New transitivity of Atanassov’s intuitionistic fuzzy sets in a decision making model

Pękala, Barbara ; Grochowalski, Piotr ; Szmidt, Eulalia

International Journal of Applied Mathematics and Computer Science, Tome 31 (2021) no. 4, pp. 563-576.

Résumé

Atanassov’s intuitionistic fuzzy sets and especially his intuitionistic fuzzy relations are tools that make it possible to model effectively imperfect information that we meet in many real-life situations. In this paper, we discuss the new concepts of the transitivity problem of Atanassov’s intuitionistic fuzzy relations in an epistemic aspect. The transitivity property reflects the consistency of a preference relation. Therefore, transitivity is important from the point of view of real problems appearing, e.g., in group decision making in preference procedures. We propose a new type of optimistic and pessimistic transitivity among the alternatives (options) considered and their use in the procedure of ranking the alternatives in a group decision-making problem.

Détail

Keywords: optimistic transitivity, pessimistic transitivity, preference relations, optimistic intuitionistic fuzzy negations, pessimistic intuitionistic fuzzy negations
Mots-clés : przechodniość optymistyczna, przechodniość pesymistyczna, relacja preferencji

Bibliographie

[1] [1] Asiain,M.J., Bustince, H., Mesiar, R., Kolesarova, A. and Takac, Z. (2018). Negations with respect to admissible orders in the interval-valued fuzzy set theory, IEEE Transactions on Fuzzy Systems 26(2): 556–568.

[2] [2] Atanassov, K.T. (1999). Intuitionistic Fuzzy Sets: Theory and Applications, Springer, Heidelberg.

[3] [3] Atanassov, K.T. (2008). On the intuitionistic fuzzy implications and negations, in P. Chountas et al. (Eds), Intelligent Techniques and Tools for Novel System Architectures, Springer, Berlin, pp. 381–394.

[4] [4] Atanassov, K.T. (2012). On Intuitionistic Fuzzy Sets Theory, Springer, Heidelberg.

[5] [5] Atanassov, K.T. (2016). Mathematics of intuitionistic fuzzy sets, in C. Kahraman et al. (Eds), Fuzzy Logic in Its 50th Year: New Developments, Directions and Challenges, Springer, Berlin, pp. 61–86.

[6] [6] Beliakov, G., Bustince Sola, H., James, S., Calvo, T. and Fernandez, J. (2012). Aggregation for Atanassov’s intuitionistic and interval valued fuzzy sets: The median operator, IEEE Transactions on Fuzzy Systems 20(3): 487–498.

[7] [7] Bentkowska, U. (2018). New types of aggregation functions for interval-valued fuzzy setting and preservation of pos-B and nec-B-transitivity in decision making problems, Information Sciences 424: 385–399.

[8] [8] Bentkowska, U., Bustince, H., Jurio, A., Pagola, M. and Pekala, B. (2015). Decision making with an interval-valued fuzzy preference relation and admissible orders, Applied Soft Computing 35: 792–801.

[9] [9] Burillo, P. and Bustince, H. (1995). Intuitionistic fuzzy relations: Effect of Atanassov’s operators on the properties of the intuitionistic fuzzy relation, Mathware and Soft Computing 2(2): 117–148.

[10] [10] Burillo, P. and Bustince, H. (1996). Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets, Fuzzy Sets Systems 78(3): 305–316.

[11] [11] Deschrijiver, G., Cornelis, C. and Kerre, E.E. (2004). On the representation of intuitionistic fuzzy t-norms and t-conorms, IEEE Transactions on Fuzzy Systems 12(1): 45–61.

[12] [12] Deschrijver, G. and Kerre, E. (2003). On the relationship between some extensions of fuzzy set theory, Fuzzy Sets and Systems 133(2): 227–235.

[13] [13] Drygaś, P. (2011). Preservation of intuitionistic fuzzy preference relations, Proceedings of the 7th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-11), Aix-les-Bains, France, pp. 554–558.

[14] [14] Dubois, D., Godo, L. and Prade, H. (2014). Weighted logics for artificial intelligence an introductory discussion, International Journal of Approximate Reasoning 55(9): 1819–1829.

[15] [15] Dubois, D. and Prade, H. (1988). Possibility Theory, Plenum Press, New York.

[16] [16] Dubois, D. and Prade, H. (2012). Gradualness, uncertainty and bipolarity: making sense of fuzzy sets, Fuzzy Sets and Systems 192: 3–24.

