Abstract
When one estimates the importance of alternatives under rational choice, it is natural to avoid self-contradiction from the viewpoint of psychology. Due to the vagueness encountered in a manner analogous to human thought, decision makers always exhibit limited rationality. The judgements could be expressed as interval-valued comparison matrices within the framework of analytic hierarchy process. In this study, for additive reciprocal matrices (ARMs), three axiomatic properties are proposed to characterize the additive consistency and the multiplicative consistency under fully rational behavior. For interval additive reciprocal matrices (IARMs), the concept of weak consistency is used to capture the limited rationality. By weakening some axiomatic properties of consistent ARMs, the reasonable properties of IARMs with weak consistency are presented. Two kinds of IARMs satisfying the properties of weak consistency are analyzed and some comparisons are offered. It is observed that the consistency of ARMs can be defined exactly and characterized by using the axiomatic properties. The properties of characterizing the consistency degree of IARMs should be captured by weakening the axiomatic ones of consistent ARMs. The proposed approach visualizes the development process starting from cardinal consistency of numeric-valued preference relations to weak consistency of interval-valued comparison matrices.
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References
Brunelli, M. (2015). Introduction to the analytic hierarchy process. New York: Springer.
Cabrerizo, F. J., Morente-Molinera, J. A., Pedrycz, W., Taghavi, A., & Herrera-Viedma, E. (2018). Granulating linguistic information in decision making under consensus and consistency. Expert Systems With Applications, 99, 83–92.
Chiclana, F., Herrera-Viedma, E., Alonso, S., & Herrera, F. (2009). Cardinal consistency of reciprocal preference relations: A characterization of multiplicative transitivity. IEEE Transactions on Fuzzy Systems, 17(1), 14–23.
Dong, Y. C., Li, C. C., Chiclana, F., & Herrera-Viedma, E. (2016). Average-case consistency measurement and analysis of interval-valued reciprocal preference relations. Knowledge-Based Systems, 114, 108–117.
Dubois, D., & Prade, H. (1988). Possibility theory. New York: Plenum Press.
Fedrizzi, M., & Brunelli, M. (2010). On the priority vector associated with a reciprocal relation and a pairwise comparison matrix. Soft Computing, 14, 639–645.
Herrera-Viedma, E., Herrera, F., Chiclana, F., & Luque, M. (2004). Some issues on consistency of fuzzy preference relations. European Journal of Opernational Research, 154, 98–109.
Krejčí, J. (2017). On additive consistency of interval fuzzy preference relations. Computers & Industrial Engineering, 107, 128–140.
Krejčí, J. (2019). On extension of multiplicative consistency to interval fuzzy preference relations. Operational Research: An International Journal, 19, 783–815.
Li, C. C., Dong, Y. C., Xu, Y. J., Chiclana, F., Herrera-Viedma, E., & Herrera, F. (2019). An overview on managing additive consistency of reciprocal preference relations for consistency-driven decision making and fusion: Taxonomy and future directions. Information Fusion, 52, 143–156.
Liu, B. D. (2010). Uncertainty theory: A branch of mathematics for modelling human uncertainty. Berlin: Springer.
Liu, W., Dong, Y., Chiclana, F., Cabrerizo, F. J., & Herrera-Viedma, E. (2017). Group decision-making based on heterogeneous preference relations with self-confidence. Fuzzy Optimization and Decision Making, 16(4), 429–447.
Liu, X. W., Pan, Y. W., Xu, Y. J., & Yu, S. (2012). Least square completion and inconsistency repair methods for additively consistent fuzzy preference relations. Fuzzy Sets and Systems, 198, 1–19.
Liu, F., Pedrycz, W., Wang, Z. X., & Zhang, W. G. (2017). An axiomatic approach to approximation-consistency of triangular fuzzy reciprocal preference relations. Fuzzy Sets and Systems, 322, 1–18.
Liu, F., Peng, Y. N., Yu, Q., & Zhao, H. (2018). A decision-making model based on interval additive reciprocal matrices with additive approximation-consistency. Information Sciences, 422, 161–176.
Meng, F. Y., Tan, C. Q., & Chen, X. H. (2017). Multiplicative consistency analysis for interval fuzzy preference relations: A comparative study. Omega, 68, 17–38.
Saaty, T. L. (1980). The analytic hierarchy process. New York: McGraw-Hill.
Simon, H. A. (1961). Models of man (2nd ed.). New York: Wiley.
Tanino, T. (1984). Fuzzy preference orderings in group decision-making. Fuzzy Sets and Systems, 12, 117–131.
Ureña, M. R., Chiclana, F., Fujita, H., & Herrera-Viedma, E. (2015). Confidence-consistency driven group decision making approach with incomplete reciprocal intuitionistic preference relations. Knowledge-Based Systems, 89, 86–96.
Ureña, M. R., Chiclana, F., Morente-Molinera, J. A., & Herrera-Viedma, E. (2015). Managing incomplete preference relations in decision making: A review and future trends. Information Sciences, 302(1), 14–32.
Wang, Z. J., Lin, J., & Liu, F. (2019). Axiomatic property based consistency analysis and decision making with interval multiplicative reciprocal preference relations. Information Sciences, 491, 109–137.
Xu, Z. S. (2004). On compatibility of interval fuzzy preference relations. Fuzzy Optimization and Decision Making, 3, 217–225.
Xu, Z. S., & Chen, J. (2008). Some models for deriving the priority weights from interval fuzzy preference relations. European Journal of Operational Research, 184, 266–280.
Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338–353.
Acknowledgements
The work was supported by the National Natural Science Foundation of China (Nos. 71571054, 71871072), 2017 Guangxi high school innovation team and outstanding scholars plan, the Guangxi Natural Science Foundation for Distinguished Young Scholars (No. 2016GXNSFFA380004), and the Innovation Project of Guangxi Graduate Education (No. YCSW2019045).
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Liu, F., Zhang, JW., Yu, Q. et al. On weak consistency of interval additive reciprocal matrices. Fuzzy Optim Decis Making 19, 153–175 (2020). https://doi.org/10.1007/s10700-020-09314-z
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DOI: https://doi.org/10.1007/s10700-020-09314-z