2. Construct a pairwise comparison matrix. Starting from the second layer of the hierarchical structure model, for the factors subordinate to (or affecting) the factors in the same layer, the paired comparison method is adopted, and the comparison scale is 1-9, and the paired comparison matrix is constructed until the bottom layer.
3. Calculate the weight vector and check the consistency. For each pairwise comparison matrix, the maximum feature root and the corresponding feature vector are calculated, and the consistency index, random consistency index and consistency ratio are used for consistency test. If the test passes, the feature vector (normalized) is the weight vector; If it fails, it needs to be reconstructed into a paired comparison matrix.
4. Calculate the combination weight vector and do the combination consistency test. Calculate the combination weight vector of the lowest layer to the target, and do the combination consistency test according to the formula. If the test passes, the decision can be made according to the result represented by the combined weight vector, otherwise, the model needs to be reconsidered or the paired judgment matrix with large consistency ratio needs to be reconstructed.
The factors involved in the problem are layered: the highest level (the purpose of solving the problem); Intermediate layer (measures taken to achieve the overall goal, standards that must be considered, etc.). ). It can also be called policy layer, constraint layer, criterion layer, etc. ); The lowest level (various measures and schemes to solve problems, etc.). Put all the factors to be considered at an appropriate level. The relationship between these factors is clearly expressed through the hierarchical structure diagram.
[Example 1] Shopping Mode
When a customer buys a TV set, he takes eight standards as the evaluation basis of four TV sets sold in the market, and establishes the following analytic hierarchy process model:
[Example 2] Mode of selecting cadres
According to the five criteria of selecting cadres: moral character, ability, seniority, age and mass relations, the following hierarchical analysis model is formed for three cadres: y 1, y2 and y3. According to the five criteria for selecting cadres: morality, ability, seniority, age and mass relations, the following hierarchical analysis model is formed.
[edit]
Construction of Pairwise Comparison Matrix
When comparing the importance of the I-th element and the J-th element relative to a certain factor at the next higher level, it is described by quantitative relative weight aij. Let * * * have n elements to participate in the comparison, which is called pairwise comparison matrix.
The value of aij in pairwise comparison matrix can refer to Satty's suggestion and be assigned according to the following scale. Aij takes the value between 1-9 and its reciprocal.
Aij = 1, and element I and element j are equally important to the factors at the next higher level;
Aij = 3, element I is slightly more important than element J;
Aij = 5, element I is more important than element j;
Aij = 7, element I is much more important than element J;
Aij = 9, element I is more important than element j;
Aij = 2n, n= 1, 2, 3, 4. Is the importance of elements I and J between Aij = 2n? Between 1 and aij = 2n+ 1;
, n= 1, 2, ..., 9, if and only if aji = n.
The characteristics of pairwise comparison matrix are: (Note: when i=j, aij = 1)
In case 2, five conditions are considered in selecting cadres: moral character x 1, talent x2, seniority x3, age x4 and mass relations x5. The pairwise comparison matrix obtained by decision makers using the pairwise comparison method is as follows:
A 14 = 5 means that the ratio of the importance of morality to age is 5, that is, decision makers think that morality is more important than age.
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Conduct consistency test
It is concluded theoretically that if A is a completely consistent pairwise comparison matrix, there should be
But in fact, it is impossible to satisfy many of the above equations when constructing pairwise comparison matrices. Therefore, the paired comparison matrices must have certain consistency, that is, the paired comparison matrices are allowed to have certain inconsistency.
From the analysis, it can be seen that the eigenvalue with the largest absolute value is equal to the dimension of the completely consistent paired comparison matrix. The consistency requirement of pairwise comparison matrix is transformed into the requirement that the eigenvalue with the largest absolute value is not much different from the matrix dimension.
Check the consistency of the pairwise comparison matrix A as follows:
Calculate and measure the paired comparison matrix a (n >; 1 square matrix) inconsistency index CI:
RI is obtained as follows: for a fixed n, a comparison matrix A is generated by mechanism, where aij is randomly selected from 1, 2, …, 9, 1/2, 1/3, …, 1/9. Such a is inconsistent, and a large enough sample should be taken.
?
? n
?
?
? 1
?
?
? 2
?
?
? three
?
?
? four
?
?
? five
?
?
? six
?
?
? seven
?
?
? eight
?
?
? nine
?
?
? Indonesia
?
?
? 0
?
?
? 0
?
?
? 0.58
?
?
? 0.90
?
?
? 1. 12
?
?
? 1.24
?
?
? 1.32
?
?
? 1.4 1
?
?
? 1.45
?
Precautions:
Find out the standard RI for testing the consistency of pairwise comparison matrix A from the relevant data: RI is called the average random consistency index, which is only related to the matrix order n. ..
Calculate the random consistency ratio CR of the pairwise comparison matrix A according to the following formula:
The judgment method is as follows: When Cr
For example, the matrix of Comparative Example 2.
Through calculation, it is found that RI= 1. 12,
This shows that A is not a uniform matrix, but A has satisfactory consistency, and A's inconsistency is acceptable.
At this time, the eigenvector corresponding to the maximum eigenvalue of A is U = (-0.8409, -0.4658, -0.095 1, -0. 1733, -0. 1920). This vector is also needed by the problem. Usually, a vector should be standardized so that all its components are greater than zero and the sum of the components is equal to 1. The normalized feature vector becomes u = (0.475, 0.263, 0.05 1, 0. 103, 0. 126) z, and this vector is called weight vector after standardization. This reflects that when selecting cadres, policy makers attach the most importance to moral conditions, followed by talent, mass relations, age factors and qualifications. The relative importance of each factor is determined by each component of the weight vector u.
