Dynamic thresholds of geometric consistency index associated with pairwise comparison matrix
Abstract
Pairwise comparison matrix (PCM) has been widely employed in the multi-criteria decision-making (MCDM) problems to rank the criteria and alternatives according to the considered criteria in Analytic Hierarchy Process (AHP). The PCM should have the acceptable consistency before deriving a priority vector from it. Approximate thresholds of geometric consistency index (GCI) and consistency ratio (CR) have been proposed to test whether the PCM has the acceptable consistency. However, approximate thresholds of GCI and CR always suffer from some criticisms and disagreements in existing literature. In this paper, we try to induce dynamic thresholds of GCI by combining hypothesis testing and random index (RI), which vary with the order of the PCM, significance level and assessment level of decision maker. The induced dynamic thresholds of GCI may explain different (or conflicting) results obtained by approximate thresholds of GCI and CR and avoid the unnecessary revisions of some judgments of the PCM for the desired consistency. Finally, several numerical examples and real-world decision-making problems are examined and compared with existing decision-making methods to illustrate the performance of dynamic thresholds of GCI.
Keyword : analytic hierarchy process, pairwise comparison matrix, geometric consistency index, dynamic thresholds
How to Cite
Lin, C., Kou, G., Peng, Y., & Hefni, M. A. (2022). Dynamic thresholds of geometric consistency index associated with pairwise comparison matrix. Technological and Economic Development of Economy, 28(4), 1137–1157. https://doi.org/10.3846/tede.2022.16544
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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Amenta, P., Lucadamo, A., & Marcarelli, G. (2020). On the transitivity and consistency approximate thresholds of some consistency indices for pairwise comparison matrices. Information Sciences, 507, 274–287. https://doi.org/10.1016/j.ins.2019.08.042
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Barzilai, J., & Golany, B. (1994). AHP rank reversal, normalization and aggregation rules. INFOR: Information Systems and Operational Research, 32(2), 57–63. https://doi.org/10.1080/03155986.1994.11732238
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Benítez, J., Delgado-Galván, X., Izquierdo, J., & Pérez-García, R. (2011). Achieving matrix consistency in AHP through linearization. Applied Mathematical Modelling, 35(9), 4449–4457. https://doi.org/10.1016/j.apm.2011.03.013
Bernardo, J. M., & Smith, A. F. M. (1994). Bayesian theory. Wiley. https://doi.org/10.1002/9780470316870
Bozóki S., & Rapcsák, T. (2008). On Saaty’s and Koczkodaj’s inconsistencies of pairwise comparison matrices. Journal of Global Optimization, 42(2), 157–175. https://doi.org/10.1007/s10898-007-9236-z
Bozóki, S., Dezső, L., Poesz, A., & Temesi, J. (2013). Analysis of pairwise comparison matrices: an empirical research. Annals of Operations Research, 211(1), 511–528. https://doi.org/10.1007/s10479-013-1328-1
Brugha, C. (1996). The structure of qualitative decision-making: Implications for the analytical hierarchy process. In The International Symposium on the Analytic Hierarchy Process (pp. 190–201). https://doi.org/10.13033/isahp.y1996.076
Brugha, C. M. (2004). Structure of multi-criteria decision-making. Journal of the Operational Research Society, 55(11), 1156–1168. https://doi.org/10.1057/palgrave.jors.2601777
Brunelli, M. (2016). A technical note on two inconsistency indices for preference relations: A case of functional relation. Information Sciences, 357, 1–5. https://doi.org/10.1016/j.ins.2016.03.048
Brunelli, M. (2017). Studying a set of properties of inconsistency indices for pairwise comparisons. Annals of Operations Research, 248(1), 143–161. https://doi.org/10.1007/s10479-016-2166-8
Brunelli, M. (2018). A survey of inconsistency indices for pairwise comparisons. International Journal of General Systems, 47(8), 751–771. https://doi.org/10.1080/03081079.2018.1523156
Brunelli, M., & Fedrizzi, M. (2015). Axiomatic properties of inconsistency indices for pairwise comparisons. Journal of the Operational Research Society, 66(1), 1–15. https://doi.org/10.1057/jors.2013.135
Cavallo, B. (2020). Functional relations and Spearman correlation between consistency indices. Journal of the Operational Research Society, 71(2), 301–311. https://doi.org/10.1080/01605682.2018.1516178
Cavallo, B., & D’Apuzzo, L. (2009). A general unified framework for pairwise comparison matrices in multicriterial methods. International Journal of Intelligent Systems, 24(4), 377–398. https://doi.org/10.1002/int.20329
Cavallo, B., & D’Apuzzo, L. (2010). Characterizations of consistent pairwise comparison matrices over abelian linearly ordered groups. International Journal of Intelligent Systems, 25(10), 1035–1059. https://doi.org/10.1002/int.20438
Crawford, G., & Williams, C. (1985). A note on the analysis of subjective judgment matrices. Journal of Mathematical Psychology, 29(4), 387–405. https://doi.org/10.1016/0022-2496(85)90002-1
Csató, L. (2018a). Characterization of an inconsistency ranking for pairwise comparison matrices. Annals of Operations Research, 261, 155–165. https://doi.org/10.1007/s10479-017-2627-8
Csató, L. (2018b). Characterization of the row geometric mean ranking with a group consensus axiom. Group Decision and Negotiation, 27, 1011–1027. https://doi.org/10.1007/s10726-018-9589-3
Csató, L. (2019a). Axiomatizations of inconsistency indices for triads. Annals of Operations Research, 280(1–2), 99–110. https://doi.org/10.1007/s10479-019-03312-0
Csató, L. (2019b). A characterization of the Logarithmic Least Squares Method. European Journal of Operational Research, 276(1), 212–216. https://doi.org/10.1016/j.ejor.2018.12.046
Csató, L., & Petróczy, D. G. (2020). On the monotonicity of the eigenvector method. European Journal of Operational Research, 292(1), 230–237. https://doi.org/10.1016/j.ejor.2020.10.020
Dadkhah, K. M., & Zahedi, F. (1993). A mathematical treatment of inconsistency in the analytic hierarchy process. Mathematical and Computer Modelling, 17(4), 111–122. https://doi.org/10.1016/0895-7177(93)90180-7
Devore, T. L. (2000). Probability and statistics. Thomson Learning.
