1. Analytical Hierarchy Process
Hyman Chapter 9.3

2. Relative Preferences (Pairwise Comparison)
Can be used in the pairwise comparison for a specific criterion but with a finer relative scale as shown in the following table
Relative Scale for Preference in Pairwise Comparison
Judgment of Preference
Numerical Rating
Extremely preferred
Very strongly preferred
Strongly preferred
Moderately preferred
Equally preferred 9 8 7 6 5 4 3 2 1

3. Pairwise Comparison (Torque criterion)
Three options A, B, and C
A is strongly preferred over B (scale of 5)
A is very strongly preferred over C (scale of 7)
B is strongly/moderately preferred over C (4) A B C 1 5 7
1/5 4
1/7
1/4
Sum
1.343
6.25 12 A B C
Average
1/1.343=0.745
0.8
0.583
0.709
1/5/1.343=0.149
0.16
0.333
0.214
1/7/1.343=0.106
0.04
0.084
0.077
1/3( )=0.077

4. Relative Importance (Pairwise Comparison)
Similar scale can also be used in the relative comparison between criteria (ex. how important in considering torque vs. bearing loads in design; it is the “k” factor in Hyman chapter 9, figure 9.2)
Relative Scale for importance in Comparison of Criteria
Judgment of Importance
Numerical Rating
Extremely more important
Very strongly more important
Strongly more important
Moderately more important
Equally important 9 8 7 6 5 4 3 2 1

5. Pairwise Comparison Between Criteria
Three criteria slippage (S), bearing loads (BL), and torque (T)
BL is moderately more important than S (scale of 3)
BL is moderately/equally important than T (scale of 2)
S is equally important then T (1) S BL T 1
1/3 3 2
1/2
Sum 5
1.83 4 S BL T
Average
1/5=0.2
0.182
0.25
0.211
3/5=0.6
0.546
0.5
0.549
0.273
0.241
1/3( )=0.241

6. Overall Weighed Scale Overall Preference
Different design weighed scale
Overall Preference
Slippage
Bearing Loads
Torque A
0.379
0.319
0.709 B
0.425
0.074
0.214 C
0.196
0.607
0.077
Criteria weighed scale importance to overall design
Average
Slippage
0.211
Bearing Loads
0.549
Torque
0.241
Inferior design
A = (0.211)(0.379) + (0.549)(0.319) + (0.241)(0.709) = 0.426
B = (0.211)(0.425) + (0.549)(0.074) + (0.241)(0.214) = 0.182
C = (0.211)(0.196) + (0.549)(0.607) + (0.241)(0.077) = 0.393