To find the eigenvalue of A, we can use MATLAB statement to find the eigenvalue of A: [y, d] = EIG (a), d is the eigenvalue of paired comparison matrix, and the column of Y is the corresponding eigenvector.
In practice, the maximum eigenvalue λmax(A) of the pairwise comparison matrix A = (aij) and the approximate value of the corresponding eigenvector can be calculated by the following method.
definition
It can be approximately regarded as the eigenvector corresponding to the maximum eigenvalue of a.
calculate
It can be approximately regarded as the maximum eigenvalue of a, and in practice λ can be used to judge the consistency of matrix A.
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Hierarchical comprehensive sequencing and decision-making
Now, to completely solve the problem of Example 2, it is necessary to choose a candidate that is most suitable for the above five conditions from three candidates: Y 1, Y2 and Y3. In this regard, the three candidates y = y 1, y2, y3 are compared in terms of moral character (x 1), ability (x2), seniority (x3), age (x4) and mass relations (x5).
Firstly, the virtues of the three candidates are compared in pairs, and a paired comparison array is obtained.
After calculation, the weight vector of B 1
ωx 1(Y)= 0
(0.082，0.244，0.674)z
So the inconsistency of B 1 is acceptable. Ω x1(y) can be directly regarded as each candidate's score in morality.
Similarly, the talents, qualifications, ages and mass relations of the three candidates are also compared in pairs.
Through calculation, the corresponding weight vector is
Can be used as each candidate's talent score, seniority score, age score, mass relations score. Through inspection, it can be seen that the inconsistency among B2, B3, B4 and B5 is acceptable.
Finally, calculate the total score of each candidate. Total score y 1
According to the calculation formula, the total score ω (y 1) of y1is actually the conditional score ω x 1 (y 1), ω x2 (y 1), ..., ω x5 (y 1). In the same way, we can get the scores of Y2 and Y3 as follows.
ωz(y2)= 1
0.243，ωz(y3) = 0.452
?
?
?
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? 0.457
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? 0.263
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?
? 0.05 1
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? 0. 103
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? 0. 126
?
?
? aggregate score
?
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? Y 1
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? 0.082
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? 0.606
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? 0.429
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? 0.636
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? 0. 167
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? 0.305
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? Y2
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? 0.244
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? 0.265
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? 0.429
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? 0. 185
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? 0. 167
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? 0.243
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? Y3
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? 0.674
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? 0. 129
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? 0. 143
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? 0. 179
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? 0.667
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? 0.452
?
That is, ranking: y3 > y1> Y2
After comparison, it can be concluded that the candidate y3 is the first cadre candidate.
[edit]
Application examples of analytic hierarchy process
For example, someone is going to buy a refrigerator. After learning about six different types of refrigerators on the market, he often doesn't directly compare the refrigerators as a whole when deciding which style to buy, because there are many incomparable factors, but chooses some intermediate indicators to investigate. For example, the capacity, refrigeration level, price, model, power consumption, foreign reputation, after-sales service, etc. Then consider the advantages and disadvantages of various models of refrigerators under the above intermediate standards. With this help, the purchase decision is finally made. In decision-making, the six kinds of refrigerators are generally inconsistent in ranking the intermediate standards. Therefore, decision makers should first estimate the importance of these seven standards and give a ranking, then find out the ranking weight of six kinds of refrigerators for each standard, and finally get the ranking weight for the overall goal by synthesizing these information data, that is, buying refrigerators. With this weight vector, decision-making is easy.
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Application program of analytic hierarchy process
When using analytic hierarchy process to make a decision, you need to go through the following four steps:
1. Establish the hierarchical structure of the system;
2, constructing a pairwise comparison judgment matrix; (Positive reciprocal matrix)
3. For a certain standard, calculate the weight of each alternative element;
4. Calculate the ranking weight of the current layer elements relative to the overall goal.
5. Check the consistency.
[edit]
Problems needing attention in applying analytic hierarchy process
If the selected elements are unreasonable, the meaning is unclear, or the relationship between the elements is incorrect, the quality of the results of AHP method will be reduced, and even the decision of AHP method will fail.
In order to ensure the rationality of the hierarchical structure, we need to grasp the following principles:
1, grasp the main factors when decomposing and simplifying the problem, and don't miss too much;
2. Pay attention to the strong and weak relationship between the compared elements. Elements that are too different cannot be compared at the same level.
[edit]
Application example of analytic hierarchy process
1. Establish a hierarchical structure;
2, constructing a pairwise comparison judgment matrix; (Positive reciprocal matrix)
After comparing the indexes in pairs, the relative advantages and disadvantages of the evaluation indexes are arranged according to the proportion of 9 percentiles, and the judgment matrix of the evaluation indexes is constructed in turn.
3. For a certain standard, calculate the weight of each alternative element;
There are two methods to calculate the weight of judgment matrix, namely geometric average method (root method) and canonical column average method (sum method).
(1) geometric average method (root method)
Calculating the product of each element mi of each row of the judgment matrix A;
Calculate the n-th root of mi;
Normalized vector;
This vector is the required weight vector.
(2) Normative column average method (summation method)
Calculating the sum of each element mi of each row of the judgment matrix A;
Normalize the sum of elements in each row of;
This vector is the required weight vector.
Calculate the maximum eigenvalue of matrix a? maximum
For any i= 1, 2, …, n, where is the i-th element of vector AW.
(4) Consistency test
After constructing the judgment matrix, it is necessary to calculate the relative weight of each element in a certain criterion layer according to the judgment matrix and check the consistency. Although the consistency of judgment is not required when constructing the judgment matrix A, it is not allowed to deviate too much from the consistency of judgment. Therefore, it is necessary to check the consistency of judgment matrix A.