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Minimization of the Perron eigenvalue of incomplete pairwise comparison matrices by Newton iteration. (English) Zbl 1349.91092
Summary: Pairwise comparison matrices are of key importance in multi-attribute decision analysis. A matrix is incomplete if some of the elements are missing. The eigenvector method, to derive the weights of criteria, can be generalized for the incomplete case by using the least inconsistent completion of the matrix. If inconsistency is indexed by CR, defined by Saaty, it leads to the minimization of the Perron eigenvalue. This problem can be transformed to a convex optimization problem. The paper presents a method based on the Newton iteration, univariate and multivariate. Numerical examples are also given.
MSC:
91B06 Decision theory
49M15 Newton-type methods
90B50 Management decision making, including multiple objectives
Software:
BRENT; Matlab; fminbnd
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References:
[1] K. Ábele-Nagy, Incomplete pairwise comparison matrices in multi-attribute decision making (In Hungarian, Nem teljesen kitöltött páros összehasonlítás mátrixok a többszempontú döntésekben), Master’s Thesis, Eötvös Loránd University, Budapest, 2010. ⇒61
[2] S. Bozóki, J. Fülöp, L. Rónyai, On optimal completion of incomplete pairwise comparison matrices, Math. Comput. Modelling52, 1-2, (2010) 318-333. ⇒60, 61, 62, 63, 66 · Zbl 1201.15012
[3] R. P. Brent, Algorithms for Minimization without Derivatives, Prentice-Hall, Englewood Cliffs, NJ, 1973. ⇒63 · Zbl 0245.65032
[4] M. Brunelli, Introduction to the Analytic Hierarchy Process, SpringerBriefs in Operations Research, Springer, New York, 2015. ⇒59 · Zbl 1309.91006
[5] G. E. Forsythe, M. A. Malcolm, C. B. Moler, Computer Methods for Mathematical Computations, Prentice-Hall, Englewood Cliffs, NJ, 1976. ⇒63 · Zbl 0361.65002
[6] J. Fülöp, An optimization approach for the eigenvalue method, IEEE 8th International Symposium on Applied Computational Intelligence and Informatics (SACI 2013), Timişoara, Romania, 23-25 May 2013. ⇒69
[7] P. T. Harker, Alternative modes of questioning in the Analytic Hierarchy Process. Math. Model.9, 3 (1987) 353-360. ⇒59 · Zbl 0626.90001
[8] P. T. Harker, Derivatives of the Perron root of a positive reciprocal matrix: with application to the Analytic Hierarchy Process. Appl. Math. Comput.22, 2-3 (1987) 217-232. ⇒61, 63, 65, 70 · Zbl 0619.15017
[9] P. T. Harker, Incomplete pairwise comparisons in the Analytic Hierarchy Process. Math. Model.9, 11 (1987) 837-848. ⇒59
[10] A. Ishizaka, M. Lusti, How to derive priorities in AHP: a comparative study. Cent. Eur. J. Oper. Res., 14, 4 (2006) 387-400. ⇒62 · Zbl 1122.90367
[11] M. Kwiesielewicz, The logarithmic least squares and the generalised pseudoinverse in estimating ratios. European J. Oper. Res.93, 3 (1996) 611-619. ⇒ 62 · Zbl 0912.90005
[12] D. G. Luenberger, Y. Ye, Linear and Nonlinear Programming (3rd Edition), Series: International Series in Operations Research & Management Science, 116, Springer, New York, 2008. ⇒62 · Zbl 1207.90003
[13] Matlab Documentation: fminbnd . ⇒63
[14] T. L. Saaty, A scaling method for priorities in hierarchical structures. J. Math. Psych., 15, 3 (1977) 234-281. ⇒59 · Zbl 0372.62084
[15] T. L. Saaty, The Analytic Hierarchy Process, McGraw-Hill, New York, 1980. ⇒59, 66 · Zbl 0587.90002
[16] S. Shiraishi, T. Obata, M. Daigo, Properties of a positive reciprocal matrix and their application to AHP. J. Oper. Res. Soc. Jpn.41, 3 (1998) 404-414. ⇒60 · Zbl 1003.15019
[17] S. Shiraishi, T. Obata, On a maximization problem arising from a positive reciprocal matrix in AHP. Bull. Inform. Cybernet.34, 2 (2002) 91-96. ⇒60 · Zbl 1270.91019
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