Analytical Hierarchy Process in Mathematical Modeling (Review/Short Version of Speed Pass)

V2.0(2023-07-26): Supplement the code

V1.0(2023-07-25): Some necessary formulas are missing, and the explanation is not clear enough

algorithm thinking

Introducing: Subjective Scoring Problems

In general, it is used in small sample data. If there are other options, other algorithms can be used instead.

By establishing a hierarchical structure, multiple subjective judgments are transformed into objective pairwise comparisons, and qualitative judgments that are difficult to quantify are transformed into operable element comparisons. Therefore, it is often used to solve evaluation problems.

Hierarchical structure

• target layer
• Standard layer
• Program layer

for example:

Choice of travel destination (covers multiple factors)

Choice of school (covers multiple factors)

Goal: Solve evaluation problems

Use scoring ideas to solve

Weight --> Importance thinking

• Scoring Method Realization Problem Realization Form

Features of evaluation questions

Determine evaluation indicators and form an evaluation system

Questions to think about:

• What is the purpose of the evaluation
• We have a branch plan to achieve this goal
• Evaluation Criteria or Indicators

Evaluation questions background search engine

Google—>bing—>Baidu—>Zhihu—>WeChat

Specify indicator weight

Divide and Conquer: Pairwise Comparisons

• Scale satisfaction on a scale of 1-9

Abstraction of evaluation form - judgment matrix

• The meaning expressed by Aij is, compared with the index j, the importance of i

• When i=j, the index is the same, set to 1

• Aij and satisfy Aij*Aji =1 (positive reciprocal matrix)

• Consistent matrix: satisfy Aij*Ajk=1

  Rows and columns are in multiples

consistency check

The judgment matrices constructed in the AHP are all consistent matrices with multiple relations

Before calculating the weight of the judgment matrix, a consistency check must be carried out

A consistent matrix has one eigenvalue of n and the rest of the eigenvalues ​​are 0.

in conclusion:

When the n-order positive and reciprocal matrix A is a straight matrix, if and only if the largest eigenvalue λ=n, λ>n when it is inconsistent

1. Calculate the consistency index CI

where λ is the maximum value of the eigenvalues ​​of the judgment matrix

n is the order of the judgment matrix 

2. Find the corresponding average random consistency index RI

constructed using random sampling

RI table

 

3. Calculate the consistency ratio CR

CR=CI/RI

If CR<0.1, it can be considered that the consistency of the judgment matrix is ​​acceptable, otherwise the judgment matrix needs to be corrected

Weights need to be normalized

Calculate the ratio as an arithmetic ratio

If the judgment matrix is ​​not a positive and reciprocal matrix, then all the data can be used

• Calculate the weight of each column
• arithmetic mean

Weight solution method

Arithmetic average method to find weight

1. Normalize the judgment matrix by column
2. Adding the normalized columns
3. Divide each element of the added vector by n to get the weight vector

The weight vector obtained by the arithmetic mean method

Geometric mean method to find weight

1. Multiply the elements of A by row to get a new column vector
2. Raise each component of the new vector to the nth power
3. normalize

Eigenvalue method

tip: Generally speaking, you will choose to use the eigenvalues ​​​​to solve the weights

A consistent matrix has one eigenvalue n and the rest eigenvalues ​​are 0

step:

1. Find the largest eigenvalue of matrix A and its corresponding eigenvector
2. Find the eigenvector and normalize to get our weight

 

Obtain the evaluation result through the weight matrix

Use Excel to solve the evaluation results

• Use F4 to lock the indicator weight column

AHP steps

Step 1: Hierarchy Diagram

Target layer Objective

Criterion layer

Program layer Plan

Generated with SmartArt

The method that comes with PPT

Generated by Edraw icon

or similar professional software

Step 2: Construct the Judgment Matrix

Construct judgment matrix OC

C1-C2-C3-C4-C5 can generate five judgment matrices

Step 3: The judgment matrix passes the consistency test

(The test can only be used if the weight is passed)

The calculation of weight in the game can use the eigenvalue method more

Highlights: In order to ensure the robustness of the results

This paper uses three methods to calculate the weight, and then calculates the scores of each scheme according to the weight, and performs sorting and comprehensive analysis to avoid the deviation caused by using one method

Step 4: Calculate the synthetic weight of each layer element to the system target, and sort them

Limitations of AHP

• There should not be too many decision-making layers, otherwise the error between the judgment matrix and the consistency matrix will be too large

Code Implementation of Analytical Hierarchy Process

This blog is very clear