# AHP

AHP stands for Analytic Hierarchy Process.

Many Multi-Variable Analysis and Data Mining methods analyze matrix data that rows are for indexes and columns are for samples. But AHP uses the data of pair-data among indexes and among samples.

## Evaluation by Total Points

If there are indexes "Cost","Speed" and "Design", the total points (sum) is the easy way to evaluate.

And if we think that "Design is not so important.", weighting is used. For example,

This method is called "Multi-Criteria Analysis".

AHP is one of the methods of total points with weighting. The last formulation of AHP is same to above.

### Range of AHP as Analysis

If we have the data of each point and weights, we do not need AHP.

AHP includes the methods to get those data.

## Get Data by Compairing

Plans of projects include complicated parts and unclear parts. So comparison all plans at the same time is difficult.

But it is not so difficult for us to compare two plans. AHP uses this ability of us.

### Get Weighting Data

If we estimate that "A1 is more that A2 about 3", "A1 is more than A3 about 5" and "A2 is more than A3 about 2", the matrix is below.

The position of "3","5" and "2" and inverses of them is important to make this matrix.

The eigen vectors of the matrix is (0.93,0.33,0.17). And then we get that "(A1,A2,A3) =(0.93,0.33,0.17)".

### Get Points Data

The estimation of points of each index is similar to weighting.

In this example, "(Cost1,Cost2,Cost3) =(0.87,0.38,0.33)".

The way to get the points of "Speed" and "Design" is same. We can calculate total points after the all estimation.

## Logic of AHP

It was difficult for me to understand the reason of using inverses. Evaluation of Pairs is one of the answers.

So if we can estimate correct ratio of pairs, the calculation more correct.

But it is the real approch to use the difference because estimation of the ratio is more difficult than the difference.

## Calculation of Eigen Vectors by R

It is not easy to calculate eigen vectors by hands.

For example, if we calculate the matrix above the function of R is
eigen(matrix(c(1,1/3,1/5,3,1,1/2,5,2,1),3,3))

## ISM and DEMATEL

ISM stands for Interpretive Structural Modeling. DEMATEL stands for Decision Making Trial and Evaluation Laboratory.

AHP uses pair-data to calculate the weights of variables. But ISM and DEMATEL use the data to analyze network structural relationship among variables.

ISM uses 0 and 1 data as the information of causes and effects. DEMATEL uses some level as the information of causes and effects.