学步园

AHP is a theory which related closed to the practical world.

I think it is easy to understand. It is analogous to the “Divide and Conquer ” method in algorithm, and it has its application in a concept named Pyramid Principle.

I found some of the Chinese textbooks intimidating; however, this tutor (link http://people.revoledu.com/kardi/tutorial/AHP/index.html)is absolutely perfect and the following parts is my note of reading the tutorial.

Let’s have a look at the real-world example. Suppose you need to find a best apartment, then you will face the several criteria:

1 Price,

2 Down payment,

3 Distance from shops,

4 Distance from work/school

5 Neighbor's Friendliness

And you will face several alternatives:

Apartment A,

apartment B,

apartment C,

apartment D…

It is difficult to decide which alternative best satisfy the above criteria, because you have several instead of one.

AHP deals with the problem of multi criteria decision making (MCDM).

AHP is a method to derive ratio scales from paired comparisons. Why does AHP deal with paired comparisons?

because it is more practical. Technically, we can give the rank of each alterative for each of the criteria, and we can sum the rank of each of the alternative to decide which is better.

It is an easy to understand, but hard to implement. When the alternative become large in number, it will be barely possible to give ranks.

As a result, comparing paired criteria replaced the above method. Instead of giving the rank, we just have to compare every paired criteria derived from the set of criteria.

And then comes the most confusing part of the method.

The paired comparison is given subjectively by people. In order to tell whether the paired comparison is proper or not, there should be certain judgment. And the judgment is like below:

First, we have to do some calculation.

a reciprocal matrix ->

normalize the sum of each column ->

averaging across the rows->

priority vector(which is an approximation of the normalized eigen vector)

Second, we have to use a formula to compute CI(consistent index):

Prof. Saaty proved that for consistent reciprocal matrix, the largest Eigen value is equal to the size of comparison matrix , =n. Then he gave a measure of consistency, called Consistency Index as deviation or degree of consistency using the following formula

To decide whether the comparison is consistent or not, we have to see if the value of CI is proper, according to the table below

Table 8: Random Consistency Index (  )

Third, make comparison matrices for each choice

Forth, compute the overall composite weight of each alternative choice. It is just normalization of linear combination of multiplication between weight and priority vector. And the result is shown.

In general, AHP deals with problems in the field of multi criteria decision making. The input is the paired comparisons of the criteria. Then it is transformed as a priority vector indicating the weight of the criteria. Finally, we can compute the weight of each choice.

Using Analytic Hierarchy Process (AHP), you can convert ordinal scale to ratio scale and even  check its consistency.

The way to calculate eigen vecotr in AHP is just an approximation. The sacrifice of accuracy  brings more efficiency.

Then transitive proeprty is the basis of consistency.That is, if A is better than B, B better  than C, then it will be not proper if we find A is no better than C.