Mathematical modeling -- analytic hierarchy process model

One 、 Theoretical knowledge

1. Three issues that should be given priority by using analytic hierarchy process

(1) What is the goal of our evaluation ？

eg: The best destination for college students

(2) What are the solutions to achieve this goal ？

eg: Beijing 、 Shanghai 、 Hangzhou

(3) What is the criterion of evaluation ？

eg: cost 、 traffic 、 accommodation 、 diet 、 scenery

2. Positive reciprocal matrix ( Judgment matrix ) And consistency matrix

(1) Positive reciprocal matrix ( Judgment matrix ) Conditions met

Diagonal lines are 1;Aij*Aji=1;Aij>0.

give an example ：

1	1/7	3
7	1	5
1/3	1/5	1

(2) Uniform matrix

If the positive reciprocal matrix satisfies Aij*Ajk=Aik, It is called consistent matrix （ Or it can be understood that the corresponding row or column is proportional ）.

give an example ：

1	1/2	3
2	1	6
1/3	1/6	1

3. How to calculate the weight

(1) Calculate the weight with the arithmetic average method

First step ： Normalize the judgment matrix according to columns （ Each element is divided by the sum of its columns ）

The second step ： Add the normalized columns （ Sum up by line ）

The third line ： Divide each element of the vector after adding by n You can get the weight vector

summary ： Hypothetical judgment matrix A=   , Then the weight vector obtained by the arithmetic mean method is ：

Example ： It is known that

scenery	Hangzhou	Shanghai	Suzhou
Hangzhou	1	2	4
Shanghai	1/2	1	2
Suzhou	1/4	1/2	1

  Calculate according to the first column ： Hangzhou ：1/(1+1/2+1/4)=0.57

                              Shanghai ：(1/2)/(1+1/2+1/4)=0.29

                                Suzhou ：(1/4)/(1+1/2+1/4)=0.14

  Calculate according to the second column ： Hangzhou ：2/(2+1+1/2)=0.57

                              Shanghai ：1/(2+1+1/2)=0.29

                                Suzhou ：(1/2)/(2+1+1/2)=0.14

  Calculate according to the third column ： Hangzhou ：4/(4+2+1)=0.57

                              Shanghai ：2/(4+2+1)=0.29

                                Suzhou ：1/(4+2+1)=0.14

So in the column of scenery, the weight of the three ： Hangzhou ：（0.57+0.57+0.57)/3=0.57

                                                  Shanghai ：（0.29+0.29+0.29)/3=0.29

                                                  Suzhou ：（0.14+0.14+0.14)/3=0.14

Be careful ： If it is a consistent matrix, only one column can be calculated ; If it is a general judgment matrix, you need to calculate a column to find the average .

(2) Calculate the weight by geometric average method

First step ： Multiply the matrix by rows to get a new column vector

The second step ： Open each component of the new vector n Power

The third step ： The weight vector can be obtained by normalizing the column vector

(3) Calculate the weight by eigenvalue method

First step ： Find out the maximum eigenvalue of the judgment matrix and its corresponding eigenvector

The second step ： The weight can be obtained by normalizing the obtained feature vector

for example ：

Be careful ： In the above three methods, if all the weights calculated do not add up to 1 It's because a person cut the score when calculating ！！！

4. Steps of consistency inspection （ If it is a consistency matrix, this step is not necessary ）

(1) First step ： Calculate the consistency index CI

        

(2) The second step ： Find the corresponding average random consistency index RI

n	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
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

(3) The third step ： Calculate the consistency ratio CR

If CR<0.1, Then it can be considered that the consistency of the judgment matrix is acceptable ; Otherwise, the judgment matrix needs to be modified （ You can modify flexibly ）.

Two 、 The problem solving steps

First step ： Analyze the relationship between various factors in the system , Establish the hierarchical relationship of the system

The second step ： Compare the importance of each element at the same level with respect to a certain criterion in the upper level , Construct judgment matrix

	scenery	cost	live	diet	traffic
scenery	1	1/2	4	3	3
cost	2	1	8	6	6
live	1/4	1/8	1	3/4	3/4
diet	1/3	1/6	4/3	1	1
traffic	1/3	1/6	4/3	1	1

Then use the above three methods to solve the weight to get Index weight

Suppose the weight obtained above is ：

	Index weight
scenery	0.2636
cost	0.4758
live	0.0538
diet	0.0981
traffic	0.1087

The third step ： There is a judgment matrix to calculate the relative weight of the compared elements for the criterion , And conduct consistency inspection （ Only when the consistency test passes the weight can it be used ）

scenery	Hangzhou	Shanghai	Suzhou
Hangzhou	1	2	4
Shanghai	1/2	1	2
Suzhou	1/4	1/2	1

cost	Hangzhou	Shanghai	Suzhou
Hangzhou	1	2	4
Shanghai	1/2	1	2
Suzhou	1/4	1/2	1

live	Hangzhou	Shanghai	Suzhou
Hangzhou	1	2	4
Shanghai	1/2	1	2
Suzhou	1/4	1/2	1

diet	Hangzhou	Shanghai	Suzhou
Hangzhou	1	2	4
Shanghai	1/2	1	2
Suzhou	1/4	1/2	1

traffic	Hangzhou	Shanghai	Suzhou
Hangzhou	1	2	4
Shanghai	1/2	1	2
Suzhou	1/4	1/2	1

  The calculation method of weight is the same as above , Refer to the above for the method of consistency inspection .

