Inefficient weights from pairwise comparison matrices with - PowerPoint PPT Presentation

Inefficient weights from pairwise comparison matrices with arbitrarily small inconsistency Sándor Bozóki Institute for Computer Science and Control Hungarian Academy of Sciences (MTA SZTAKI), Corvinus University of Budapest 15 September 2014 Inefficiency – p. 1/14

Definitions and notations PCM n denotes the set of pairwise comparison matrices of size n × n . λ max ( A ) denotes the dominant eigenvalue of pairwise comparison matrix A of size n × n . w EM ( A ) , also called EM weight vector , denotes the right eigenvector of A corresponding to λ max ( A ) . n w EM ( A ) is usually normalized to 1, that is, w EM ( A ) � = 1 . i i =1 � � w EM ( A ) X EM ( A ) = X EM def = is the consistent i w EM ( A ) j i,j =1 ,...,n pairwise comparison matrix generated by w EM ( A ) . It is the approximation of A by the eigenvector method. Inefficiency – p. 2/14

Inconsistency index CR Saaty defined the inconsistency index as λ max ( A ) − n = λ max ( A ) − n CR ( A ) def n − 1 = , λ n × n λ n × n max − n max − n n − 1 where λ n × n max denotes the average value of the maximal eigenvalue of randomly generated pairwise comparison matrices of size n × n such that each element a ij ( i < j ) is chosen from the ratio scale 1 / 9 , 1 / 8 , . . . , 1 / 2 , 1 , 2 , . . . , 9 with equal probability. CR ( A ) is a positive linear transformation of λ max ( A ) . CR ( A ) ≥ 0 and CR ( A ) = 0 if and only if A is consistent. Saaty suggested the rule of acceptability CR < 0 . 1 . Inefficiency – p. 3/14

Inefficiency Example of Blanquero, Carrizosa and Conde (2006, p. 282):       6 . 01438057 6 . 01438057 1 2 6 2       1 / 2 4 . 26049429 4 . 26049429 1 4 3       w EM = w ∗ = , , . A =             1 / 6 1 / 4 1 / 2 1 . 003 1 1             1 / 2 1 / 3 2 . 0712416 2 . 0712416 2 1 x EM | a i 3 − x EM i a i 3 x ∗ | | a i 3 − x ∗ i 3 | i 3 i 3 i 3 1 6 6.01438057 5.99639139 0.01438057 0.00360859 2 4 4.26049429 4.24775103 0.26049429 0.24775103 3 1 1 1 0 0 4 2 2.07124160 2.06504646 0.07124160 0.06504646 Inefficiency – p. 4/14

(Internal) inefficiency Let A = [ a ij ] i,j =1 ,...,n ∈ PCM n and w = ( w 1 , w 2 , . . . , w n ) T be a positive weight vector. Definition. w is called inefficient if there exists a weight n ) T such that vector w ′ = ( w ′ 1 , w ′ 2 , . . . , w ′ | a ij − w ′ i /w ′ j | ≤ | a ij − w i /w j | for all i, j , and there exist k, ℓ such that | a kℓ − w ′ k /w ′ ℓ | < | a kℓ − w k /w ℓ | . Definition. w is called internally inefficient if there exists a n ) T such that weight vector w ′ = ( w ′ 1 , w ′ 2 , . . . , w ′ a ij ≤ w ′ i /w ′ j ≤ w i /w j if a ij ≤ w i /w j , and a ij ≥ w ′ i /w ′ j ≥ w i /w j if a ij ≥ w i /w j are fulfilled for all i, j , and there exist k, ℓ such that w ′ k /w ′ ℓ < w k /w ℓ if a kℓ ≤ w k /w ℓ , and w ′ k /w ′ ℓ > w k /w ℓ if a kℓ ≥ w k /w ℓ . Inefficiency – p. 5/14