Dixit, P. D. (2018). Entropy production rate as a criterion for inconsistency in decision theory. Journal of Statistical Mechanics: Theory and Experiment, 2018(5), 053408. https://doi.org/10.1088/1742-5468/aac137
Duszak, Z., & Koczkodaj, W. (1994). Generalization of a new definition of consistency for pairwise comparisons. Information Processing Letters, 52(5), 273–276. https://doi.org/10.1016/0020-0190(94)00155-3
Escobar, M. T., & Moreno-Jiménez, J. M. (2000). Reciprocal distributions in the analytic hierarchy process. European Journal of Operational Research, 123(1), 154–174. https://doi.org/10.1016/S0377-2217(99)00086-7
Farley, M., & Trow, S. (2005). Losses in water distribution networks. A practitioner’s guide to assessment, monitoring and control. IWA Publishing. https://doi.org/10.2166/9781780402642
Fedrizzi, M., & Ferrari, F. (2018). A chi-square-based inconsistency index for pairwise comparison matrices. Journal of the Operational Research Society, 69(7), 1125–1134. https://doi.org/10.1080/01605682.2017.1390523
Fedrizzi, M., & Giove, S. (2007). Incomplete pairwise comparisons and consistency optimization. European Journal of Operational Research, 183(1), 303–313. https://doi.org/10.1016/j.ejor.2006.09.065
Gass, S. I., & Rapcsák, T. (2004). Singular value decomposition in AHP. European Journal of Operational Research, 154(3), 573–584. https://doi.org/10.1016/S0377-2217(02)00755-5
Georgiou, D., Mohammed, E. S., & Rozakis, S. (2015). Multi-criteria decision making on the energy supply configuration of autonomous desalination units. Renewable Energy, 75, 459–467. https://doi.org/10.1016/j.renene.2014.09.036
Grzybowski, A. Z. (2016). New results on inconsistency indices and their relationship with the quality of priority vector estimation. Expert Systems with Applications, 48(C), 130. https://doi.org/10.1016/j.eswa.2015.08.049
Hamchaoui, S., Boudoukha, A., & Benzerra, A. (2015). Drinking water supply service management and sustainable development challenges: Case study of Bejaia, Algeria. Journal of Water Supply: Research and Technology-Aqua, 64(8), 937–946. https://doi.org/10.2166/aqua.2015.156
Ishizaka, A., & Labib, A. (2011). Review of the main developments in the analytic hierarchy process. Expert Systems with Applications, 38(11), 14336–14345. https://doi.org/10.1016/j.eswa.2011.04.143
Jin, F., Ni, Z., Chen, H., & Li, Y. (2016). Approaches to decision making with linguistic preference relations based on additive consistency. Applied Soft Computing, 49, 71–80. https://doi.org/10.1016/j.asoc.2016.07.045
Jin, F., Ni, Z., Langari, R., & Chen, H. (2020). Consistency improvement-driven decision-making methods with probabilistic multiplicative preference relations. Group Decision and Negotiation, 29, 371–397. https://doi.org/10.1007/s10726-020-09658-2
Jin, F., Liu, J., Zhou, L., & Martínez, L. (2021). Consensus-based linguistic distribution large-scale group decision making using statistical inference and regret theory. Group Decision and Negotiation, 30, 813–845. https://doi.org/10.1007/s10726-021-09736-z
Johnson, R. A., & Wichern, D. W. (1998). Applied multivariate statistical analysis. Prentice Hall.
Karapetrovic, S., & Rosenbloom, E. A. (1999). Quality control approach to consistency paradoxes in AHP. European Journal of Operational Research, 119(3), 704–718. https://doi.org/10.1016/S0377-2217(98)00334-8
Koczkodaj, W. W. (1993). A new definition of consistency for pairwise comparisons. Mathematical and Computer Modelling, 18(7), 79–84. https://doi.org/10.1016/0895-7177(93)90059-8
Koczkodaj, W., & Szwarc, R. (2014). On Axiomatization of inconsistency indicators for pairwise comparisons. Fundamenta Informaticae, 132(4), 485–500. https://www.deepdyve.com/lp/ios-press/on-axiomatization-of-inconsistency-indicators-for-pairwise-comparisons-KvdxIkyigt
Koczkodaj, W. W., & Urban, R. (2018). Axiomatization of inconsistency indicators for pairwise comparisons. International Journal of Approximate Reasoning, 94, 18–29. https://doi.org/10.1016/j.ijar.2017.12.001
Kou, G., & Lin, C. (2014). A cosine maximization method for the priority vector derivation in AHP. European Journal of Operational Research, 235(1), 225–232. https://doi.org/10.1016/j.ejor.2013.10.019
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