AHP Method: Example

This example is extracted from the study by Saaty (1987) to demonstrate the calculation procedure followed in Analytic Hierarchy Process (AHP).

Problem: The author’s teenage son graduated from high school with good grades and high SAT scores. He has to choose the college for his future study. The author used the AHP model in this process. The author’s son was accepted from Swarthmore College, Northwestern University, the University of Michigan, Vanderbilt University, and Carnegie-Mellon University (two blocks from his home). The cost of the college was not included in the criteria since it was almost similar to all the colleges. The son chose Northwestern University, the college which has slightly less reputation and learning environment than that of Swarthmore College. His nature was more artistic and also had already chosen a dorm room in Northwestern that was facing Lake Michigan.

Four different criteria were looked at. Based on the observation, his son preferred the “Location” further away from home. The “Reputation” was looked at slightly more than“Academics” and “Ambience” was little more important than either Academics or Location.

Step 1: Identify the Criteria

-          To choose the best college, following criteria were identified: location, ambience, reputation, and academics.

Step 2: Criteria Comparison Matrix [C]

a. Set up a matrix (n × n)

         Where, n = number of criteria

-          In this example, n = 4. Therefor, (4 x 4) matrix is set up below.

-          It is to be noted that the diagonal elements are 1. Because the comparison between location and location would be equal ranking.

b. Pairwise Comparison

-          Pairwise comparison of the elements i (left side of the matrix) is performed with the elements j (top) of the matrix.

-          When making a comparison the ranking is done based on the question: “Which criteria is more important and by how much more. Table 1 below provides us the fundamental scale values for ranking the criteria.

-          The criteria comparison matrix for this example is computed as below:

-          In this matrix, Ambience is ranked as having very strong importance over Location. The reciprocal value 1/7 is entered in (1,2) position

-          Similarly, Reputation is ranked as having essential or strong importance over Location. Therefore, Location vs. Reputation ranking equals the reciprocal of 5 i.e., 1/5.

-          Similar approach is followed throughout the matrix

-          The completed matrix is denoted as

[C] = Criteria Comparison Matrix = 

Note: 0.0535 = Average of (0.0556, 0.0723, 0.0476, 0.0385). Similarly other criteria weights are calculated.

-          Based on the calculation above, it can be seen that the Ambience is the most important criteria (0.4870 being the highest criteria weight).

Step 4. Consistency Index

-          The consistency of the ranking is checked

Step 5. Pairwise Comparison of Schools in Level 3 of Hierarchy is performed with respect to the Criteria in Level 2 of Hierarchy

-          The pairwise comparison of Swarthmore, Northwestern, U. Michigan, Vanderbilt, and CMU colleges are performed as in Step 1. It is to be noted that this comparison is with respect to their perceived desirability according to LOCATION.

-          Calculations below are done similar from Step 1 through Step 4.

Reference:

Saaty, R. W. (1987). "The analytic hierarchy process—what it is and how it is used." Mathematical Modelling, 9(3), 161-176.