      1 / 9 1 / 9 1 / 8 0 . 1281 0 . 1281 1 4 9       1 / 4 1 / 8 1 / 4 1 / 7 1 / 5 0 . 0180 0 . 0206 1              1 / 2   0 . 3028   0 . 3471  9 8 1 8 4 w EM = , w ∗ =       , . A =       1 / 9 1 / 8 1 / 3 0 . 1237 0 . 1237  4 1 7                  1 / 4 1 / 7 1 / 5 0 . 1440 0 . 1440 9 7 1             0 . 2835 0 . 3249 8 5 2 3 5 1 Approximations are X EM X ∗     7 . 13 0 . 42 1 . 03 0 . 88 0 . 45 6.22 0.36 1 . 03 0 . 88 0.39 1 1     0 . 14 0 . 05 0 . 14 0 . 12 0 . 06 0.16 0 . 05 0.16 0.14 0 . 06 1 1          2 . 36 16 . 86 2 . 44 2 . 10 1 . 06   2.71 16 . 86 2.80 2.40 1 . 06  1 1     ,     0 . 96 6 . 88 0 . 40 0 . 85 0 . 43 0 . 96 6.01 0.35 0 . 85 0.38 1 1             1 . 12 8 . 02 0 . 47 1 . 16 0 . 50 1 . 12 7.00 0.41 1 . 16 0.44 1 1         2 . 21 15 . 78 0 . 93 2 . 29 1 . 96 2.53 15 . 78 0 . 93 2.62 2.25 1 1 Inefficiency – p. 6/14

  1 p p p . . . p p 1 /p 1 q 1 . . . 1 1 /q      1 /p 1 /q 1 q . . . 1 1    . . . . . ...   . . . . . A ( p, q ) = , . . . . .     . . . . . ...   . . . . . . . . . .      1 /p 1 1 1 . . . 1 q    1 /p q 1 1 . . . 1 /q 1 Proposition. Let q be positive and q � = 1 . Then w EM is internally inefficient, therefore inefficient. Furthermore, CR inconsistency can be arbitrarily small if q is close enough to 1 . Sketch of the proof. If p > 1 , then with w ∗ = ( p, 1 , . . . , 1) T we have x EM < x ∗ 1 j = p ( j = 2 , 3 , . . . , n ) . 1 j Inefficiency – p. 7/14

Eigenvector method as the solution of optimization problems: Fichtner’ metric min max and max min problems of Perron and Frobenius Inefficiency – p. 8/14

Fichtner’ metric Theorem (Fichtner, 1984) Let δ : PCM n × PCM n → R be as follows: � n � � 2 + | λ max ( A ) − λ max ( B ) | � δ ( A , B ) def w EM ( A ) − w EM ( B ) � � = + � i i 2( n − 1) i =1 + χ ( A , B ) | λ max ( A ) + λ max ( B ) − 2 n | , 2( n − 1) where � 0 if A = B , χ ( A , B ) = 1 if A � = B . Then, δ is a metric in PCM n with the following properties: Inefficiency – p. 9/14

Fichtner’ metric (a) for every A ∈ PCM n , X EM ( A ) is the optimal solution of the problem min { δ ( A , X ) | X is consistent } ; (b) min { δ ( A , X ) | X is consistent } = δ ( A , X EM ( A ) ) = λ max ( A ) − n . n − 1 Note that Fichtner’s metric is not continuous. Inefficiency – p. 10/14

Theorem (Perron, Frobenius). Let A ∈ PCM n , and the largest eigenvalue of A be denoted by λ max . Then n n � � a ij w j a ij w j j =1 j =1 max min ≤ λ max ≤ min 1 ≤ i ≤ n max w i w i w ∈ R n w ∈ R n 1 ≤ i ≤ n + + where w = ( w 1 , w 2 , . . . , w n ) . Furthermore, both inequalities hold with equality if and only if w = κ w EM , where κ is an arbitrary positive number. Inefficiency – p. 11/14

Conclusion. Optimality with respect to reasonable and nice objective functions does not exclude inefficiency. Inefficiency – p. 12/14

Main references Bozóki, S. (2014): Inefficient weights from pairwise comparison matrices with arbitrarily small inconsistency, Optimization , 63 (12):1893–1901. Blanquero, R., Carrizosa, E., Conde, E. (2006): Inferring efficient weights from pairwise comparison matrices, Mathematical Methods of Operations Research 64 (2):271–284. Fichtner, J. (1984): Some thoughts about the Mathematics of the Analytic Hierarchy Process, Report 8403, Universität der Bundeswehr München, Fakultät für Informatik, Institut für Angewandte Systemforschung und Operations Research, Werner-Heisenberg-Weg 39, D-8014 Neubiberg, F .R.G. Inefficiency – p. 13/14

Thank you for attention. bozoki.sandor@sztaki.mta.hu http://www.sztaki.mta.hu/ ∼ bozoki Inefficiency – p. 14/14