Analytic hierarchy process

Analytic hierarchy process (AHP)

Models and methods for making decisions on complex systems that are difficult to quantify completely

step ：

1. Build a hierarchical model
2. Structural judgment （ Compare in pairs ） matrix
3. Hierarchical single sorting and its consistency test
4. Hierarchical total ranking and its consistency test

According to the interrelation, the goal of the decision 、 Factors to consider （ Decision accurate measurement ） And the decision object
At the top ： The purpose of the decision 、 The problem to be solved
Middle layer ： Factors to consider 、 Criteria for decision making
The lowest floor ： Alternatives in decision making
For two adjacent layers , Call the high level the target level , The bottom layer is the factor layer

Compare in pairs （ Uniform matrix method ）
Solve the problem that qualitative determination of weight is not easy to be accepted The pairwise comparison matrix represents the comparison of the relative importance of all factors in this layer to a factor in the previous layer The elements on the matrix are represented by 1-9 The scaling method gives

1. Don't compare all the factors together , It's a comparison between two
2. Take a relative measure of the matter , In order to minimize the difficulty of comparing the most different factors , To improve accuracy

Scale table of pairwise comparison matrix

scale	meaning
1	Of equal importance
3	A little important
5	Obviously important
7	Strongly important
9	Extremely important
2 4 6 8	The median value of the above two adjacent judgments
Reciprocal	factors i And j The judgment of comparison is aij, be j And i Comparison aji=1/aij

Inconsistencies in paired comparisons ： Allow inconsistencies , But determine the allowable range of inconsistencies

Uniformity

A theorem ：n The only nonzero eigenvalue of an order uniform matrix is n
Theorem 2 ：n Order positive reciprocal matrix A The largest characteristic root of λ≥n, If and only if λ=n when A Is a uniform matrix
because λ Continuous dependence on a_ij, be λ Than n The bigger the more ,A The more serious the inconsistency is . The eigenvector corresponding to the maximum eigenvalue is used as the weight vector of the influence degree of the compared factor on a factor in the upper layer , The greater the degree of inconsistency , The greater the judgment error caused . So you can use λ-n Measured by the size of the value A The degree of inconsistency .

From theoretical analysis ： If A Is a completely consistent judgment matrix , Should have

But in fact, it is impossible to satisfy many of the above equations when constructing the pairwise comparison matrix . Therefore, it is required that the pairwise comparison matrix has a certain consistency , That is, a certain degree of inconsistency in the pairwise comparison matrix can be allowed .

The analysis shows that , For completely consistent pairwise comparison matrix , The maximum eigenvalue of its absolute value is equal to the dimension of the matrix . Consistency requirements for pairwise comparison matrix , Translate into requirements ： The eigenvalue with the largest absolute value of the matrix has little difference with the dimension of the matrix

Consistency check

Use consistency metrics and consistency ratios <0.1 And random consistency index , Yes A The process of inspection .
n The only nonzero eigenvalue of an order uniform matrix is n
Define consistency indicators ：
$$CI=(\lambda -n)/(n-1)$$

When CI=0, There is complete consistency
When CI The bigger it is , The more serious the inconsistency
To measure CI Size , Introduce random consistency index RI：
Random construction 500 A judgment matrix , obtain 500 Consistency indicators CI, The random consistency index formula can be obtained ：

Random consistency index RI give the result as follows

Define consistency ratio ：
$$CR=CI/RI$$

General , When the consistency ratio CR<0.1 when , Think A The degree of inconsistency is within the allowable range , Have satisfactory consistency , Pass the consistency test .

Simplified calculation of maximum eigenvalue and eigenvector of positive reciprocal array

Root method He fa Power law

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Judgment matrix M It is a positive reciprocal array , That is, the following conditions are met ：

further , A positive reciprocal matrix that precisely satisfies the following conditions is called a consistency matrix

Intuitive understanding ： If i Yes j The importance of is a,j Yes k The importance of is b, that i Yes k The importance of should be a*b, Similar to transitivity .

The consistency matrix has the following properties ： If consistency matrix R The maximum eigenvalue of λ_max The corresponding eigenvector is

Combined with the construction of judgment matrix a_ij To express factors i Relative to factors j Importance , and a_ij=w_i/w_j, So you can put w_i And w_j As a factor i And factors j The absolute importance of , I.e. factors i And factors j The weight of , thus W Is the weight vector of each factor . You also have to do the vector W Normalize it ： Each weight is divided by the weight sum as its own value , The final sum is 1.

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The consistency ratio of the total ranking of the hierarchy

When CR<0.1 when , It is considered that the hierarchical total ranking passes the consistency test . The hierarchical total ranking has satisfactory consistency , Otherwise, it is necessary to readjust the element values of the judgment matrix with high consistency ratio