Balancing Pairwise Comparison Matrices by Transitive Matrices

• Autorzy: Farkas A.

Identyfikatory

DOI

10.3233/FI-2016-1343

• Języki publikacji: EN

Abstrakty

EN

We discuss the development and use of a recursive rank-one residue iteration (triple R-I) to balancing pairwise comparison matrices (PCMs). This class of positive matrices is in the center of interest of a widely used multi-criteria decision making method called analytic hierarchy process (AHP). To find a series of the ‘best’ transitive matrix approximations to the original PCM the Newton-Kantorovich (N-K) method is employed for the solution to the formulated nonlinear problem. Applying a useful choice for the update in the iteration, we show that the matrix balancing problem can be transformed to minimizing the Frobenius norm. Convergence proofs for this scaling algorithm are given. A comprehensive numerical example is included to illustrate the useful features to measuring and reducing perturbation errors and inconsistency of a PCM as a result of the respondents’ judgments on the pairwise comparisons.

Słowa kluczowe

EN

numerical mathematics; matrix balancing; diagonal similarity scaling; pairwise comparison matrix

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2016
• Tom: Vol. 144, nr 3/4
• Strony: 397--417
• Opis fizyczny: Bibliogr. 26 poz.

Twórcy

autor

Farkas A.

• Faculty for Business and Economics, Óbuda University, 1084 Budapest, Tavaszmező út 17, Hungary

Bibliografia

• [1] FarkasA, Rózsa P, Stubnya E. Transitivematrices and their applications. Linear Algebra and Its Applications. 1999; 302-303(1): 423–433. doi:10.1016/S0024-3795(99)00208-6.
• [2] Saaty TL. A scaling method for priorities in hierarchical structures. Journal of Mathematical Psychology. 1977; 15(3): 234–281. doi:10.1016/0022-2496(77)90033-5.
• [3] Osborne EE. On pre-conditioning of matrices. Journal of the Association for Computing Machinery. 1960; 7(4): 338–345. doi:10.1145/321043.321048.
• [4] Hartfiel DJ. Concerning diagonal similarity of irreducible matrices. Proceedings of the American Mathematical Society. 1971; 30(3): 413–425. Available from: http://www.jstor.org/stable/2037708. doi:10.2307/2037708.
• [5] Brualdi RA, Parter SV, Schneider H. The diagonal equivalence of a nonnegative matrix to a stochastic matrix. Journal of Mathematical Analysis and Applications. 1966; 16(1): 31–50. doi:10.1016/0022-247X(66)90184-3.
• [6] Saunders BD, Schneider MH. Flows on graphs and applied to diagonal similarity and diagonal equivalence for matrices. Discrete Mathematics. 1978; 24(2): 205–220. doi: 101016/0012-365X(78)90200-5.
• [7] Eaves BC, Hoffman AJ, Rothblum UG, Schneider H. Line-sum-symmetric scaling of square nonnegative matrices. Mathematical Programming Studies. 1985; 25: 124–141. doi:10.1007/BFb0121080.
• [8] Schneider MH, Zenios SA. A comparative study of algorithms for matrix balancing. Operations Research. 1990; 38(3): 439–455. doi:10.1287/opre.38.3.439.
• [9] Chen TY. Preconditioning sparse matrices for computing eigenvalues and solving linear systems of equations PhD thesis. University of California at Berkeley. 2001. Available from http://www.cs.pomona.edu/~tzuyi/Papers/phd_thesis.
• [10] Genma K, Kato Y, Sekitani K. Matrix balancing problemand binary AHP. Journal of the Operations Research Society of Japan. 2007; 50(4): 515–539. ISSN:0453-4514.
• [11] Jensen RE. Comparisons of consensus methods for priority ranking problems. Decision Sciences. 1986; 17(2):195–211. doi:10.1111/j.1540-5915.1986.tb00221.x.
• [12] Takahashi I. AHP applied to binary and ternary comparisons. Journal of the Operations Research Society of Japan. 1990; 33(3): 199–206.
• [13] Kalantari B, Khachiyan L, Shokoufandeh A. On the complexity of matrix balancing. SIAM Journal on Matrix Analysis and Applications. 1997; 18(2): 450–463. doi:10.1137/S0895479895289765.
• [14] Jensen RE. An alternative scaling method for priorities in hierarchical structures. Journal of Mathematical Psychology. 1984; 28(3): 317–332. doi:10.1016/0022-2496(84)90003-8.
• [15] Chu ATW, Kalaba RE, Springarn K. A comparison of two methods for determining the weights of belonging to fuzzy sets. Journal of Optimization Theory and Applications. 1979; 27(4): 531–538. doi:10.100/BF00933438.
• [16] Blankmeyer E. Approaches to consistency adjustments. Journal of Optimization Theory and Applications. 1987; 54(3): 479–488. doi:10007/BF00940197.
• [17] Bozóki S, Lewis RH. Solving the least-squares problem in the AHP for 3 × 3 and 4 × 4 matrices. Central European Journal of Operations Research. 2005; 13(3): 255–270. Available from: http://www.ams.org/mathscinet-getitem?mr=2169885
• [18] Fülöp J. A method for approximating pairwise comparison matrices by consistent matrices. Journal of Global Optimization. 2008; 42(3): 423–442. doi:10.1007/S10898-008-9303-0.
• [19] Farkas A, Lancaster P, Rózsa P. Consistency adjustments for pairwise comparison matrices. Numerical Linear Algebra with Applications. 2003; 10(8): 689–700. doi:10.1002/nla.318.
• [20] Kantorovich LV. On Newton’s method for functional equations. Doklady Akademii Nauk SSSR. 1948; 59: 1237–1240.
• [21] Nesterov Y, Polyak BT. Cubic regularization of a Newton scheme and its global performance. CORE Discussion Papers. 2003; 41:17. Available from: http://www.core.ucl.ac.be/services/psfiles/dp03/dp2003-41.pdf.
• [22] Polyak BT. Newton-Kantorovich method and its global convergence. Journal of Mathematical Sciences. 2006; 133(4): 1513–1523. doi:10.1007/s10958-006-0066-1.
• [23] Farkas A. Recursive Least-Squares Algorithm for SR Matrices in Mathematica Code. 2012;p. 6. Computer Program, ´Obuda University, Budapest 2012. Available from: http://www.kgk.uni-obuda.hu/farkas_andras.
• [24] Kalmár L. On Cauchy’s convergence test. Acta Mathematica Academiae Scientiarum Hungaricae. 1950; 1(1): 109–112. ISSN:1588-2632. doi:10.1007/BF02022556.
• [25] Farkas A, Rözsa P. A recursive least-squares algorithm for pairwise comparison matrices. Central European Journal of Operations Research. 2013; 21(4): 817–843. doi:10.1007/s10100-012-0262-7.
• [26] Johnson CR, Beine WB, Wang TJ. Right-left asymmetry in an eigenvector ranking procedure. Journal of Mathematical Psychology. 1979; 19(1): 61–64. doi:10.1016/0022-2496(79)90005-1.