Abstract
This paper forms both theoretical and practical innovation basis for decision making process in micro and macro economics. The decision making problem considered here is to rank n alternatives from the best to the worst, using the information given by the decision maker(s) in the form of an \(n\times n\) pairwise comparisons matrix. Here, we deal with pairwise comparisons matrices with fuzzy elements. Fuzzy elements of the pairwise comparisons matrix are applied whenever the decision maker is uncertain about the value of his/her evaluation of the relative importance of elements in question. We investigate pairwise comparisons matrices with elements from abelian linearly ordered group (alo-group) over a real interval which is a generalization of traditional multiplicative or additive approaches. The concept of reciprocity and consistency of pairwise comparisons matrices with fuzzy elements is well known. Here, we extend these concepts, namely to the strict as well as strong consistency of pairwise comparisons matrices with fuzzy elements (PCF matrices). We derive the necessary and sufficient conditions for strict/strong consistency and investigate their properties as well as some consequences to the problem of ranking the alternatives. Illustrating examples are presented and discussed.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Thurstone, L.L.: A law of comparative judgement. Psychol. Rev. 34, 278–286 (1927)
Saaty, T.L.: Scaling method for priorities in hierarchical structures. J. Math. Psychol. 15(3), 234–281 (1977)
Vaidya, O.S., Sushil, K.: Analytic hierarchy process: an overview of applications. Eur. J. Oper. Res. 169, 1–29 (2006)
Kou, G., Ergu, D., Lin, C.S., Chen, Y.: Pairwise comparison matrix in multiple criteria decision making. Technol. Econ. Dev. Econ. 22(5), 738–765 (2016)
Entani, T., Sugihara, K.: Uncertainty index based interval assignment by Interval AHP. Eur. J. Oper. Res. 219(2), 379–385 (2012)
Ohnishi, S., Yamanoi, T., Imai, H.: A weights representation for fuzzy constraint-based AHP. In: IPMU 2008 (2008). http://www.gimac.uma.es/ipmu08/proceedings/papers/036-OhnishiYamanoiImai.pdf
Zhang, H.: Group decision making based on multiplicative consistent reciprocal preference relations. Fuzzy Sets Syst. 282, 31–46 (2016)
Ramik, J., Korviny, P.: Inconsistency of pairwise comparison matrix with fuzzy elements based on geometric mean. Fuzzy Sets Syst. 161, 1604–1613 (2010)
Ramik, J.: Pairwise comparison matrix with fuzzy elements on alo-group. Inf. Sci. 297, 236–253 (2015)
Bilgic, T., Turksen, I.B.: Measurement of membership functions: theoretical and empirical work. In: Dubois, D., Prade, H. (eds.) Fundamentals of Fuzzy Sets, pp. 195–227. Kluwer Academic Publ., New York (2000)
Ramik, J., Vlach, M.: Generalized concavity in optimization and decision making. Kluwer Academic Publishers, Boston-Dordrecht-London (2001)
Bourbaki, N.: Algebra II. Springer, Heidelberg (1998)
Cavallo, B., D’Apuzzo, L.: Reciprocal transitive matrices over abelian linearly ordered groups: characterizations and application to multi-criteria decision problems. Fuzzy Sets Syst. 266, 33–46 (2015)
Acknowledgment
This paper was supported by the Ministry of Education, Youth and Sports Czech Republic within the Institutional Support for Long-term Development of a Research Organization in 2017.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this paper
Cite this paper
RamÃk, J. (2018). Strict and Strong Consistency in Pairwise Comparisons Matrix with Fuzzy Elements. In: Czarnowski, I., Howlett, R., Jain, L. (eds) Intelligent Decision Technologies 2017. IDT 2017. Smart Innovation, Systems and Technologies, vol 72. Springer, Cham. https://doi.org/10.1007/978-3-319-59421-7_27
Download citation
DOI: https://doi.org/10.1007/978-3-319-59421-7_27
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-59420-0
Online ISBN: 978-3-319-59421-7
eBook Packages: EngineeringEngineering (R0)