Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices

Transcription

1 Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices Kristóf Ábele-Nagy Eötvös Loránd University (ELTE); Corvinus University of Budapest (BCE) Sándor Bozóki Computer and Automation Research Institute, Hungarian Academy of Sciences (MTA SZTAKI); Corvinus University of Budapest (BCE) 15 December, 2010 Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 1/39

2 Outline Pairwise comparison matrix Incomplete pairwise comparison matrix Eigenvalue optimization Cyclic coordinates Newton s method in one variable Newton s method in higher dimensions Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 2/39

3 Given n objects with weights w 1,w 2,w 3,...,w n. The pairwise comparison matrix is defined as follows: w 1 1 w 1 w w 2 w w n w 2 w w w w w n w 3 w 3 w w 1 w w n, w n w where w n w 1 for any i,j,k indices. w n w 2 w ij > 0, w ij = 1 w ji, w ij = w ik w kj. Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 3/39

4 In real decision situations, weights are unknown, but pairwise comparisons can be made: 1 a 12 a a 1n a 21 1 a a 2n A = a 31 a a 3n, a n1 a n2 a n where a ij > 0, a ij = 1 a ji. for i,j = 1,...,n. The aim is to determine the weight vector w = (w 1,w 2,...,w n ) R n +. Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 4/39

5 In the Eigenvector Method (EM) the approximation w EM of w is defined by Aw EM = λ max w EM, where λ max denotes the maximal eigenvalue, also known as Perron eigenvalue, of A and w EM denotes the the right-hand side eigenvector of A corresponding to λ max. By Perron s theorem, w EM is positive and unique up to a scalar multiplication. The most often used normalization is n wi EM = 1. i=1 Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 5/39

6 Saaty defined the inconsistency ratio as CR = λ max n n 1 RI n, where λ max is the Perron eigenvalue of the complete pairwise comparison matrix given by the decision maker, and RI n is defined as λ max n n 1, where λ max is an average value of the Perron eigenvalues of randomly generated n n pairwise comparison matrices. It is well known that λ max n and equals to n if and only if the matrix is consistent, i.e., the transitivity property holds. It follows from the definition that CR is a positive linear transformation of λ max. According to Saaty, larger value of CR indicates higher level of inconsistency and the 10%-rule (CR 0.10) separates acceptable matrices from unacceptable ones. Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 6/39

7 Incomplete pairwise comparison matrix ( = pairwise comparison matrix with missing elements) A = 1 a a 1n 1/a 12 1 a /a a 3n /a 1n 1/a 3n Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 7/39

8 Incomplete pairwise comparison matrix ( = pairwise comparison matrix with missing elements) A = 1 a 12 x 1... a 1n 1/a 12 1 a /x 1 1/a a 3n /a 1n 1/a 3n Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 8/39

9 Incomplete pairwise comparison matrix ( = pairwise comparison matrix with missing elements) A = 1 a 12 x 1... a 1n 1/a 12 1 a x d 1/x 1 1/a a 3n /a 1n 1/x d 1/a 3n... 1, Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 9/39

10 Incomplete pairwise comparison matrix ( = pairwise comparison matrix with missing elements) A = 1 a 12 x 1... a 1n 1/a 12 1 a x d 1/x 1 1/a a 3n /a 1n 1/x d 1/a 3n... 1, where x 1,x 2,...,x d R +. Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 10/39

11 Based on the idea above, Shiraishi, Obata and Daigo considered the eigenvalue optimization problems as follows. In case of one missing element, denoted by x, the λ max (A(x)) to be minimized: min x>0 λ max(a(x)). In case of more than one missing elements, arranged in vector x, the aim is to solve min λ max(a(x)). x>0 Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 11/39

12 Graph representation of a pairwise comparison matrix Given A incomplete pairwise comparison matrix of size n n. Graph G = (V,E) is defined as follows: V = {1, 2,...,n} E = {e(i,j) a ij (and a ji ) are given and i j} Special case: all the comparisons are given, the corresponding graph is K n. Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 12/39

13 Graph representation of a pairwise comparison matrix Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 13/39

14 Theorem (B., Fülöp, Rónyai, 2010): The optimal solution of the eigenvalue minimization problem min λ max(a(x)). x>0 is unique if and only if the graph G corresponding to the incomplete pairwise comparison matrix is connected. If graph G corresponding to the incomplete pairwise comparison matrix is connected, then by using the exponential parametrization x 1 = e y 1,x 2 = e y 2,...x d = e y d, the eigenvalue minimization problem is transformed into a strictly convex optimization problem. Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 14/39

15 Example Q = 1 2 x 1/ /x 1/4 1 λ max (Q(x)) and, by using the exponential scaling x = e t, λ max (Q(e t )) are plotted. Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 15/39

16 Algorithms for solving the eigenvalue minimization problem min x>0 λ max(a(x)). cyclic coordinates with Matlab s function fminbnd cyclic coordinates univariate Newton s method multivariate Newton s method Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 16/39

17 Method of cyclic coordinates M(x) = /3 1/4 1/5 1 x 1 5 x 2 3 x 3 1/7 1/3 1/x 1 1 x 4 3 x 5 6 x 6 1/7 1/5 1/x 4 1 x 7 1/4 x 8 1/8 1/6 1/x 2 1/3 1/x 7 1 x 9 1/5 x 10 1/6 1/3 1/x 5 4 1/x 9 1 x 11 1/6 3 1/x 3 1/6 1/x 8 5 1/x 11 1 x /x 6 8 1/x /x 12 1 Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 17/39