Assume that the weights obtained above are ：

	Beijing	Shanghai	Hangzhou
scenery	0.5954	0.2764	0.1283
cost	0.0819	0.2363	0.6817
live	0.4286	0.4286	0.1429
diet	0.6337	0.1919	0.1744
traffic	0.1667	0.1667	0.6667

Step four ： Calculate the composite weight of each layer element to the system target , And sort

	Index weight	Beijing	Shanghai	Hangzhou
scenery	0.2636	0.5954	0.2764	0.1283
cost	0.4758	0.0819	0.2363	0.6817
live	0.0538	0.4286	0.4286	0.1429
diet	0.0981	0.6337	0.1919	0.1744
traffic	0.1087	0.1667	0.1667	0.6667

Beijing ：0.2636*0.5945+0.4758*0.0819+0.0538*0.4286+0.0981*0.6337+0.1087*0.1667=0.2992604

Shanghai ：0.2636*0.2764+0.4758*0.2363+0.0538*0.4286+0.0981*0.1919+0.1087*0.1667=0.24529494

Hangzhou ：0.2636*0.1283+0.4758*0.6817+0.0538*0.1429+0.0981*0.1744+0.1087*0.6667=0.45543969

therefore , in summary , Hangzhou is the best , Beijing comes second , Shanghai finally .

3、 ... and 、Matlab Code demonstration —— Calculate the required weight

%  Analytic hierarchy process —— Find the weight of a given matrix 

%  First step ： Input judgment matrix 
clear;
clc;
A=input('Please enter your number:');
% A =[1 1 4 1/3 3;
%  1 1 4 1/3 3;
%  1/4 1/4 1 1/3 1/2;
%  3 3 3 1 3;
%  1/3 1/3 2 1/3 1];

Error=0;
[xx,yy]=size(A);

%  Judge whether it is a square matrix 、 Whether the dimension is less than or equal to 1
if xx~=yy || xx<=1
    Error=1;
end

%  Judge whether the elements in the matrix are less than 0
if Error==0
    if min(min(A))<=0
        Error=2;
    end
end

%  Determine whether the dimension exceeds 15
if Error==0
    if size(A,1)>15
        Error=3;
    end
end

%  Judge whether it is satisfied Aij*Aji=1
if Error==0
    if A' .* A~=ones(size(A,1))
        Error=4;
    end
end

if Error==0
    if size(A,1)~=2  % If the dimension is greater than 2 To judge consistency , If 2 There is no need to judge consistency , Because at this time, it must be a consistent matrix 
        %  The second step ： Judge consistency 
        [E,F]=eig(A);  % obtain A All eigenvalues of the matrix form a column vector ,E Is the eigenvector matrix ,F Is the eigenvalue matrix , Diagonals are eigenvalues , Others are 0
        Fmax=max(max(F));  % Get the maximum eigenvalue 
        n=size(A,1);
        CI=(Fmax-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];
        CR=CI/RI(n);
        if CR<0.1
            disp('CI=');
            disp(CI);
            disp('CR=');
            disp(CR);
            disp(' The consistency of the matrix is acceptable ！');
        else
            disp(' The consistency of the matrix is unacceptable ！');
        end
    end

    %  The third step ： Calculate the weight 

    %  Method 1 ： Calculate the weight with the arithmetic average method 
    Sum_A=sum(A,1);  % Sum the columns to get a row vector 
    SUM_A=repmat(Sum_A,size(A,1),1);  % use repmat Function to construct a matrix whose elements are sum 
    Stand_A=A./SUM_A;  % Each element is divided by the sum of its column to form a normalization matrix , Notice that this is  ./, No /
    disp(' The weight obtained by the arithmetic mean method is ：');
    disp(sum(Stand_A,2)/n);

    %  Method 2 ： Calculate the weight by geometric average method 
    A_prod=prod(A,2);
    A_prod=A_prod.^(1/n);
    disp(' The weight obtained by geometric average method is ：');
    disp(A_prod/sum(A_prod));

    %  Method 3 ： Calculate the weight by eigenvalue method 
    [x,y]=find(F~=0,1);  % An eigenvalue of the uniform matrix is n, The rest are 0
    %[x,y]=find(F==Fmax,1);
    disp(' The weight obtained by the eigenvalue method is ：');
    disp(E(:,y)/sum(E(:,y)));
elseif Error==1
    disp(' The input matrix is not a square matrix or the dimension is less than or equal to 1 !');
elseif Error==2
    disp(' The input matrix has elements less than or equal to 0 !');
elseif Error==3
    disp(' The dimension of the input matrix is greater than 15 !');
elseif Error==4
    disp(' The input matrix does not meet Aij*Aji=1 !');
end