Ranking Alternatives by Pairwise Comparisons Matrix and Priority Vector


  • Jaroslav Ramík




pairwise comparisons matrix, priority vector, ranking alternatives, analytic hierarchy process, AHP


The decision making problem considered here is to rank n alternatives from the best to the worst, using information given by the decision maker(s) in the form of an n×n pairwise comparisons (PC) matrix. We investigate pairwise comparisons matrices with elements from a real interval which is a traditional multiplicative approach used in Analytic hierarchy process (AHP). Here, we deal with two essential elements of AHP: measuring consistency of PC matrix and the method of eliciting the priority vector by which the final ranking of alternatives is derived. Classical approaches introduced by T. Saaty in AHP are compared with later approaches based on the AHP criticism published in the literature. Advantages and disadvantages of both approaches are highlighted and discussed.

JEL Codes - C44


Bana e Costa, C. A., and Vansnick, J. C., 2008. A critical analysis of the eigenvalue method used to derive priorities in the AHP. European Journal of Operational Research, 187(3), 1422-1428. doi: http://dx.doi.org/10.1016/j.ejor.2006.09.022

Barzilai, J., 1997. Deriving weights from pairwise comparison matrices. The Journal of the Operational Research Society, 48(12), 1226-1232. doi: http://dx.doi.org/10.1057/palgrave.jors.2600474

Barzilai, J., 1998. Consistency Measures for Pairwise Comparison Matrice. J. Multi-Crit. Decision Analysis, 7(1), 123-132.

Gavalec, M., Ramik, J., and Zimmermann, K., 2014. Decision making and Optimization - Special Matrices and Their Applications in Economics and Management. Switzerland, Cham-Heidelberg-New York-Dordrecht-London: Springer Internat. Publ.

Greco, S., Ehrgott, M., and Figueira, J. R., 2016. Multiple Criteria Decision Making. Heidelberg, New York, Dordrecht, London: Springer.

Koczkodaj, W. W., 1993. A new definition of consistency of pairwise comparisons. Mathematical and Computer Modelling, 18(7), 79-84. doi: http://dx.doi.org/10.1016/0895-7177(93)90059-8

Ramik, J., and Perzina, R., 2015. Educational Microsoft Excel Add-ins Solving Multicriteria Decision Making Problems. Paper presented at the CSEDU 2015 - 7th International Conference on Computer Supported Education.

Saaty, T. L., 1977. A Scaling Method for Priorities in Hierarchical Structure. Journal of Mathematical Psychology, 15(3), 234-281. doi: http://dx.doi.org/10.1016/0022-2496(77)90033-5

Saaty, T. L., 1991. Multicriteria decision making - the Analytical Hierarchy Process. Pittsburgh: RWS Publications.

Saaty, T. L., and Vargas, L. G., 1984. Comparison of eigenvalue, logarithmic least squares and least squares methods in estimating ratios. Mathematical Modelling, 5(5), 309-324. doi: http://dx.doi.org/10.1016/0270-0255(84)90008-3

Saaty, T. L., Vargas, L. G., and Whitaker, R., 2009. Addressing Criticisms of the AHP. International Journal of the Analytic Hierarchy Process, 1(1), 121-134.

Thurstone, E. L. L., 1927. A Law of Comparative Judgments. Psychological Review, 34(4), 273-286. doi: http://dx.doi.org/10.1037/h0070288

Whitaker, R., 2007. Criticisms of the Analytic Hierarchy Process: Why they often make no sense. Mathematical and Computer Modelling, 46(7-8), 948-961. doi: http://dx.doi.org/10.1016/j.mcm.2007.03.016




How to Cite

Ramík, J. (2017). Ranking Alternatives by Pairwise Comparisons Matrix and Priority Vector. Scientific Annals of Economics and Business, 64, 85–95. https://doi.org/10.1515/saeb-2017-0040