18 Method of cyclic coordinates Let x (k) i denote the value of x i in the k-th step of the iteration, which has d (in the example, d = 12) substeps for each k. For k = 0 : Let the initial points be equal to 1 for every variable: while x (0) i := 1 (i = 1, 2,...,d). max i=1,2,...,d xk i xk 1 i > T x (k) i := arg min λ max (M(x (k) x 1,...,x(k) i i 1,x i,x (k 1) i+1,...,x (k 1) d )), i = 1, 2,...,d next k end while Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 18/39

19 Method of cyclic coordinates Focus on min x i λ max (M(x (k) 1,...,x(k) i 1,x i,x (k 1) i+1,...,x (k 1) d )) Matlab s function fminbnd solves is directly and fast. Univariate Newton s method can be also applied. Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 19/39

20 min x>0 λ max(a(x)) Let x = e t and L(t) = λ max (e t ). t n+1 = t n L (t n ) L (t n ) = t n 2 λ max (x) ( x) 2 λ max (x) x e t n + λ max (x) x. By Harker, formal derivatives λ max(x) x known. and 2 λ max (x) ( x) 2 are Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 20/39

21 Harker s formula for the first derivative λ max(x) x ( ) λmax (A) i > j = a ij ( [y(a) i x(a) j ] [y(a) ) jx(a) i ] [a ij ] 2 where vectors x(a), y(a) are the right-hand side and left-hand side eigenvectors of A, respectively. Normalization y(a) T x(a) = 1 is applied. Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 21/39

22 Harker s formula for the second derivative 2 λ max (A) a ij a kl = (x(a)y(a) T ) li Q + jk + (x(a)y(a)t ) jk Q + li if i k or j l, (x(a)y(a)t ) ki Q + jl + (x(a)y(a)t ) jl Q + ki [a kl ] 2 (x(a)y(a)t ) lj Q + ik + (x(a)y(a)t ) ik Q + lj [a ij ] 2 + (x(a)y(a)t ) kl Q + il + (x(a)y(a)t ) il Q + kj [a ij ] 2 [a kl ] 2 Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 22/39

23 Harker s formula for the second derivative 2 λ max (A) a ij a kl = 2(x(A)y(A)T ) ij [a ij ] 3 + 2(x(A)y(A) T ) ji Q + ii 2 (x(a)y(a)t ) ii Q + jj + (x(a)y(a)t ) jj Q + ii [a ij ] 2 +2 (x(a)y(a)t ) ij Q + ij [a ij ] 4 if i = k and j = l, where Q = λ max (A)I A and Q + denotes the Moore-Penrose pseudoinverse of Q. Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 23/39

24 Multivariate Newton s method t n+1 = t n [HL(t n )] 1 L(t n ). Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 24/39

25 Computational results: All methods mentioned above are fast enough for typical matrices in multi-attribute decision making; in the talk s example (8 8 matrix, 12 variables, 20 cycles with f minbnd: 0.3 seconds, 1-variable Newton: 5.3 seconds); as a test, in a matrix, variables, 1 cycle takes 1.5 hours with fminbnd. Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 25/39

26 Applications of the results: A generalization of the Eigenvector Method for the incomplete case CR-inconsistency can be computed during the filling in process, as soon as a connected graph is given User may get an automatic warning in case of misprints, detected as a high jump in CR-inconsistency. Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 26/39

27 Questions: How many comparisons are needed, if less than n(n 1)/2? Thresholds for warning the user? Other inconsistency indices, presented by Attila Poesz on Monday (session 2A). Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 27/39

28 References 1/2 Ábele-Nagy, K. [2010]: Incomplete pairwise comparison matrices in multi-attribute decision making, Master s Thesis, Eötvös Loránd University. Bozóki, S., Fülöp, J., Rónyai, L. [2010]: On optimal completions of incomplete pairwise comparison matrices, Mathematical and Computer Modelling, 52, pp Harker, P.T. [1987]: Incomplete pairwise comparisons in the analytic hierarchy process. Mathematical Modelling, 9(11), pp Harker, P.T. [1987]: Derivatives of the Perron root of a positive reciprocal matrix: with application to the Analytic Hierarchy Process. Applied Mathematics and Computation, 22, pp Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 28/39

29 References 2/2 Saaty, T.L. [1977]: A scaling method for priorities in hierarchical structures, Journal of Mathematical Psychology, 15, pp Saaty, T.L. [1980]: The analytic hierarchy process, McGraw-Hill, New York. Shiraishi, S., Obata, T., Daigo, M. [1998]: Properties of a positive reciprocal matrix and their application to AHP. Journal of the Operations Research Society of Japan, 41(3), pp Shiraishi, S., Obata, T. [2002]: On a maximization problem arising from a positive reciprocal matrix in AHP. Bulletin of Informatics and Cybernetics, 34(2), pp Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 29/39

30 Thank you for attention. bozoki Newton s method in eigenvalue optimization for incomplete pairwise comparison matrices p. 30/39

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