Modeling Planning learning articles (1) - Analytic Hierarchy

Article Directory

Background and analysis of application level
Principles and steps of the Analytic Hierarchy Process

principle
Method steps

(1) establish a hierarchy model
(2) Comparison matrix structure
(3) single-level sorting and consistency check
(4) The total level sorting and consistency test

to sum up
Application examples
Reference links

Background and analysis of application level

Analytic Hierarchy is the United States 70 years Satie in resolving "the contribution of various industrial sectors of national welfare and power distribution," a hierarchical Quanshi issues raised heavy decision analysis methods.
The essence of the method is to use less quantitative information to analyze the mathematical thinking process of decision-making, so as to multi-objective, multi-criteria decision-making or non-structural complex problem solving methods to provide organized and simple solution.

Principles and steps of the Analytic Hierarchy Process

principle

AHP depending on the nature of the problem and the overall goal to be achieved, the problem is decomposed into component elements, and in accordance with the interrelated influence between the factors and the affiliation will be factors at different levels of aggregation combined to form a multi-level analysis of the structural model and ultimately make the problem boils down to the lowest level (for program decisions, measures) or scheduled to determine the relative merits of the order of relative importance weights of the highest level (overall goal) is.

Method steps

(1) establish a hierarchy model

The goal of the decision-making factors (decision criteria) to consider and decision-making objects into the top, middle and lowest layers according to the relationship between them, draw hierarchy diagrams.

• the highest level: the purpose of decision-making, problem to be solved.

• lowest level: alternatives during decision-making.

• middle layer: factors to consider, the decision-making criteria.

• For the adjacent two-story, high-rise called for the target layer, the lower layer is a factor.

Determine the relative merits or value of the order is scheduled to be right
to employment options after graduation, for example
if graduate employment factors to consider are:
1, whether to play their own talents
2, treatment of the income
3, the reputation of the unit
4, the working environment
5, city
6, the prospect
that according to the relevant hierarchy these can be set up as follows:
Here Insert Picture Description

(2) Comparison matrix structure

Determine the merits of the order or relative weights to be scheduled

When right between the factors determining the level of each weight, if only qualitative results, it is often not easy to be accepted by others, thus Santy et al: consistent matrix method, namely:

Do not put all the factors that put them together, but pairwise comparisons with each other
At this time, use of the relative scale, to minimize by different properties such

Su difficult compared with each other, in order to improve accuracy.

References 1-9 scale method in comparison Santy:
Here Insert Picture Description
easy to get comparison matrix elements properties are: $Age = 1 / Aji$ (matrix A of aij denotes the i-th row j-th column element)
to select tourism Example: target layer and which is a standard level

we can set at will the $C1、C2、、、Cn$ importance of the object O so as to obtain a matrix comparison.
Here Insert Picture Description
For comparison matrix: it allows inconsistent, but need to determine the extent of the inconsistency.
If it is exactly the same situation was: W can be used as a sort vector

(3) single-level sorting and consistency check

For (2), how to determine the extent of the inconsistency of the array, it is necessary given the consistency check.
Quote two theorems:

Theorem 1: The only consistent non-zero-order matrix wherein n is the n root

Theorem 2: n-order matrix A Reciprocal of maximum Eigenvalue $l>=n,$ , if and only if l = n A consistent array.
Introduced

Here Insert Picture Description

Can ci <= 0.1, the having satisfactory consistency, or (consistency ratio): $CR=CI/RI<=0.1时$ 具有满意的一致性。（RI是一个查表值）

同时，正互反阵最大特征根特征值可以进行简化计算。

（4）层次总排序及其一致性检验

偷懒直接贴PPT
Here Insert Picture Description

总结

通过对层次分析法的基本了解，不难发现层次分析法对人们的思维过程进行了加工整理，提出了一套系统分析问题的方法，为科学管理和决策提供了较有说服力的依据。
但很明显的缺点是，整个分析过程似乎都是依赖于人的主观判断思维，一来不够客观，二来两两比较全部人为完成，还是非常耗费精力的，尤其是当候选方案比较多的时候。

应用举例

仍然以毕业生就业为例：

有一个毕业生为挑选合适的工作。经双方恳谈，已有三个单位表示愿意录用某毕业生，该毕业生考虑的因素有6个，研究课题、发展前途、待遇、同事情况、地理位置和单位名气。

那么这六个因素就是准则层，三个单位就是方案层，最后要求的就是应该去哪个单位。

1）准则层判断矩阵（主观性）
Here Insert Picture Description

2）方案层判断矩阵（主观性）
Here Insert Picture Description

3）分别针对每一个B，判断C1、C2、C3之间的相对大小），进行一致性检验

一致性检验Matlab 程序如下：



clc,
clear
 fid=fopen(‘txt3.txt’,’r’);
n1=6;
n2=3; 
a=[];
 for i=1:n1
  tmp=str2num(fgetl(fid)); a=[a;tmp];
   %读准则层判断矩阵 
   end 
   for i=1:n1
    str1=char([‘b’,int2str(i),’=[];’]);
str2=char([‘b’,int2str(i),’=[b’,int2str(i),’;tmp];’]); 
eval(str1); 
for j=1:n2
tmp=str2num(fgetl(fid)); 
eval(str2); %读方案层的判断矩阵
 end
-173-
 end 
ri=[0,0,0.58,0.90,1.12,1.24,1.32,1.41,1.45]; 
%一致性指标 
[x,y]=eig(a);
 lamda=max(diag(y)); 
 num=find(diag(y)==lamda);
w0=x(:,num)/sum(x(:,num)); 
cr0=(lamda-n1)/(n1-1)/ri(n1) 
for i=1:n1
[x,y]=eig(eval(char([‘b’,int2str(i)]))); lamda=max(diag(y));
num=find(diag(y)==lamda); w1(:,i)=x(:,num)/sum(x(:,num));
cr1(i)=(lamda-n2)/(n2-1)/ri(n2); 
end 
cr1, ts=w1*w0, cr=cr1*w0

1 1 1 4 1 1/2

1 1 2 4 1 1/2

1 1/2 1 5 3 1/2

1/4 1/4 1/5 1 1/3 1/3

1 1 1/3 3 1 1

2 2 2 3 3 1

1 1/4 1/2

4 1 3

2 1/3 1

1 1/4 1/5

4 1 1/2

5 2 1

1 3 1/3

1/3 1 1/7