Pairwise comparison matrices and efficient weight vectors Sndor - PowerPoint PPT Presentation

Pairwise comparison matrices and efficient weight vectors Sándor BOZÓKI Institute for Computer Science and Control Hungarian Academy of Sciences (MTA SZTAKI), Corvinus University of Budapest December 14, 2016 Efficiency – p. 1/32

Multi Criteria Decision Making Analytic Hierarchy Process Criterion tree Pairwise comparison matrix Efficiency – p. 2/32

Criterion tree Efficiency – p. 3/32

Pairwise comparison matrix In practical problems   1 a 12 a 13 . . . a 1 n a 21 1 a 23 . . . a 2 n       a 31 a 32 1 . . . a 3 n A = ,   . . . . ...  . . . .  . . . .     a n 1 a n 2 a n 3 . . . 1 is given, where for any i, j = 1 , . . . , n indices 1 a ij > 0 , a ij = a ji . The aim is to find the w = ( w 1 , w 2 , . . . , w n ) ⊤ ∈ R n + weight vector. Efficiency – p. 4/32

Weighting methods Eigenvector Method (Saaty): Aw = λ max w . Least Squares Method (LSM): n n � 2 � a ij − w i � � min w j i =1 j =1 n � w i = 1 , w i > 0 , i = 1 , 2 , . . . , n. i =1 Logarithmic Least Squares Method (LLSM): n n � 2 � log a ij − log w i � � min w j i =1 j =1 n � w i = 1 , w i > 0 , i = 1 , 2 , . . . , n. i =1 Efficiency – p. 5/32

      1 1 4 9 0 . 404518 0 . 436173       1 1 7 5 0 . 436173 0 . 436173       , w EM = , w ∗ =             1 / 4 1 / 7 1 4 0 . 110295 0 . 110295             1 / 9 1 / 5 1 / 4 1 0 . 049014 0 . 049014   1 0 . 9274 3 . 6676 8 . 2531   1 . 0783 1 3 . 9546 8 . 8989 � w EM �   i =   w EM   0 . 2727 0 . 2529 1 2 . 2503 j     0 . 1212 0 . 1124 0 . 4444 1   1 3 . 9546 8 . 8989 1   1 3 . 9546 8 . 8989 � w ′ � 1   i = .   w ′   0 . 2529 0 . 2529 1 2 . 2503 j     0 . 1124 0 . 1124 0 . 4444 1 Efficiency – p. 6/32

      1 1 4 9 0 . 404518 0 . 436173       1 1 7 5 0 . 436173 0 . 436173       , w EM = , w ∗ =             1 / 4 1 / 7 1 4 0 . 110295 0 . 110295             1 / 9 1 / 5 1 / 4 1 0 . 049014 0 . 049014   1 0 . 9274 3 . 6676 8 . 2531   1 . 0783 1 3 . 9546 8 . 8989 � w EM �   i =   w EM   0 . 2727 0 . 2529 1 2 . 2503 j     0 . 1212 0 . 1124 0 . 4444 1   1 3 . 9546 8 . 8989 1   1 3 . 9546 8 . 8989 � w ′ � 1   i = .   w ′   0 . 2529 0 . 2529 1 2 . 2503 j     0 . 1124 0 . 1124 0 . 4444 1 Efficiency – p. 7/32

      1 1 4 9 0 . 404518 0 . 436173       1 1 7 5 0 . 436173 0 . 436173       , w EM = , w ∗ =             1 / 4 1 / 7 1 4 0 . 110295 0 . 110295             1 / 9 1 / 5 1 / 4 1 0 . 049014 0 . 049014   1 0 . 9274 3 . 6676 8 . 2531   1 . 0783 1 3 . 9546 8 . 8989 � w EM �   i =   w EM   0 . 2727 0 . 2529 1 2 . 2503 j     0 . 1212 0 . 1124 0 . 4444 1   1 3 . 9546 8 . 8989 1   1 3 . 9546 8 . 8989 � w ′ � 1   i = .   w ′   0 . 2529 0 . 2529 1 2 . 2503 j     0 . 1124 0 . 1124 0 . 4444 1 Efficiency – p. 8/32

      1 1 4 9 0 . 404518 0 . 436173       1 1 7 5 0 . 436173 0 . 436173       , w EM = , w ∗ =             1 / 4 1 / 7 1 4 0 . 110295 0 . 110295             1 / 9 1 / 5 1 / 4 1 0 . 049014 0 . 049014   1 0 . 9274 3 . 6676 8 . 2531   1 . 0783 1 3 . 9546 8 . 8989 � w EM �   i =   w EM   0 . 2727 0 . 2529 1 2 . 2503 j     0 . 1212 0 . 1124 0 . 4444 1   1 3 . 9546 8 . 8989 1   1 3 . 9546 8 . 8989 � w ′ � 1   i = .   w ′   0 . 2529 0 . 2529 1 2 . 2503 j     0 . 1124 0 . 1124 0 . 4444 1 Efficiency – p. 9/32

The multi-objective optimization problem is as follows: �� � � � a ij − x i � � min � � x j � i � = j x i > 0 ∀ i Efficiency – p. 10/32

