virgin Tech AOE 3-2 Analytical Hierarchy Process A systematic method for comparing a list of objectives or alternatives When used in the systems engineering process. AhP can be a powerful tool for comparing alternative design concepts Reference: Ernest h. forman Decision bt Objectives, http:/mdm.gwu.edu/forman/dbo.pdf
Analytical Hierarchy Process • A systematic method for comparing a list of objectives or alternatives • When used in the systems engineering process, AHP can be a powerful tool for comparing alternative design concepts • Reference Reference Reference: Ernest H. Forman, Decision by Objectives, http://mdm.gwu.edu/Forman/DBO.pdf • A systematic method for comparing a list of objectives or alternatives • When used in the systems engineering process, AHP can be a powerful tool for comparing alternative design concepts • Reference Reference Reference: Ernest H. Forman, Decision by Objectives, http://mdm.gwu.edu/Forman/DBO.pdf
virgin Tech AHP AOE 3-2 Assume that a set of objectives has been established (VSD, OH), and that we are trying to establish a normalized set of weights to be used when comparing alternatives using these objectives e For simplicity we assume that there are 4 objectives: O,O2,O3, and O4
AHP • Assume that a set of objectives has been established (VSD, OH), and that we are trying to establish a normalized set of weights to be used when comparing alternatives using these objectives. • For simplicity, we assume that there are 4 objectives: O1, O2, O3, and O4. • Assume that a set of objectives has been established (VSD, OH), and that we are trying to establish a normalized set of weights to be used when comparing alternatives using these objectives. • For simplicity, we assume that there are 4 objectives: O1, O2, O3, and O4
virgin Tech AHP AOE 3-2 Form a pairwise comparison matrix A, where the number in the ih row and ih column gives the relative importance of o, as compared with Use a 1-9 scale. with v 1 if the two objectives are equal in importance ai=3 if O, is weakly more important than O Fi,=5 if O, is strongly more important than O i=7 if O; is very strongly more important than O ai=9 if O, is absolutely more important than O a= 1 3 if o is weakly more important than o
AHP • Form a pairwise comparison matrix A, where the number in the ith row and jth column gives the relative importance of Oi as compared with Oj • Use a 1–9 scale, with – aij = 1 if the two objectives are equal in importance – aij = 3 if Oi is weakly more important than Oj – aij = 5 if Oi is strongly more important than Oj – aij = 7 if Oi is very strongly more important than Oj – aij = 9 if Oi is absolutely more important than Oj – aij = 1/3 if Oj is weakly more important than Oi • Form a pairwise comparison matrix A, where the number in the ith row and jth column gives the relative importance of Oi as compared with Oj • Use a 1–9 scale, with – aij = 1 if the two objectives are equal in importance – aij = 3 if Oi is weakly more important than Oj – aij = 5 if Oi is strongly more important than Oj – aij = 7 if Oi is very strongly more important than Oj – aij = 9 if Oi is absolutely more important than Oj – aij = 1/3 if Oj is weakly more important than Oi
virgin Tech AHP AOE 3-2 Thus we might arrive at the following matrix 11/51/31/71「10000.20003330.143 51355000100300500 31/3 3|30000.3331000300 71/51/3170000.2000.331.00 To normalize the weights, compute the sum of each column and then divide each column by the corresponding sum Using an overbar to denote normalization, we get 0.0630.11500710.016 0.3130.5770.6430.547 0.1880.1920.2140.328 04380.1150.0710.109
AHP • Thus we might arrive at the following matrix: • To normalize the weights, compute the sum of each column and then divide each column by the corresponding sum • Using an overbar to denote normalization, we get: • Thus we might arrive at the following matrix: • To normalize the weights, compute the sum of each column and then divide each column by the corresponding sum • Using an overbar to denote normalization, we get: = = 7.000 0.200 0.333 1.000 3.000 0.333 1.000 3.000 5.000 1.000 3.000 5.000 1.000 0.200 0.333 0.143 7 1 / 5 1 / 3 1 3 1 / 3 1 3 5 1 3 5 1 1 / 5 1 / 3 1 / 7 A = 0.438 0.115 0.071 0.109 0.188 0.192 0.214 0.328 0.313 0.577 0.643 0.547 0.063 0.115 0.071 0.016 A
virgin Tech AHP AOE 3-2 00630.1150.0710.016 0.3130.5770.6430.547 A= 0.1880.1920.2140.328 04380.1150.0710.109 The numbers in the second row are generally larger than the rest of the numbers, except for the case of column 1 This indicates some inconsistency in the comparisons used in the original matrix Ideally the 4 normalized columns would all be identical if the pairwise comparisons were consistent In practice, one can compute a consistency measure using the eigenvalues of the normalized comparison matrix
AHP • The numbers in the second row are generally larger than the rest of the numbers, except for the case of column 1 • This indicates some inconsistency in the comparisons used in the original matrix • Ideally, the 4 normalized columns would all be identical if the pairwise comparisons were consistent • In practice, one can compute a consistency measure using the eigenvalues of the normalized comparison matrix. • The numbers in the second row are generally larger than the rest of the numbers, except for the case of column 1 • This indicates some inconsistency in the comparisons used in the original matrix • Ideally, the 4 normalized columns would all be identical if the pairwise comparisons were consistent • In practice, one can compute a consistency measure using the eigenvalues of the normalized comparison matrix. = 0.438 0.115 0.071 0.109 0.188 0.192 0.214 0.328 0.313 0.577 0.643 0.547 0.063 0.115 0.071 0.016 A
virgin Tech AHP AOE 3-3 The next step is to compute the average values of each row and use these as the weights in the Objective Hierarchy For this example, the weights would be =|00660.5200.2310.183 Note that by construction W.三 These weights would be used in summing the measures as required in the evaluation of the Objective Hierarchy
AHP • The next step is to compute the average values of each row and use these as the weights in the Objective Hierarchy • For this example, the weights would be: • Note that by construction, • These weights would be used in summing the measures as required in the evaluation of the Objective Hierarchy. • The next step is to compute the average values of each row and use these as the weights in the Objective Hierarchy • For this example, the weights would be: • Note that by construction, • These weights would be used in summing the measures as required in the evaluation of the Objective Hierarchy. [ ]T w = 0.066 0.520 0.231 0.183 ∑ = = 4 1 1. i wi