Detailed explanation of data envelopment analysis (DEA) (taking the 8th Ningxia provincial competition as an example)

Catalog

One 、 Basic introduction

1.1 principle

1.2CCR Model

 1.3BCC Model

  Two 、 Code

2.1MATLAB Code

 2.2Python Code

3、 ... and 、 case analysis

3.1 Case introduction

3.2 case analysis

3.3 Case solving

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One 、 Basic introduction

1.1 principle

There are many models for data envelopment analysis , Mainly for ：CCR Model ,BBC Model 、 Cross model 、A&P Model . Look at the various models of this method , The specific mathematical reasoning process of each model is basically the same , The standard linear programming formulas are also similar , The difference between them is mainly reflected in the different conditions applicable to each model . Beyond seas , This method was used in banks and hospitals earlier 、 Evaluation of the efficiency of cities and other aspects . In recent years , This method has also achieved a lot of application results in many fields of social economy in China . Because the method has relatively loose requirements for the evaluation object , Apply it to evaluate the same type DMU The relative effectiveness of the dominant position , It is difficult to be replaced by other methods .

1.2CCR Model

CCR A model is a reference to DMU adopt “ Input a certain amount of production factors , And produce a certain number of products ” To judge the relative rationality and effectiveness of each unit . From the point of view of investing resources , At the current level of output , Compare the use of invested resources , Take this as the basis for benefit evaluation , This pattern is called “ Input oriented mode ”.

Define the decision making unit j The efficiency evaluation index of is ：

  For the above formula, the weight coefficient can be appropriately taken v and u, bring , Right. Three decision-making units are used for efficiency evaluation , Generally speaking The larger it is, the greater it is Be able to get relatively more output with relatively less input . Therefore, only the maximum value of demand , You can explore Here n individual DUM Is it relatively optimal .

 1.3BCC Model

 BCC The model discusses efficiency from the perspective of output , That is, under the same input level , Compare the achievement of output resources , This pattern is called “ Input oriented mode ”. What you get is “ Technical benefits ”,DEA=1 be called “ Technology works ”, Optimal solution It is a decision-making unit j Of “ Technical benefits ”.

  Two 、 Code

2.1MATLAB Code

clear
clc
format long
data=[14.40 0.65 31.30 3621.00 0.00
16.90 0.72 32.20 3943.00 0.09
15.53 0.72 31.87 4086.67 0.07
15.40 0.76 32.23 4904.67 0.13
14.17 0.76 32.40 6311.67 0.37
13.33 0.69 30.77 8173.33 0.59
12.83 0.61 29.23 10236.00 0.51
13.00 0.63 28.20 12094.33 0.44
13.40 0.75 28.80 13603.33 0.58
14.00 0.84 29.10 14841.00 1.00]';
 
X=data([1:3],:);%X For input variables 
Y=data([4:5],:);%Y Is the output variable 
[m,n]=size(X);
s=size(Y,1);
A=[-X' Y'];% Because the objective function is minimized , there -X It turns into seeking the maximum 
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);% front 3 Listed as input coefficient , after 2 Listed as output coefficient 
   E(i,i)=Y(:,i)'*w(m+1:m+s,i);% Output value * Output coefficient 
end
theta=diag(E)';
fprintf(' use DEA The relative evaluation result of this method is ：\n');
disp(theta);

 2.2Python Code

import gurobipy
import pandas as pd

#  Display data in pages ,  Set to  False  Paging is not allowed 
pd.set_option('display.expand_frame_repr', False)

#  Maximum number of columns to display ,  Set to  None  Show all columns 
pd.set_option('display.max_columns', None)

#  Maximum number of rows to display ,  Set to  None  Show all rows 
pd.set_option('display.max_rows', None)

class DEA(object):
   def __init__(self, DMUs_Name, X, Y, AP=False):
      self.m1, self.m1_name, self.m2, self.m2_name, self.AP = X.shape[1], X.columns.tolist(), Y.shape[1], Y.columns.tolist(), AP
      self.DMUs, self.X, self.Y = gurobipy.multidict({DMU: [X.loc[DMU].tolist(), Y.loc[DMU].tolist()] for DMU in DMUs_Name})
      print(f'DEA(AP={AP}) MODEL RUNING...')

