[mathematical modeling] analytic hierarchy process (AHP)

Analytic hierarchy process

purpose

Evaluation questions , If you choose which scheme is the best , Which athlete or employee performed better . Usually, specific data are not given .

One 、 Model is introduced

A typical example

This is a typical topic using analytic hierarchy process , No specific data are given , It is required to take certain measures to evaluate , Choose the best plan , The keyword has been highlighted .
We need to consider three issues ：
The goal is 、 programme 、 Rules .
The objectives and plans are generally known , The guidelines require us to collect data to fill in .

Our goal is to complete the following form , Weighted calculation .

1. Build a hierarchy

This figure needs to be presented in the paper

2. Compare the importance of elements in the same level with respect to a certain criterion in the previous level , Construct pairwise comparison matrix （ Judgment matrix ）.

Matrix for calculating weights ！

The following table uses 1-9 A table of importance

2.1 O-C matrix （ Determine the weight of indicators ）

The table is about cross / vertical Satisfaction data

2.2 Criterion layer — Judgment matrix of scheme layer

（1） The weight of each scenic spot under a certain factor （ For example, the weight of the three scenic spots in terms of scenery （ Or satisfaction score ））
（2） The elements in the analytic hierarchy process judgment matrix can only be 1-9 And their countdown .
（3） The data in this matrix should be filled in in combination with the actual situation , Make full use of the data in the topic .

3. The relative weight of the compared element relative to the criterion is calculated from the judgment matrix （ Satisfaction degree ）, Consistency check

Before using the judgment matrix to calculate the weight , It must be checked for consistency .
Our early enough matrix is Positive reciprocal matrix , But not necessarily Uniform matrix .

3.1 Consistency check

principle ： It is used to test whether there is too much difference between the judgment matrix and the consistency matrix constructed by us .

First step ： Calculate the consistency index CI

Maximum eigenvalue - Matrix order / Matrix order -1

The second step Find the corresponding average consistency index RI（n<=10)

The third step Calculate the consistency ratio CR

CR>1 when , Try to adjust to the consistency matrix （ The lines are in multiples ）

3.2 Calculate weight

There are three ways , It is recommended to use all （ Enhance robustness ）

Arithmetic average

Written expression
Formula expression

The geometric average method

Eigenvalue method

4. The weight matrix is obtained by summarizing the results

have access to EXCEL Calculate the data

Some limitations

Two 、 Code demonstration

To be continued

3、 ... and 、 Model expansion

1. There are multiple criteria layers —— Layer by layer
2. Each quantity in the criterion layer does not correspond to all schemes —— The corresponding setting is 0？
3. There is only one criterion layer , But a criterion only corresponds to a specific scheme （ One to many ）