AHP (Analytic Hierarchy Process) for Decision making

Everybody has to make some decisions at some point in time of their life . These decisions become more critical when there are a lot of things on stake. So, it is very important to make sure that one is taking the right decision . Here, I am going to explain an analytical way to choose an alternative from the available choices considering both tangibles as well as intangibles factors. 

AHP stands for Analytical hierarchal process. It is one of the most used decision making techniques. In AHP technique, main issue is broken down to sub-issues, which are comparatively easy to solve. There are two fundamental approaches to solve a problem: 1- Deductive Approach, 2- Inductive Approach. AHP combines both the approaches in one integrated, logic framework.

The analytic hierarchy process (AHP) was developed by Thomas L. Saaty. And was designed to solve complex problems involving multi criteria. It is also known as a Multi Criteria Decision Making (MCDM) technique.

AHP provides numerical values to intangibles criteria’s using a specially designed 9 point scale of measurement. Thus, AHP is capable of incorporating both subjective (intangibles) and objective (tangibles) measures of evaluations in decision making.

Below is the step by step approach for AHP.

Step 1- Design the hierarchy and find put the alternatives available

Before choosing the best alternatives, it’s necessary to know all the possible alternatives.  We should design the hierarchy as in above figure to avoid the missing of any alternative which can be useful. According to Saaty, a useful way to structure the hierarchy is to work down from the goal as far as possible and then work up from the alternatives until the levels of two are linked in such a way as to make comparisons possible.

Step 2- Pairwise comparison of various choices

Now from step 1, we have the alternatives (choices) available with us. The opinions of subject experts and decision makers are sought, and both subjective and objective data are collected that corresponds to the problem structure and in the pair-wise comparison of various choices on a specifically designed scale.

    

Pair-Wise Comparison Scale of Relative Importance
Scale	Numerical Rating		Reciprocal
Extremely Preferred		9.00	1/9
Very Strong to Extremely Preferred		8.00	1/8
Very Strongly Preferred		7.00	1/7
Strongly to Very Strongly Preferred		6.00	1/6
Strongly Preferred		5.00	1/5
Moderately to Strongly Preferred		4.00	1/4
Moderately Preferred		3.00	1/3
Equally to Moderately Preferred		2.00	1/2
Equally Preferred		1.00	1

Step 3- Finding Weight (Eigen Values) of sub-criteria and alternatives corresponding to each criteria

 The values obtained from the step 2 are arranged into a N*N matrix. Where N can be the number of sub-criteria for a particular criteria and alternatives available. The diagonal elements are 1. The Criterion in row k is rated higher than the criterion in column j if the value of (k,j) is greater than 1; otherwise, criterion in column j is rated higher than criterion in row k. The (j,k) element in the matrix is reciprocal of the (k,j) element.

Now prepare a normalised matrix by dividing the each element of a criterion column by the sum of the criterion column.

Now find out the Eigen vector (weight) by averaging the criterion rows.

Step 4- Consistency check

Consistency checks means if alternative A is strongly preferred than alternative B and alternative B is strongly preferred than alternative C. Then alternative A must be at least strongly preferred than alternative C. If it’s not happening than our matrix is not consistent.

The consistency of a matrix of order n is evaluated by the consistency index, (CI) that is calculated as follows:

                                           CI = (ʎ_max – n)/ (n-1)

Where ʎ_max   is the maximum Eigen value of the judgement matrix.

This CI can be compared with the CI of a random matrix (RI). CI/RI is called consistency ratio (CR). The RI values are fixed as per the table below.

If CR is less than 10%, we assume that matrix is consistent.

Random Consistency Index (RCI)
n	1	2	3	4	5	6	7	8	9	10
RI	0	0	0.58	0.9	1.12	1.24	1.32	1.41	1.45	1.49

Step 5- Repeat step 2, 3 and 4 for Criteria available

Now we can repeat step 2, 3 and 4 for criteria to get the weights of each criteria. While doing the pairwise comparison of criteria, we can give the preference as per our requirement. This preference can change from company to company and even from person to person.

Step 6- Generate Global Rating

The rating (weight) of each alternative is multiplied by the weights of sub-criteria and aggregated to result in local rating for each criterion. The local ratings are then multiplied by weights of criteria and aggregated to generate the global ratings.