Enhancing data consistency in decision matrix: adapting Hadamard model to mitigate judgment contradiction. (English) Zbl 1338.91057

Summary: Cardinal and ordinal inconsistencies are important and popular research topics in the study of decision making with pair-wise comparison matrices (PCMs). Few of the currently-employed tactics are capable of simultaneously dealing with both cardinal and ordinal inconsistency issues in one model, and most are heavily dependent on the method chosen for weight (priorities) derivation or the obtained closest matrix by optimization method that may change many of the original values. In this paper, we propose a Hadamard product induced bias matrix model, which only requires the use of the data in the original matrix to identify and adjust the cardinally inconsistent element(s) in a PCM. Through graph theory and numerical examples, we show that the adapted Hadamard model is effective in identifying and eliminating the ordinal inconsistencies. Also, for the most inconsistent element identified in the matrix, we develop innovative methods to improve the consistency of a PCM. The proposed model is only dependent on the original matrix, is independent of the methods chosen to derive the priority vectors, and preserves most of the original information in matrix A since only the most inconsistent element(s) need(s) to be modified. Our method is much easier to implement than any of the existing models, and the values it recommends for replacement outperform those derived from the literature. It significantly enhances matrix consistency and improves the reliability of PCM decision making.


91B06 Decision theory
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
Full Text: DOI