Efficiency (Pareto optimality) Let A = [ a ij ] i,j =1 ,...,n be an n × n pairwise comparison matrix and w = ( w 1 , w 2 , . . . , w n ) ⊤ be a positive weight vector. Definition: weight vector w is called efficient, if there exists n ) ⊤ such that no positive weight vector w ′ = ( w ′ 1 , w ′ 2 , . . . , w ′ � � � a ij − w ′ � � � a ij − w i � � for all 1 ≤ i, j ≤ n, i � � � ≤ � � � � w ′ w j � � � j � a kℓ − w ′ � � � � � a kℓ − w k for some 1 ≤ k, ℓ ≤ n. k � � � � � < � � � � w ′ w ℓ � ℓ Efficiency – p. 11/32

An efficient weight vector cannot be improved such that every element of the pairwise comparison matrix is approximated at least as good, and at least one element is approximated strictly better. Efficiency – p. 12/32

Test of efficiency Given pairwise comparison matrix A and weight vector w , our goal is check whether w is efficient. Efficiency – p. 13/32

Let v i = log w i , 1 ≤ i ≤ n, and b ij = log a ij , 1 ≤ i, j ≤ n, � � � � a ij < w i � I = ( i, j ) � w j � � � � a ij = w i � J = ( i, j ) , i < j � w j � min − s ij ( i,j ) ∈ I for all ( i, j ) ∈ I , y j − y i ≤ − b ij for all ( i, j ) ∈ I , y i − y j + s ij ≤ v i − v j for all ( i, j ) ∈ J , y i − y j = b ij for all ( i, j ) ∈ I , s ij ≥ 0 y 1 = 0 Variables are y i , 1 ≤ i ≤ n and s ij ≥ 0 , ( i, j ) ∈ I. Efficiency – p. 14/32

� min − s ij ( i,j ) ∈ I for all ( i, j ) ∈ I , y j − y i ≤ − b ij for all ( i, j ) ∈ I , y i − y j + s ij ≤ v i − v j for all ( i, j ) ∈ J , y i − y j = b ij for all ( i, j ) ∈ I , s ij ≥ 0 y 1 = 0 Theorem (Bozóki, Fülöp, 2016): The optimum value of the linear program above is at most 0 and it is equal to 0 if and only if weight vector w is efficient. Denote the optimal solution to the LP above by ( y ∗ , s ∗ ) ∈ R n + | I | . If weight vector w is inefficient, then weight vector exp( y ∗ ) is efficient and dominates w internally. Efficiency – p. 15/32

Pairwise Comparison Matrix Calculator The efficiency of a weight vector can be tested at pcmc.online If the weight vector is found to be inefficient, then a dominating efficient weight vector is provided. PCMC deals with incomplete pairwise comparison matrices, too. Efficiency – p. 16/32

Characterization of efficiency Definition: Let A = [ a ij ] i,j =1 ,...,n ∈ PCM n and w = ( w 1 , w 2 , . . . , w n ) ⊤ be a positive weight vector. Directed graph ( V, − → E ) A , w is defined as follows: V = { 1 , 2 , . . . , n } and � − → � w i � arc( i → j ) � E = ≥ a ij , i � = j . � w j � Theorem (Blanquero, Carrizosa and Conde, 2006): Weight vector w is efficient if and only if ( V, − → E ) A , w is strongly connected, that is, there exist directed paths from i to j and from j to i for all pairs of i � = j nodes. Efficiency – p. 17/32

    1 2 6 2 6 . 01438057     1 / 2 1 4 3 4 . 26049429     w EM = A = ,         1 / 6 1 / 4 1 1 / 2 1         1 / 2 1 / 3 2 1 2 . 0712416     1 1 . 41 6 . 01 2 . 90 6 . 01438057     0 . 71 1 4 . 26 2 . 06 4 . 26049429     X EM = w EM =         0 . 1663 0 . 23 1 0 . 48 1         0 . 34 0 . 49 2 . 07 1 2 . 0712416 Efficiency – p. 18/32

Efficiency of the principal right eigenvector Efficiency – p. 19/32

Special cases Efficient principal right eigenvector: simple perturbed PCM double perturbed PCM Inefficient principal right eigenvector: PCM with arbitrarily small inconsistency Numerical examples Efficiency – p. 20/32

      1 1 4 9 0 . 404518 0 . 436173       1 1 7 5 0 . 436173 0 . 436173       , w EM = , w ∗ =             1 / 4 1 / 7 1 4 0 . 110295 0 . 110295             1 / 9 1 / 5 1 / 4 1 0 . 049014 0 . 049014   1 0 . 9274 3 . 6676 8 . 2531   1 . 0783 1 3 . 9546 8 . 8989 � w EM �   i =   w EM   0 . 2727 0 . 2529 1 2 . 2503 j     0 . 1212 0 . 1124 0 . 4444 1   1 3 . 9546 8 . 8989 1   1 3 . 9546 8 . 8989 � w ′ � 1   i = .   w ′   0 . 2529 0 . 2529 1 2 . 2503 j     0 . 1124 0 . 1124 0 . 4444 1 Efficiency – p. 21/32