Analytical Hierarchy Process

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Analytical Hierarchy Process Hyman Chapter 9.3

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

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(0.106+0.04+0.084)=0.077

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

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(0.2+0.273+0.25)=0.241

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