[1] Ali, I.; Cook, W.; Kress, M., On the minimum violations ranking of a tournament, Management Science, 32, 6, : 660-672, (1986) · Zbl 0601.90003
[2] Altuzarra, A.; Moreno-Jiménez, J. M.; Salvador, M., Consensus building in AHP-group decision making: A Bayesian approach, Operations Research, 58, 6, 1755-1773, (2010) · Zbl 1232.91116
[3] Bana e Costa, C. A.; Vansnick, J. C., A critical analysis of the eigenvalue method used to derive priorities in AHP, European Journal of Operational Research, 187, 1422-1428, (2008) · Zbl 1137.91350
[4] Barzilai, J., Deriving weights from pairwise comparison matrices: the additive case, Operations Research Letters, 9, 6, 407-410, (1999) · Zbl 0711.90007
[5] Birnbaum, M. H., Testing for intransitivity of preferences predicted by a lexicographic semi-order, Organizational Behavior and Human Decision Processes, 104, 1, 96-112, (2007)
[6] Cao, D.; Leung, L. C.; Law, J. S., Modifying inconsistent comparison matrix in analytic hierarchy process: A heuristic approach, Decision Support Systems, 44, 944-953, (2008)
[7] Choo, E.; Wedley, W., A common framework for deriving preference values from pairwise comparison matrices, Computer and Operations Research, 31, 6, 893-908, (2004) · Zbl 1043.62063
[8] Diaye, M. A.; Urdanivia, M. W., Violation of the transitivity axiom may explain why, in empirical studies, a significant number of subjects violate GARP, Journal of Mathematical Psychology, 53, 6, 586-592, (2009) · Zbl 1182.91067
[9] Ergu, D.; Kou, G.; Peng, Y.; Shi, Y., A simple method to improve the consistency ratio of the pair-wise comparison matrix in ANP, European Journal of Operational Research, 213, 1, 246-259, (2011) · Zbl 1237.90220
[10] Forman, E. H.; Gass, S. I., The analytic hierarchy process—an exposition, Operations Research, 49, 4, 469-486, (2001) · Zbl 1163.90300
[11] Gass, S. I., Tournaments, transitivity and pairwise comparison matrices, Journal of the Operational Research Society, 49, 616-624, (1998) · Zbl 1131.90381
[12] Genest, C.; Zhang, S. S., A graphical analysis of ratio-scaled paired comparison data, Management Science, 42, 3, 335-349, (1996) · Zbl 0884.90003
[13] González-Pachón, J.; Romero, C., A method for dealing with inconsistencies in pairwise comparisons, European Journal of Operational Research, 158, 351-361, (2004) · Zbl 1067.90070
[14] Gower, J. C., The analysis of asymmetry and orthogonality, (Barra, J.-R.; Brodeau, F.; Romier, G.; Van Cutsem, B., Recent Developments in Statistics, (1977), North-Holland Amsterdam), 109-123 · Zbl 0366.62058
[15] Grošelj, P.; Zadnik Stirn, L., Acceptable consistency of aggregated comparison matrices in analytic hierarchy process, European Journal of Operational Research, 223, 2, 417-420, (2012) · Zbl 1292.91053
[16] Harker, P. T., Derivatives of the Perron root of a positive reciprocal matrix: with applications to the analytic hierarchy process, Applied Mathematics and Computation, 22, 217-232, (1987) · Zbl 0619.15017
[17] Herman, M. W.; Koczkodaj, W. W., Monte Carlo study of pairwise comparison, Information Processing Letters, 57, 1, 25-29, (1996) · Zbl 1004.68550
[18] Hu, J.; Mehrotra, S., Robust and stochastically weighted multiobjective optimization models and reformulations, Operations Research, 60, 4, 936-953, (2012) · Zbl 1342.90074
[19] 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. · Zbl 1305.91082
[20] Kwiesielewicz, M.; van Uden, E., Inconsistent and contradictory judgements in pairwise comparison method in the AHP, Computer and Operation Research, 31, 5, 713-719, (2004) · Zbl 1048.90121
[21] Li, H. L.; Ma, L. C., Detecting and adjusting ordinal and cardinal inconsistencies through a graphical and optimal approach in AHP models, Computers and Operations Research, 34, 780-798, (2007) · Zbl 1125.90029
[22] Lin, C., A revised framework for deriving preference values from pairwise comparison matrices, European Journal of Operational Research, 176, 2, 1145-1150, (2007) · Zbl 1110.90052
[23] Liu, F., Zhang, W. G., & Zhang, L. H. (2014). Consistency analysis of triangular fuzzy reciprocal preference relations, European Journal of Operational Research, 235(3), 718-726. · Zbl 1305.91095
[24] Ma, L. C.; Li, H. L., Using gower plots and decision balls to rank alternatives involving inconsistent preferences, Decision Support Systems, 51, 712-719, (2011)
[25] Mikhailov, L.; Knowles, J., Priority elicitation in the AHP by a Pareto envelope-based selection algorithm. multiple criteria decision making for sustainable energy and transportation systems, Lecture Notes in Economics and Mathematical Systems, 634, 3, 249-257, (2010) · Zbl 1184.90157
[26] Pahikkala, T.; Waegeman, W.; Tsivtsivadze, E.; Salakoski, T.; De Baets, B., Learning intransitive reciprocal relations with kernel methods, European Journal of Operational Research, 206, 3, 676-685, (2010) · Zbl 1188.68234
[27] Raiffa, H.; Keeney, R., Decisions with multiple objectives: Preferences and value tradeoffs, Decisions with Multiple Objectives: Preferences and Value Tradeoffs, (1976), Cambridge University Press · Zbl 0488.90001
[28] Saaty, T. L. (1972). An eigenvalue allocation model in contingency planning (vol. 19, p. 72). University of Pennsylvania.
[29] Saaty, T. L., The analytical hierarchy process, (1980), McGraw-Hill New York
[30] Saaty, T. L., Axiomatic foundation of the analytic hierarchy process, Management Science, 32, 7, 841-855, (1986) · Zbl 0596.90003
[31] Saaty, T. L., How to make a decision: the analytic hierarchy process, Interfaces, 24, 19-43, (1994)
[32] Saaty, T. L., Decision making with dependence and feedback: the analytic network process, (1996), RWS Publications Pittsburgh, Pennsylvania, ISBN 0-9620317-9-8
[33] Saaty, T. L., Decision-making with the AHP: why is the principal eigenvector necessary, European Journal of Operational Research, 145, 1, 85-91, (2003) · Zbl 1012.90015
[34] Saaty, T. L., The analytic network process, decision making with the analytic network process, (International Series in Operations Research & Management Science, Vol. 95, (2006), Springer), 1-26
[35] Siraj, S. (2011). Preference elicitation from pairwise comparisons in multi-criteria decision making doctoral dissertation. University of Manchester).
[36] Siraj, S.; Mikhailov, L.; Keane, J., A heuristic method to rectify intransitive judgments in pairwise comparison matrices, European Journal of Operational Research, 216, 420-428, (2012) · Zbl 1237.90121
[37] Thurstone, L., A law of comparative judgment, Psychological Review, 34, 4, 273, (1927)
[38] Tsetlin, I.; Winkler, R. L., Decision making with multiattribute performance targets: the impact of changes in performance and target distributions, Operations Research, 55, 2, 226-233, (2007) · Zbl 1167.91335
[39] Vansnick, J.-C., On the problem of weights in multiple criteria decision making (the noncompensatory approach), European Journal of Operational Research, 24, 2, 288-294, (1986) · Zbl 0579.90059
[40] 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
[41] Zhü, K. (2014). Fuzzy analytic hierarchy process: Fallacy of the popular methods, European Journal of Operational Research, 236(1), 113-125.
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.