Data Envelopment Analysis (DEA) - BCC Model

Written in front:
The blogger participated in mathematical modeling competitions more than ten times during college, and the winning grades were all second prize or above. In order to help more students avoid detours on the road to mathematical modeling, we have gathered common mathematical model algorithms for mathematical modeling in this column, hoping to be helpful to students who want to participate in mathematical modeling competitions.

1 Introduction
2. Model establishment
3. Model solution

1 Introduction

The CCR model of data envelopment analysis has been introduced above. The CCR model is a radial EDA model under the assumption of constant returns to scale (CRS), that is, λ \lambda in the $value$ $\lambda \$ geqslant $l ⩾ 0.$ For specific model principles, please refer to the link:Data Envelopment Analysis - CCR Model.
However, in the actual production process, the scale returns of production technology are not CRS. If the CRS assumption is adopted, the technical efficiency obtained is not entirely pure technical efficiency, but a comprehensive efficiency that includes scale efficiency components.
Generally speaking, the returns to scale of production technology go through three stages: increasing returns to scale (IRS), constant returns to scale (CRS), and diminishing returns to scale (DRS). If it is impossible to determine which stage the research sample is in, then the variable returns to scale (VRS) model should be selected to evaluate technical efficiency, that is, $\lambda in the model$ $\sum \lambda=$ 1 $\sum l = 1$ . At this time, the technical efficiency derived from the VRS model is pure technical efficiency.

2. Model establishment

The BCC model is the radial DEA model under the assumption of variable returns to scale (VRS). The difference between it and the CCR model is the addition of the equality constraint $\sum \lambda=1$ .
Input-oriented BCC dual model:
$\min \theta$ $s.t.\left\{\begin{array}{c}\sum_{i=1}^{n} \lambda_{i} x_{i j} \leqslant \theta x_{i j} \\ \sum_{i=1}^{n} \lambda_{i} y_{i r} \geqslant y_{k r} \\ \sum_{i=1}^{n} \lambda_{i}=1 \\ \lambda_{i} \geqslant 0, j=1, \cdots, m ; r=1, \cdots, 3 q\end{array}\right.$
Among them, $\cdots, n$ .
Output-oriented BCC dual model:
$\max \phi$ $s.t.\left\{\begin{array}{c}\sum_{i=1}^{n} \lambda_{i} x_{i j} \leqslant x_{i j} \\ \sum_{i=1}^{n} \lambda_{i} y_{i r} \geqslant \phi y_{k r} \\ \sum_{i=1}^{n} \lambda_{i}=1 \\ \lambda_{i} \geqslant 0, j=1, \cdots, m ; r=1, \cdots, 3 q\end{array}\right.$ Among them, $\cdots, n$ 。

3. Model solution

Let’s take the following question as an example:
A certain city’s education committee needs to evaluate six key middle schools, and the corresponding indicators are as shown in the table. The per-student investment and the percentage of non-low-income families in the table are input indicators, and the per-student writing score and per-student science and technology score are output indicators. Please evaluate which schools are relatively effective based on these indicators.
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Based on the model established above, the MATLAB program for writing the input-oriented BCC model is as follows:

%投入导向BCC
w=[];
for i=1:n
    f=[zeros(1,n) 1];
    A=[X -X(:,i); -Y zeros(q,1)];
    b=[zeros(1,m) -Y(:,i)']';
    Aeq=[ones(1,n) 0];
    beq=1;
    LB=[zeros(n+1,1)];
    UB=[];
    w(:,i)=linprog(f,A,b,Aeq,beq,LB,UB);
end
BCC_IN=w(n+1,:)'

The results obtained are: 1, 0.9804, 1, 0.9395, 1,
1. The MATLAB program of the output-oriented BCC model is as follows:

%产出导向BCC
w=[];
for i=1:n
    f=[zeros(1,n) -1];
    A=[X zeros(m,1); -Y Y(:,i)];
    b=[X(:,i)' zeros(1,q)]';
    Aeq=[ones(1,n) 0];
    beq=1;
    LB=[zeros(n+1,1)];
    UB=[];
    w(:,i)=linprog(f,A,b,Aeq,beq,LB,UB);
end
BCC_OUT=1./w(n+1,:)'

The results obtained are: 1, 0.9948, 1, 0.9466, 1, 1

Observing the results of the input-oriented BCC model and the output-oriented BCC model, we can find that the input and output of schools A, C, E, and F are relatively effective.