# Analytical hierarchy process, why do you think?

Analytical hierarchy process (AHP). A decision-making method that considers multiple criteria, both quantitative and qualitative. AHP converts qualitative criteria into quantitative criteria. AHP also breaks down complex problems into several sub-problems that are simpler, logical, and organized according to a hierarchy.

AHP Stages:

1. Define the problem clearly.
2. Analyzing and disaggregating the problem into sub-problems, factors, criteria, and decision variables
3. Organize the problem hierarchy according to relevant sub-problems, factors, criteria, or decision variables.
4. Develop a pairwise comparison matrix.
5. Comparing sub-problems, factors, criteria, and decision variables on a scale of 1 to 9
6. Calculating the priority scale between sub-problems, factors, criteria, and decision variables
7. Evaluating the level of consistency in determining comparisons between sub-problems, factors, criteria, and decision variables

## Analytical hierarchy process: Developing a Pairwise Comparison Matrix

Determining the scale of comparison

AHP is designed based on pairwise comparisons expressed on a scale of 1 to 9.

1: equally preferred
2: there is a slight moderate
3: tends to
4: tends to be somewhat strong
5: strong
6 : strong to very strong
7 : very strong
8 : very strong to extreme
9 : extreme

Sample Issue:
In the framework of an agribusiness development program, Bogor District needs to determine the leading commodities. The three leading commodities in the district are rice, corn, and soybeans. The selection takes into account three criteria: agroecosystem suitability, income, and environmental sustainability.

## Analytical hierarchy process: Issues

How to score in the pairwise comparison matrix:

Agroecosystem (row) with agroecosystem (column) is worth 1. Because the criteria are the same. So are the other criteria (green shading).

The sentence used is “which is more important between income (criteria in the row) and agroecosystem (criteria in the column)?”. The answer to this example question is that income is more important than agroecosystem (score 8). Yellow shading

Since income is stronger than agroecosystem (value 8), conversely, the value of agroecosystem (row) should be 1/8 compared to income (column) (red shading). These values must be consistent to get the correct values.

And so on until we get the following pairwise comparison matrix with a total down:

Make normalization by dividing each cel in the table with the total according to the column. For example: a value of 8 for income (row) → agroecosystem (column), divided by the total; i.e. 14. So the normalization table is obtained as follows:

The next stage performs the same steps to determine the priority order of the sub-criteria, namely rice, corn, and soybeans in each criterion.

After obtaining the priority numbers for each criterion, enter them into the table as follows:

## Considering the order of priority values

After obtaining all the priority values, it is time to determine the leading commodity by multiplying the priority value of the goal with the priority value of the criteria, as shown in the following table:

1. Rice: (0.066 x 0.68) + (0.727 x 0.08) + (0.208 x 0.64) = 0.24
2. Corn: (0.066 x 0.26) + (0.727 x 0.21) + (0.208 x 0.27) = 0.22
3. Soybean: (0.066 x 0.06) + (0.727 x 0.72) + (0.208 x 0.09) = 0.54

From the results of the example exercise above, it is concluded that the order of superior commodities is soybean, rice, and corn.

## Consistency Check

Suppose the pairwise comparison matrix on the sustainability sub-criteria is as follows

Principal eigenvalue:

λmax = (0.639 x 1.5) + (0.274 x 4.25) + (0.087 x 11) = 3.08

Calculate consistency index (CI)

CI = (λmax – n) / (n-1);
n = matrix size (rice, corn, soybean)
CI = (3.08-3)/2 = 0.04

Calculate the consistency ratio (CR)

CR = IC / RI;
RI = Random index of consistency
CR = 0.040 / 0.58 = 0.069 = 6.9%

The smaller the CR value the more consistent it is. The value to measure is usually 10%. Above 10% is inconsistent.

Perform a consistency test on each pairwise comparison matrix.

Thank you.