   def __CCR(self):
      for k in self.DMUs:
         MODEL = gurobipy.Model()
         OE, lambdas, s_negitive, s_positive = MODEL.addVar(), MODEL.addVars(self.DMUs),  MODEL.addVars(self.m1), MODEL.addVars(self.m2)
         MODEL.update()
         MODEL.setObjectiveN(OE, index=0, priority=1)
         MODEL.setObjectiveN(-(sum(s_negitive) + sum(s_positive)), index=1, priority=0)
         MODEL.addConstrs(gurobipy.quicksum(lambdas[i] * self.X[i][j] for i in self.DMUs if i != k or not self.AP) + s_negitive[j] == OE * self.X[k][j] for j in range(self.m1))
         MODEL.addConstrs(gurobipy.quicksum(lambdas[i] * self.Y[i][j] for i in self.DMUs if i != k or not self.AP) - s_positive[j] == self.Y[k][j] for j in range(self.m2))
         MODEL.setParam('OutputFlag', 0)
         MODEL.optimize()
         self.Result.at[k, (' Benefit analysis ', ' Comprehensive technical benefits (CCR)')] = MODEL.objVal
         self.Result.at[k, (' Return to scale analysis ', ' effectiveness ')] = ' Not  DEA  It works ' if MODEL.objVal < 1 else 'DEA  Weak effective ' if s_negitive.sum().getValue() + s_positive.sum().getValue() else 'DEA  Strongly effective '
         self.Result.at[k, (' Return to scale analysis ', ' type ')] = ' The return to scale is fixed ' if lambdas.sum().getValue() == 1 else ' Increasing returns to scale ' if lambdas.sum().getValue() < 1 else ' Diminishing returns to scale '
         for m in range(self.m1):
            self.Result.at[k, (' Variance analysis ', f'{self.m1_name[m]}')] = s_negitive[m].X
            self.Result.at[k, (' Input redundancy rate ',  f'{self.m1_name[m]}')] = 'N/A' if self.X[k][m] == 0 else s_negitive[m].X / self.X[k][m]
         for m in range(self.m2):
            self.Result.at[k, (' Variance analysis ', f'{self.m2_name[m]}')] = s_positive[m].X
            self.Result.at[k, (' Under output rate ', f'{self.m2_name[m]}')] = 'N/A' if self.Y[k][m] == 0 else s_positive[m].X / self.Y[k][m]
      return self.Result

   def __BCC(self):
      for k in self.DMUs:
         MODEL = gurobipy.Model()
         TE, lambdas = MODEL.addVar(), MODEL.addVars(self.DMUs)
         MODEL.update()
         MODEL.setObjective(TE, sense=gurobipy.GRB.MINIMIZE)
         MODEL.addConstrs(gurobipy.quicksum(lambdas[i] * self.X[i][j] for i in self.DMUs if i != k or not self.AP) <= TE * self.X[k][j] for j in range(self.m1))
         MODEL.addConstrs(gurobipy.quicksum(lambdas[i] * self.Y[i][j] for i in self.DMUs if i != k or not self.AP) >= self.Y[k][j] for j in range(self.m2))
         MODEL.addConstr(gurobipy.quicksum(lambdas[i] for i in self.DMUs if i != k or not self.AP) == 1)
         MODEL.setParam('OutputFlag', 0)
         MODEL.optimize()
         self.Result.at[k, (' Benefit analysis ', ' Technical benefits (BCC)')] = MODEL.objVal if MODEL.status == gurobipy.GRB.Status.OPTIMAL else 'N/A'
      return self.Result

   def dea(self):
      columns_Page = [' Benefit analysis '] * 3 + [' Return to scale analysis '] * 2 + [' Variance analysis '] * (self.m1 + self.m2) + [' Input redundancy rate '] * self.m1 + [' Under output rate '] * self.m2
      columns_Group = [' Technical benefits (BCC)', ' Scale benefits (CCR/BCC)', ' Comprehensive technical benefits (CCR)',' effectiveness ', ' type '] + (self.m1_name + self.m2_name) * 2
      self.Result = pd.DataFrame(index=self.DMUs, columns=[columns_Page, columns_Group])
      self.__CCR()
      self.__BCC()
      self.Result.loc[:, (' Benefit analysis ', ' Scale benefits (CCR/BCC)')] = self.Result.loc[:, (' Benefit analysis ', ' Comprehensive technical benefits (CCR)')] / self.Result.loc[:,(' Benefit analysis ', ' Technical benefits (BCC)')]
      return self.Result

