# zbMATH — the first resource for mathematics

Singular value decomposition in AHP. (English) Zbl 1146.90445
Summary: The analytic hierarchy process (AHP) has been accepted as a leading multiattribute decision-aiding model both by practitioners and academics. The foundation of the AHP is the Saaty’s eigenvector method (EM) and associated inconsistency index that are based on the largest eigenvalue and associated eigenvector of an ($$n \times n$$) positive reciprocal matrix. The elements of the matrix are the decision maker’s (DM) numerical estimates of the preference of $$n$$ alternatives with respect to a criterion when they are compared pairwise using the 1–9 AHP fundamental comparison scale. The components of the normalized eigenvector provide approximations of the unknown weights of the criteria (alternatives), and the deviation of the largest eigenvector from $$n$$ yields a measure of how inconsistent the DM is with respect to the pairwise comparisons.
Singular value decomposition (SVD) is an important tool of matrix algebra that has been applied to a number of areas, e.g., principal component analysis, canonical correlation in statistics, the determination of the Moore–Penrose generalized inverse, and low rank approximation of matrices. In this paper, using the SVD and the theory of low rank approximation of a (pairwise comparison) matrix, we offer a new approach for determining the associated weights. We prove that the rank one left and right singular vectors, that is the vectors associated with the largest singular value, yield theoretically justified weights. We suggest that an inconsistency measure for these weights is the Frobenius norm of the difference between the original pairwise comparison matrix and one formed by the SVD determined weights. How this measure can be applied in practice as a means of measuring the confidence the DM should place in the SVD weights is still an open question. We illustrate the SVD approach and compare it to the EM for some numerical examples.

##### MSC:
 90B50 Management decision making, including multiple objectives
Full Text:
##### References:
 [1] Barzilai, J.; Cook, W.D.; Golany, B., Consistent weights for judgements matrices of the relative importance of alternatives, Operations research letters, 6, 131-134, (1987) · Zbl 0622.90004 [2] Barzilai, J., Deriving weights from pairwise comparison matrices, Journal of the operational research society, 48, 1226-1232, (1997) · Zbl 0895.90004 [3] Basak, I., The categorical data analysis approach for ratio model of pairwise comparisons, European journal of operational research, 128, 532-544, (2001) · Zbl 0983.90003 [4] Blankmeyer, E., Approaches to consistency adjustment, Journal of optimization theory and applications, 3, 479-488, (1987) · Zbl 0597.90049 [5] Bryson, N., A goal programming method for generating priority vectors, Journal of the operational research society, 46, 641-648, (1995) · Zbl 0830.90001 [6] Chu, A.T.W.; Kalaba, R.E.; Spingarn, K., A comparison of two methods for determining the weights of belonging to fuzzy sets, Journal of optimization theory and applications, 4, 531-538, (1979) · Zbl 0377.94002 [7] Crawford, G.; Williams, C., A note on the analysis of subjective judgment matrices, Journal of mathematical psychology, 29, 387-405, (1985) · Zbl 0585.62183 [8] De Jong, P., A statistical approach to saaty’s scaling method for priorities, Journal of mathematical psychology, 28, 467-478, (1984) · Zbl 0564.62089 [9] Donegan, H.A.; Dodd, F.J.; McMaster, T.B.M., A new approach to AHP decision-making, The Statistician, 41, 295-302, (1992) [10] Eckart, C.; Young, G., The approximation of one matrix by another of lower rank, Psychometrika, 1, 211-218, (1936) · JFM 62.1075.02 [11] Gass, S.I.; Rapcsák, T., A note on synthesizing group decisions, Decision support systems, 22, 59-63, (1998) [12] Golany, B.; Kress, M., A multicriteria evaluation of methods for obtaining weights from ratio-scale matrices, European journal of operational research, 69, 210-220, (1993) · Zbl 0800.90007 [13] Greenacre, M.J., Theory and applications of correspondence analysis, (1984), Academic Press London, Orlando · Zbl 0726.62087 [14] R.E. Jensen, Comparisons of eigenvector, least squares, chi square and logarithmic least squares methods of scaling a reciprocal matrix, Trinity University, Working Paper No. 127, 1984 [15] Jensen, R.E., An alternative scaling method for priorities in hierarchical structures, Journal of mathematical psychology, 28, 317-332, (1984/2) [16] Johnson, C.R.; Beine, W.B.; Wang, T.J., Right-left asymmetry in an eigenvector ranking procedure, Journal of mathematical psychology, 19, 61-64, (1979) [17] Kennedy, W.J.; Gentle, J.E., Statistical computing, (1980), Marcel Dekker New York, Basel · Zbl 0435.62003 [18] Mészáros, Cs.; Rapcsák, T., On sensitivity analysis for a class of decision systems, Decision support systems, 16, 231-240, (1996) [19] Mikhailov, L., A fuzzy programming method for deriving priorities in the analytic hierarchy process, Journal of the operational research society, 51, 341-349, (2000) · Zbl 1055.90560 [20] Mitrinović, D.S., Analytic inequalities, (1970), Springer-Verlag Berlin, Heidelberg, NewYork · Zbl 0199.38101 [21] Monsuur, H., An intrinsic consistency threshold for reciprocal matrices, European journal of operational research, 96, 387-391, (1996) · Zbl 0917.90012 [22] Saaty, T.L., The analytic hierarchy process, (1980), McGraw-Hill New York · Zbl 1176.90315 [23] Saaty, T.L., Eigenvector and logarithmic least squares, European journal of operational research, 48, 156-160, (1990) · Zbl 0707.90003 [24] Saaty, T.L., The analytic hierarchy process: A 1993 overview, Cejore, 2, 119-137, (1993) · Zbl 0824.90006 [25] Saaty, T.L.; Vargas, L.G., Inconsistency and rank preservation, Journal of mathematical psychology, 28, 205-214, (1984/1) · Zbl 0557.62093 [26] Saaty, T.L.; Vargas, L.G., Comparison of eigenvalue, logarithmic least squares, and least squares methods in estimating ratios, Mathematical modelling, 5, 309-324, (1984/2) · Zbl 0584.62102 [27] Schweitzer, P., An inequality concerning the arithmetic Mean, Mathematikai és physikai lapok, 23, 257-261, (1914) [28] S.M. Standard, Analysis of positive reciprocal matrices, Master of Arts Thesis, University of Maryland, 2000 [29] Stewart, G.W., Introduction to matrix computations, (1973), Academic Press New York · Zbl 0302.65021 [30] Takeda, E.; Cogger, K.; Yu, P.L., Estimating criterion weights using eigenvectors: A comparative study, European journal of operational research, 29, 360-369, (1987) · Zbl 0618.90046 [31] Tung, S.L.; Tang, S.L., A comparison of the saaty’s AHP and modified AHP for right and left eigenvector inconsistency, European journal of operational research, 106, 123-128, (1998) [32] Xu, Z.; Wei, C., A consistency improving method in the analytic hierarchy process, European journal of operational research, 116, 443-449, (1999) · Zbl 1009.90513 [33] Zahir, S., Geometry of decision making and the vector space formulation of the analytic hierarchy process, European journal of operational research, 112, 373-396, (1999) · Zbl 0946.91003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.