How to calculate the characteristic vector, weight value, CI value and other indicators in AHP?

One 、 application

AHP Analytic hierarchy process (AHP) is a research method that combines qualitative and quantitative methods to calculate decision weight for solving multi-objective complex problems . This method combines quantitative analysis with qualitative analysis , Use the experience of the decision-maker to judge the relative importance of the standards for whether the objectives can be achieved , And give the weight of each criterion of each decision-making scheme reasonably , Use weights to find out the order of advantages and disadvantages of each scheme , It can be effectively applied to those problems that are difficult to be solved by quantitative methods .

Two 、 operation

SPSSAU operation

（1） Click on SPSSAU Comprehensive evaluation ‘AHP Analytic hierarchy process ’ Button . Here's the picture

（2） Fill in the form and click start analysis

3、 ... and 、SPSSAU Analysis steps

Four 、 Case study

1. background

At present, the company hopes to organize employees to travel , Hope to meet your requirements comprehensively , So find 10 A tourism expert , On Tourism 4 A factor （ They are scenery , tickets , Traffic and congestion ） Evaluate （ Expert evaluation ）, Finally, the weight of the four influencing factors , Then combine the weight value , Yes 3 Scores are calculated for the selected scenic spots , Choose the best travel plan .

All in all 4 Evaluation factors （ That is, the criterion layer is 4 term , They are scenery , tickets , Traffic and congestion ）, share 10 A travel expert scores , use 1-5 Fractional scale method , For example A The factors are relative B Factors are very important , At this point 5 branch , that B Factors relative to A The factor is 1/5 namely 0.2 branch .A The factors are relative B Factors are more important , At this point 3 branch ;A The factors are relative B Factors are equally important , This is the case 1 branch .

share 10 Rated by tourism experts , In the end 10 The average score is calculated according to the scores of each tour , Get the final judgment matrix table , The following table ：

2. explain

The formula for judging matrix elements is ：

aij = 1, Elements i And element j The importance to the factors at the upper level is the same ;

aij = 3, Elements i Than elements j More important ;

aij = 5, Elements i Than elements j It's very important ;

(2) And vice versa

;

example ：  =3, be   =1/3;  =0.5, be   =0.5/1=2; And so on .

5、 ... and 、 analysis

Put data into analysis box ,SPSSAU The system automatically generates analysis results , as follows ：

Calculation formula

1. Eigenvector

（1） First, sum the columns of the matrix , Then normalize （  ） The results are shown in the following figure ：

The normalization formula is as follows

In style ,   Is the sum of the columns .

（2） Sum each row of the new matrix , Get the eigenvector

2. Weight value

Calculate the weight value （W） Normalize the feature vector , Here's the picture ：

example ：0.483766/4=0.120942;1.667208/4=0.416802; And so on .

3. The largest eigenvalue of a matrix

In style ,aW According to matrix  a And  W  Multiply ,n Is the order .

4. Consistency check

    The analysis results come from SPSSAU

This research constructs 4 Order judgment matrix , Corresponding to the above table, random consistency can be obtained by querying RI The value is 0.890,RI Values are used for the following consistency check calculations .

AHP Analytic hierarchy process is used to calculate the weight , And the consistency test is required ; Simply put, when constructing the judgment matrix , There may be logical errors , such as A Than B important ,B Than C important , But again C Than A important . Therefore, it is necessary to use the consistency test to check whether there is any problem .

（1）CI value

In style n Represents the order of the matrix . example ：  =0.02366;

（2）RI The value can be obtained by querying the table above .

（3）CR value

example ：0.024/0.890=0.027

6、 ... and 、 summary

Usually CR The smaller the value. , It means that the consistency of the judgment matrix is better , In general CR Less than 0.1, Then the judgment matrix meets the consistency test ; If CR Greater than 0.1, It means that there is no consistency , The judgment matrix should be properly adjusted and then analyzed again . This time for 4 The order judgment matrix is calculated CI The value is 0.024, in the light of RI The value lookup table is 0.890, So the calculation gives CR The value is 0.027<0.1, It means that the judgment matrix of this study meets the consistency test , The calculated weights are consistent .