Analytic hierarchy process

abbreviation AHP, One of the basic models .

It is mainly used to solve evaluation problems （ for example ： Which option is the best 、 Which athlete or employee performed better ）.

Evaluation questions can be used Scoring solve , Scoring is to fill in the following table

  The same color and are 1, Express The weight of a factor .

Evaluation questions （ Determine the evaluation index 、 Form an evaluation system ）

① evaluation The goal is What is it? ？

② What kinds of options are available to achieve this programme ？

③ evaluation Rules Or what the indicators are ？（ According to what evaluation is good or bad ）

General evaluation indicators can be based on Background material 、 common sense 、 Reference materials collected on the Internet Combine

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Consistency check ：

principle ： Test whether there is too much difference between our judgment matrix and consistent matrix  

After the conformance test is passed , How to calculate the weight ？

The result is the same no matter which column is used , Because the columns of the consistency matrix are proportional （ Judgment matrix may be inconsistent ） So if it is a judgment matrix, the normalization results of all columns should be calculated . after , Select the output weight of the method

Method 1 ： Calculate the weight by arithmetic average （ It can be put in the paper ）

  Method 2 ： The geometric average method

Method 3 ： Calculate the weight by eigenvalue method （ Most used ）

It is best to use all three algorithms in the competition , Then add the following paragraph to the paper ：

  Finally, fill the weight in the table , Calculate the score .

When calculating, you can use Excel Tabular calculation , You want to calculate by dragging , You need to lock the indicator weight in the formula （ For example, to lock B3, Write it directly in the formula ：$B$3, that will do ）

summary （ Analytic hierarchy process steps ）：

2. Compare the importance of each element in the same level with respect to a criterion in the previous level , Construct pairwise comparison matrix ( Judgment matrix )

 3. The relative weight of the compared element to the criterion is calculated from the judgment matrix , And conduct consistency inspection （ The weight can only be used after passing the test ）

CR>0.1 when , Adjust to the consistency matrix （ Multiply the rows of the consistent matrix ）

4. Calculate the composite weight of each layer element to the system target , And sort

Code implementation （MATLAB）：

clear;clc
disp(' Please enter the judgment matrix A： ')
A = input(' Judgment matrix A=')                 %  Input judgment matrix 
%%  Weight calculation 
%  Method 1： Calculate the weight by arithmetic average  
% First step ： Normalize the judgment matrix by column （ Each element is divided by the sum of its columns )
Sum_A = sum(A,1)                        % The matrix is summed in columns to form a row of vectors 
n = size(A,1)                           % Row of square matrix ( Column ) Count 
SUM_A = repmat(Sum_A,n,1)               %repeat matrix Abbreviation , dimension 1 Table column , the Sum_A repeat n That's ok 
Stand_A = A ./ SUM_A                    % Each element of a square matrix is divided by the sum of the columns 
% The second step :  Add the normalized columns ( Sum up by line )
sum(Stand_A,2)                        
% The third step :  Divide each element of the added vector by n The weight vector can be obtained 
disp(' The result of calculating the weight by arithmetic average method is ：');       
disp(sum(Stand_A,2) / n)
%  Method 2： Calculate the weight by geometric average method 
%  First step ： take A The elements of are multiplied by rows to get a new column vector 
Prduct_A = prod(A,2)                    % prod Functions and sum Function similar to , For multiplication and addition, respectively ,dim=2 A table dimension is a row 
%  The second step ： Open each component of the new vector n Power 
Prduct_n_A = Prduct_A .^ (1/n)     
%  The third step ： The weight vector can be obtained by normalizing the column vector  
disp(' The result of weight calculation by geometric average method is ：');
disp(Prduct_n_A ./ sum(Prduct_n_A))     %  Divide each element in this column vector by the sum of this vector 
%  Method 3： Calculate the weight by eigenvalue method 
% First step ： Find the matrix A The maximum eigenvalue of and its corresponding eigenvector 
[V,D] = eig(A)                          %V It's the eigenvector , D Is a diagonal matrix composed of eigenvalues 
Max_eig = max(D(:))
[r,c] = find(D == Max_eig , 1)          %  Look for D The position of the first element equal to the maximum characteristic value in the records its row and column .
%  The second step ： We can get our weight by normalizing the obtained eigenvector 
disp(' The result of calculating the weight by eigenvalue method is ：');        
disp( V(:,c) ./ sum(V(:,c)) )
%%  Calculate the consistency ratio CR
CI = (Max_eig - n) / (n-1);
RI=[0 0 0.52 0.89 1.12 1.26 1.36 1.41 1.46 1.49 1.52 1.54 1.56 1.58 1.59];% notes ：RI most n=15
CR=CI/RI(n);
disp(' Consistency indicators CI=');disp(CI);
disp(' Consistency ratio CR=');disp(CR);
if CR<0.10
    disp(' because CR < 0.10, So the judgment matrix A The consistency of is acceptable !');
else
    disp(' Be careful ：CR >= 0.10, Therefore, the judgment matrix A Need to change !');
end