   def analysis(self, file_name=None):
      Result = self.dea()
      file_name = 'DEA  Data envelopment analysis report .xlsx' if file_name is None else f'\\{file_name}.xlsx'
      Result.to_excel(file_name, 'DEA  Data envelopment analysis report ')

3、 ... and 、 case analysis

3.1 Case introduction

3.2 case analysis

The first problem requires us to analyze and preprocess the original data , In a wide variety of indicators , Find the right input and output indicators , Build an evaluation system , Mathematical modeling , Find the key internal indicators that have a great impact on the operational efficiency of the organization , Calculate the score of each sub index and overall operation efficiency of each maternal and child health care institution . Obviously , This is a multi index input and multi index output for the same type of unit （ department ） Systematic analysis of relative effectiveness or benefit evaluation , After consulting a large number of literatures, we choose to use data envelopment analysis （DEA） Method to solve the problem .

On the data DEA Before analysis, we need to preprocess the data , First, classify the indicators given by the original data , Find their secondary or even primary indicators , So as to divide input indicators and output indicators , Then use Excel and Python Clean up the wrong data in the data set , On the basis of consulting a large number of literatures , Eliminate relatively unimportant indicators , Create new data sets , Import MPai Data science platform , First, the data sets of each year are analyzed separately DEA analysis , Finally, calculate the technical benefit 、 Scale benefits 、 Arithmetic mean value of comprehensive technical benefits , To solve the problem .

Relative to question one , Question 2 is more in line with the actual situation , Research is more valuable , After all, the study of hospital management is inseparable from the health economy , The study of health economy is inseparable from the national macro-economy , The economic development level of the region where the maternal and child health care institutions are located 、 Population and other objective factors have a great impact on its development and operation efficiency . At present , China's macroeconomic situation has entered a new normal , The downward pressure is very high . meanwhile , The aging of the population is getting worse , Due to lack of labor , It is disadvantageous to the development of the whole economy , Economic problems are essentially population problems , The whole economy is down , It is difficult to increase investment in health care , It is difficult for hospitals to increase their income , Only strict medical cost control can be implemented . therefore , It is urgent to explore the impact of different input structure changes on the final operation efficiency .

On the basis of question one , We first visit the statistical yearbooks of the provinces , The medical and health expenses of the corresponding provinces account for GDP The proportion of maternal and child health care institutions and the degree of provincial support for maternal and child health care institutions , Look up a lot of literature , Determine impact factors , Using the good and bad solution distance method （TOPSIS） Eliminate dimensional effects , Comprehensive score evaluation shall be conducted for all maternal and child health care institutions , So as to find out the impact of different input structure changes on the final operation efficiency .

Through the analysis of problem one and problem two , We have learned that some maternal and child health care institutions are inefficient , There is an obvious waste of medical and health resources , For this , Based on our own research results , Send a letter to the corresponding manager of maternal and child health care institutions , Use the form of report to intuitively show the problems exposed to them , And give targeted opinions , Help them get faster 、 Better solve their own resource use , Improve operational efficiency , Make full use of medical and health resources .

3.3 Case solving

chart 1 2020 Benefit analysis chart of maternal and child health care institutions in

chart 2 2020 Quadrant chart of maternal and child health care institutions in

Variance analysis is based on relaxation variables ( Variance variable 、 Excess variables ) Analysis of , Reduce the input redundancy according to the redundancy situation 、 The above table shows the analysis of input and output of variance variables / Total increase and decrease ,2020 The results of variance analysis of maternal and child health care institutions in are shown in the following table 3 Shown . Variance variable : It refers to the amount of investment that can be reduced to achieve the target efficiency , It's not DEA The difference between the actual value and the target value of the effective unit , Excess variables : It refers to the output that can be increased to achieve the target efficiency , It's not DEA The difference between the target value and the actual value of the effective area .

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