \cdots & n{b}_{0t}\end{array}\right]\\ & A=\left[\begin{array}{cccccccccccc}-x_{11}& -x_{12}& \cdots & -x_{1m}& y_{11}& y_{12}& \cdots & y_{1m}& -n{b}_{11}& -n{b}_{12}& \cdots & -n{b}_{1t}\\ -x_{21}& -x_{12}& & -x_{2m}& & & & & & & & \\ \cdots & \cdots & & \cdots & & & & & & & & \\ -x_{\mathrm{n}1}& -x_{n1}& & -x_{nn}& y_{n1}& y_{n2}& \cdots & y_{nm}& -n{b}_{n1}& -n{b}_{n2}& \cdots & -n{b}_{nt}\end{array}\right]\\ & Aeq=\left[\begin{array}{llllllllllll}{x}_{01}& x_{02}& \cdots & x_{0\mathrm{m}}& 0& 0& \cdots & 0& 0& 0& \cdots & 0\end{array}\right]\\ & \mathrm{y}\mathrm{b}=\left[\begin{array}{cccccccccccc}0& 0& \cdots & 0& y_{01}& {\mathrm{y}}_{02}& \cdots & y_{0\mathrm{s}}& n{b}_{01}& n{b}_{01}& \cdots & n{b}_{0t}\end{array}\right]\end{array}$$

对偶规划
$$\begin{array}{rl}& \max z_0\\ & \text{ s.t. }\{\begin{array}{l}{\sum }_{i=1}^n{x}_{ij}{\lambda }_i+x_{0j},z_0\le 0,j=1,2\cdots m\\ {\sum }_{i=1}^n-y_{ij}{\lambda }_i⩽-y_{ij},j=1,2\cdots s\\ {\sum }_{i=1}^{n}n{b}_{ij}{\lambda }_i⩽n{b}_{ij},j=1,2\cdots t\\ {\lambda }_i\ge 0,i=1,2\cdots n\\ z_0无约束\end{array}\end{array}$$

代码实现

x=[]; y=[]; nb=[];
n=size(x,1); m=size(x',1); s=size(y',1);t=size(nb',1);
A=[-x y -nb]; b=zeros(n,1); LB=zeros(m+s+t,1); UB=[];
for i=1:n;
  f=[zeros(1,m) -y(i,:) nb(i,:)];
  Aeq=[x(i,:) zeros(1,s+t)];  beq=1;
  w(:,i)=linprog(f,A,b,Aeq,beq,LB,UB);
  yb=[y(i,:) nb(i,:)];
  E(i,i)=yb*w(m+1:m+s+t,i);
  theta(i)=E(i,i);
end
w         %权重向量
theta'    %输出相对效率值