[Mathematical Modeling] Day01 - Analytic Hierarchy Process

1. Eliciting the Analytic Hierarchy Process

1.1 Thinking questions

• What is the goal of our evaluation?
• What options do we have to achieve this goal?
• What are the evaluation criteria or evaluation indicators? (We judge good or bad by what)

Generally speaking, when we evaluate the pros and cons of a certain decision, we have already determined the goals we want to evaluate and the alternatives, that is, the way to solve the problem. The only thing we need to solve is the third problem——evaluation standard, at this time we can use the combination of 背景材料, 常识and 网上搜集到的资料to select the most suitable indicator.

1.2 Leverage from the platform

• Regarding the literature search, the recommended resources here are:
  • HowNet
  • Wanfang
  • Baidu Academic
  • Google Scholar
• Website about Kuaisou: Zong Tribe ( Click here for the website )
• About convenient search with mobile phone:
  • Google search/Baidu search (if you can't visit abroad)
  • WeChat Mini Program Search
  • Zhihu search

1.3 The idea of ​​divide and conquer

• The main body is: compare two by two, and finally calculate the weight according to the result of the comparison between two.
• The weight scale is shown in the table below
  

Let's take an example at this point, assuming we choose a scenic spot among many tourist attractions to travel, how to weigh the standards of the listed scenic spots? We only need to compare the evaluation criteria of the scenic spots in pairs The green part is the scale of the number of rows in row i and column j relative to the number of columns). Similarly, the white part is the scale of the number of rows in j row i and column relative to the number of columns. It is just the opposite of the above, so the green area is white The corresponding reciprocal of the region. 【这里的重要性可以理解为对某件事的满意度】

1.4 Consistent matrix

• Two positive and reciprocal matrices with a multiple relationship between each row (column) (positive and reciprocal matrix: matrix with each element Aij>0 and satisfying Aij×Aji=1 matrix)
• Characteristic: Aik = Aij × Ajk
• 注：在使用判断矩阵求和之前，必须要对其进行一致性检验

The following figure shows the judgment matrix and the consistency matrix, so how to find the consistency of the two matrices [that is, the similarity of the two matrices? ? ] At this time, it is necessary to check the consistency of the matrix to see whether the inconsistent results of the two matrices are within the scope of contradiction.

1.5 Consistency Check

1. to lead
   
   
2. Step
   ① Calculate 一致性指标CI
   
   ② Find the corresponding 平均随机一致性指标RI
   
   ③判断一致性比例CR
   

• If CR < 0.1, it is considered that the consistency of the judgment matrix is ​​acceptable; otherwise, CR >= 0.1, it means that the matrix is ​​inconsistent, and the judgment matrix needs to be corrected

1.6 Consistent matrix calculation weight

1. It is easier to understand directly by looking at the examples, see the figure below. In terms of scenic spots, it is assumed that the importance of Suzhou and Hangzhou is 1
2. Normalization
   

1.7 Determine the weight of the judgment matrix

• Arithmetic mean method to find weight
  • Normalize the judgment matrix by column (each element is divided by the sum of its column)
    
    
  • Add normalized columns (sum by row)
  • Divide each element in the added vector by n to get the weight vector
    
    
• geometric mean
  • Multiply the elements of A by row to get a new column vector
  • Raise each component of the new vector to the nth power
  • Normalize the column vector to get the weight vector
    

3. Eigenvalue method to find weight

A consistent matrix has one eigenvalue of 1 and the rest of the eigenvalues ​​are 0

• Find the largest eigenvalue of matrix A and its corresponding eigenvector
• The weights can be obtained by normalizing the obtained eigenvectors
  

4. When the excel table quickly calculates the weight, use F4 to lock the cell
   

2. Analytic Hierarchy Process

2.1 Definition

• The Analytic Hierarchy Process (AHP) is a comprehensive evaluation method for system analysis and decision-making created by T. L. Saaty, an American operations researcher and professor at the University of Pittsburgh in the 1970s. It is proposed on the basis of the thinking process, and it reasonably solves the qualitative problem and the quantitative processing process.

• The main feature of AHP is to transform human judgment into the comparison of the importance of several factors by establishing a hierarchical structure, thereby transforming the difficult-to-quantify qualitative judgment into the operable comparison of importance. In many cases, decision makers can directly use AHP to make decisions, which greatly improves the effectiveness, reliability and feasibility of decision-making, but its essence is a way of thinking, which decomposes complex problems into multiple components, and These factors are formed into a hierarchical structure according to the dominance relationship, and the overall ranking of the relative importance of decision-making schemes is determined by pairwise comparison. The whole process embodies the basic characteristics of human decision-making thinking, that is, decomposition, judgment, and synthesis, and overcomes the shortcomings of other methods that avoid the subjective judgment of decision-makers.

2.2 Specific steps

1. Analyze the relationship between the various factors of the system, and establish the system 层次结构- the target layer, the criterion layer, and the program layer
   [Recommended software - Yitu icon]
   
2. For the importance of each element of the same level with respect to a certain criterion in the previous level, make a pairwise comparison, and construct a pairwise comparison matrix (judgment matrix)

The judgment matrix of the equation can use the Internet to query relevant information

3. Calculate the relative weight of the compared element to the criterion from the judgment matrix, and perform a consistency check (only if the weight is passed)

• Three ways to calculate weight
  • arithmetic mean
  • geometric mean
  • Eigenvalue method

During the competition, in order to ensure the sum of the results 准确性, 稳健性it is recommended to use all three methods to calculate the weight, and then calculate the scores of each plan according to the obtained weight matrix, and perform sorting and comprehensive analysis to avoid the deviation caused by a single method and make the conclusion more accurate. Comprehensive and more effective.

• The consistency check steps are in Section 1.5 above

4. Calculate the composite weight of the system target of each layer element and sort it
   

2.3 Limitations

• There should not be too many evaluation decision-making layers. The larger n is, the greater the difference between the judgment matrix and the consistency matrix may be
• If the data of the indicators in the decision-making layer are known, we cannot use the AHP to make the data more accurate