当前位置：网站首页>Analytic hierarchy process (AHP)

Analytic hierarchy process (AHP)

2022-06-29 02:43:00 【Michael_ Lzy】

Catalog

1. Determine the evaluation index 、 Build an evaluation system

2. Get the judgment matrix

3. Hierarchical single sorting and consistency checking

（1） Hierarchical single sort ( Calculate weight )

① Sum product method ( Arithmetic average )

（2） Consistency check

4. The weight matrix is obtained by summarizing the results

5. Calculate the score of each scheme

Analytic hierarchy process is a combination of qualitative and quantitative methods 、 Systematic 、 Hierarchical analysis method .

Analytic hierarchy process (AHP) is a complex objective decision-making problem ( For example, where to travel ) As a system , Decompose goals into multiple goals or criteria , Carry out relevant calculation through qualitative indicators , A systematic method based on multi scheme optimization decision .

In a nutshell Will work with Decision making It's always about Elements Break down into goals 、 Rules 、 Plan, etc , On this basis qualitative and ration Analytical decision-making methods .

Example ：

1. Determine the evaluation index 、 Build an evaluation system

Solve evaluation problems , You need to think of the following three questions ：
1. What is the goal of our evaluation ？
2. What options do we have to achieve this goal ？
3. What are the evaluation criteria or indicators ？（ Judge good and bad by what ）

Answer with examples ：

— In general , The answer to the first two questions is obvious , According to the title, we can get .

The answer to the third question ( That is, select the evaluation index to establish the evaluation system )：
1. According to the background materials in the title 、 Common sense and reference materials collected online , Select the most appropriate indicators .
2. If you can't find the relevant literature , Determine on your own basis .

Then analyze the relationship among the factors in the system , Establish a systematic evaluation system ：

2. Get the judgment matrix

Determine the criteria layer n One factor ( indicators ),C={c1,c2,c3,...,cn}, Such as scenery 、 cost 、 Environmental Science 、 diet 、 traffic

Compare their impact on the target , Determine the proportion of the layer relative to a certain criterion .

It is difficult to compare various factors , So first compare the two indicators .

use a_i_j Express c_i Factors relative to c_j Comparison results of factors , be

$A=(a_i_j)_{n\times n}=\begin{pmatrix} a_1_1 &a_1_2 & ...& a_1_n\\ a_2_1& a_2_2 & ... &a_2_n \\ ...&... &... &... \\ a_n_1 &a_n_2 &... &a_n_n \end{pmatrix}$

A Is a judgment matrix

How to fill in the value in the judgment matrix , Use the following 1-9 Scaling method ：

for instance ：

Fill in the judgment matrix , That is to say, fill in the following table completely

Fill out this form and you actually get Judgment matrix , It's also called Positive reciprocal matrix .

Fill in the judgment matrix for each criterion ：

Every element with downward Membership （ It's called Rules ） As the first element of the judgment matrix （ In the upper left corner ）

3. Hierarchical single sorting and consistency checking

（1） Hierarchical single sort ( Calculate weight )

For the judgment matrix filled in by experts , Use a certain mathematical method to sort hierarchically .
Hierarchical single ranking refers to the relative weight of each factor of each judgment matrix against its criteria , So in essence, it is to calculate the weight coefficient .

How to calculate the weight coefficient ：
① Sum product method ( Also called arithmetic mean )
② The geometric average method
③ Eigenvalue method

① Sum product method ( Arithmetic average )

1) Normalize each column element of the judgment matrix ：

$\overline{b}_{ij}=\frac{b_{ij}}{\sum_{k=1}^{n} b_{kj}} (i,j=1,2,...,n)$ j It means column

such as ：

2) Add the normalized judgment matrix by row ：

$\bar{w_i}=\sum_{j=1}^{n}\bar{b}_{ij} (i=1,2,...,n)$

such as ：

3) For vectors $\bar{w}_i=(\bar{w}_1,\bar{w}_2,...,\bar{w}_n)^{T}$ normalization

$w_i=\frac{\bar{w}_i}{\sum_{j=1}^{n} \bar{w}_j} (i=1,2,...,n)$

such as ：

Acquired w=(1.1176,0.5175,0.0611,0.3038） Is the obtained eigenvector , That is to say, the hierarchical single sorting result of the judgment matrix ( That is, the weight coefficient )

（2） Consistency check

After the judgment matrix is obtained, the consistency test shall be carried out , Why do we need to do consistency check ？

Because the values in the judgment matrix may be contradictory , as follows

Therefore, consistency test should be carried out .

Uniform matrix
If every element in the matrix aij >0 And meet $a_i_j\times a_j_i=1$ , Then we call the matrix positive reciprocal matrix .
In analytic hierarchy process , The judgment matrices we construct are positive reciprocal matrices .

If the positive reciprocal matrix satisfies $a_i_j\times a_j_k=1$ , Then we call it a uniform matrix .

Steps of consistency inspection ：

First step ： Calculation Consistency indicators CI

$CI=\frac{\lambda _{max}-n}{n-1}$

The second step ： Find the corresponding Average random consistency index RI

The average random consistency index is multiple （500 More than once ） It is obtained by taking the arithmetic mean after repeatedly calculating the eigenvalues of the random judgment matrix . Gongmusen 、 Xushubai 1986 From 1—15 The order judgment matrix is calculated repeatedly 1000 The average random consistency index of times is as follows :

RI Just look up this table

The third step ： Calculation Consistency ratio CR

$CR=\frac{CI}{RI}$

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 .

After the conformance test is passed , Only the weight calculated in the hierarchical single sorting can be used .

CR>0.1 How to fix ？

Adjust to the consistency matrix , To multiply the rows of a uniform matrix .