Analytical Hierarchy Process Basic Model (AHP)
Analytical Hierarchy Process (AHP) is an American operations researcher professor at the University of Pittsburgh Saaty (TL Saaty) in the early 1970s, for the US Department of Defense to study the topic of "power distribution according to the contribution of various industrial sectors to national welfare" When applying network system theory and multi-objective comprehensive evaluation method, a hierarchical weight decision-making analysis method is proposed.
The characteristic of this method is that on the basis of in-depth analysis of the essence, influencing factors and internal relations of complex decision-making problems, it uses less quantitative information to mathematicize the thinking process of decision-making, thus providing multi-objective and multi-criteria It provides a simple decision-making method for complex decision-making problems with or without structural characteristics.
It is a model and method for making decisions about complex systems that are difficult to quantify completely.
Analytical Hierarchy Process Modeling
1. Overview of Analytical Hierarchy Process
Analytic Hierarchy Process (AHP) is a decision analysis method that combines qualitative and quantitative solutions to complex multi-objective problems. This method combines quantitative analysis with qualitative analysis, uses the experience of decision makers to judge the relative importance of the standards that measure whether the goals can be achieved, and reasonably gives the weight of each standard for each decision-making plan. The weights are used to calculate the pros and cons of each program, which can be more effectively applied to those problems that are difficult to solve with quantitative methods.
2. Basic principles of analytic hierarchy process
According to the nature of the problem and the overall goal to be achieved, the analytic hierarchy process decomposes the problem into different constituent factors, and gathers and combines the factors at different levels according to the interrelated influence and affiliation relationship between the factors to form a multi-level analysis structure model , so that the problem ultimately boils down to the determination of the relative weight of the lowest level (plans, measures, etc. for decision-making) relative to the highest level (total goal) or the arrangement of relative priority.
3. Steps and methods of analytic hierarchy process
1Build a hierarchical structure model
2. Construct a judgment (pairwise comparison) matrix
At this time, take Example 2 as an example for analysis
. That is, there is no obvious transmission relationship among various factors, but they always oscillate near the transmission relationship. When it exceeds a certain range, it is considered that there is a problem with pairwise comparison.
In order to determine the extent of the fluctuations, it is necessary to
first understand the exact situation
3. Hierarchical single sorting and consistency test
Since λ continuously depends on aij, the more λ is larger than n, the more serious the inconsistency of A is. The eigenvector corresponding to the largest eigenvalue is used as the weight vector of the influence degree of the compared factor on a certain factor in the upper layer. The greater the degree of inconsistency, the greater the judgment error caused. Therefore, the degree of inconsistency of A can be measured by the value of λ-n.
The RI here can be searched when it is used.
The simplified calculation of the largest eigenvalue and eigenvector of the positive and reciprocal matrix is a simplified algorithm without powerful calculation tools. Now you can use MATLAB directly for accurate solution. Just understand.
4. Overall hierarchical ordering and consistency test
At this time, analyze the pairwise comparison matrix of the plan layer to the criterion layer in Example 2
summary
Now let’s review Example 2.
Here is the code for calculating the eigenvalues, eigenvectors and consistency of the criterion layer judgment matrix.
% 层次分析法代码，此代码输入到命令行窗口后在输入矩阵A即可运行得到结果
% 使用方法
% （1）构造判断矩阵A
% （2）将下文代码复制粘贴到Matlab中即可
% 例如：A=[1 3 5;0.33 1 3;0.2 0.33,1]
disp('请输入准则层判断矩阵A(n阶)');
A=input('A=');
[n,n]=size(A);%得到矩阵的阶数，以确定RI
[V,D]=eig(A);%求得特征向量和特征值
%求出最大特征值和它所对应的特征向量
%V整个矩阵的特征向量
%D指的是矩阵的特征值，按对角线元素排列
tempNum=D(1,1);%得到特征值的初值
pos=1;%标记的第一个数
for h=1:n
if D(h,h)>tempNum
tempNum=D(h,h);
pos=h;
end
end
%找到最大的特征值及其对应的位置
w=abs(V(:,pos));%最大特征值对应的特征向量指为w
w=w/sum(w);%归一化处理（各个值只需要保持在0~1的范围之内，每个值都比上所有值之和）
t=D(pos,pos);%指的是最大特征值是多少
disp('准则层特征向量w=');disp(w);disp('准则层最大特征根t=');disp(t);
%以下是一致性检验
CI=(t-n)/(n-1);RI=[0 0 0.52 0.89 1.12 1.26 1.36 1.41 1.46 1.49 1.52 1.54 1.56 1.58 1.59 1.60 1.61 1.615 1.62 1.63];
CR=CI/RI(n);
if CR<0.10
disp('此矩阵的一致性可以接受!');
disp('CI=');disp(CI);
disp('CR=');disp(CR);
else disp('此矩阵的一致性验证失败，请重新进行评分!');
end
A complete case analysis is carried out below
[Case] Problems with the method of selecting members of multi-level elite competition analysis teams
The assumption of the model here is more important, otherwise the AHP will not be able to solve this problem.
Some teachers think that it is not appropriate to use this method to analyze and model this type of problem
Some notes on the analytic hierarchy process
Most of the above content comes from the mathematical modeling brother of station B, thank you UP. This article is also the author’s study and understanding notes.