[17] [17] Dudziak, U. and Pękala, B. (2011). Intuitionistic fuzzy preference relations, Proceedings of the 7th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-11), Aix-les-Bains, France, pp. 529–536.

[18] [18] Grzegorzewski, P., Hryniewicz, O. and Romaniuk, M. (2020). Flexible resampling for fuzzy data, International Journal of Applied Mathematics and Computer Science 30(2): 281–297, DOI: 10.34768/amcs-2020-0022.

[19] [19] Pękala, B. (2009). Preservation of properties of interval-valued fuzzy relations, Proceedings of the Joint 2009 International Fuzzy Systems Association World Congress and the 2009 European Society of Fuzzy Logic and Technology Conference, Lisbon, Portugal, pp. 1206–1210.

[20] [20] Pękala, B. (2019). Uncertainty Data in Interval-Valued Fuzzy Set Theory: Properties, Algorithms and Applications, Springer, Cham.

[21] [21] Pękala, B., Bentkowska, U., Bustince, H., Fernandez, J. and Galar, M. (2015). Operators on intuitionistic fuzzy relations, IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), Istanbul, Turkey, pp. 1–8.

[22] [22] Pękala, B., Bentkowska, U. and De Baets, B. (2016). On comparability relations in the class of interval-valued fuzzy relations, Tatra Mountains Mathematical Publications 66(1): 91–101.

[23] [23] Pękala, B., Szmidt, E. and Kacprzyk, J. (2018). Group decision support under intuitionistic fuzzy relations: The role of weak transitivity and consistency, International Journal of Intelligent Systems 33(10): 2078–2095.

[24] [24] Pradhan, R. and Pal, M. (2017). Transitive and strongly transitive intuitionistic fuzzy matrices, Annals of Fuzzy Mathematics and Informatics 13(4): 485–498.

[25] [25] Rutkowski, T., Łapa, K. and Nielek, R. (2019). On explainable fuzzy recommenders and their performance evaluation, International Journal of Applied Mathematics and Computer Science 29(3): 595–610, DOI: 10.2478/amcs-2019-0044.

[26] [26] Saminger, S., Mesiar, R. and Bodenhoffer, U. (2002). Domination of aggregation operators and preservation of transitivity, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 10(1): 11–35.

[27] [27] Szmidt, E. (2014). Distances and Similarities in Intuitionistic Fuzzy Sets, Springer, Cham.

[28] [28] Szmidt, E. and Kacprzyk, J. (2000). Distances between intuitionistic fuzzy sets, Fuzzy Sets and Systems 114(3): 505–518.

[29] [29] Szmidt, E. and Kacprzyk, J. (2006). Distances between intuitionistic fuzzy sets: Straightforward approaches may not work, 3rd International IEEE Conference on Intelligent Systems, IS06, London, UK, pp. 716–721.

[30] [30] Szmidt, E. and Kacprzyk, J. (2009). Amount of information and its reliability in the ranking of Atanassov’s intuitionistic fuzzy alternatives, in E. Rakus-Andersson et al. (Eds), Recent Advances in Decision Making, Springer, Berlin, pp. 7–19.

[31] [31] Szmidt, E. and Kacprzyk, J. (2017). A perspective on differences between Atanassov’s intuitionistic fuzzy sets and interval-valued fuzzy sets, Studies in Computational Intelligence 671: 221–237, DOI: 10.1007/978-3-319-47557-8_13.

[32] [32] Taylor, A.D. (2005). Social Choice and the Mathematics of Manipulation, Cambridge University Press, New York.

[33] [33] Xu, Y., Wanga, H. and Yu, D. (2014). Cover image weak transitivity of interval-valued fuzzy relations, Knowledge-Based Systems 63: 24–32.

[34] [34] Xu, Z. (2007). Approaches to multiple attribute decision making with intuitionistic fuzzy preference information, Systems Engineering-Theory and Practice 27(11): 62–71.

[35] [35] Xu, Z. and Yager, R.R. (2009). Intuitionistic and interval-valued intuitionistic fuzzy preference relations and their measures of similarity for the evaluation of agreement within a group, Fuzzy Optimization and Decision Making 8(2): 123–139, DOI: 10.1007/s10700-009-9056-3.

[36] [36] Zadeh, L.A. (1965). Fuzzy sets, Information and Control 8: 338–353.

[37] [37] Zapata, H., Bustince, H., Montes, S., Bedregal, B., Dimuro, G., Takac, Z. and Baczyński, M. (2017). Interval-valued implications and interval-valued strong equality index with admissible orders, International Journal of Approximate Reasoning 88: 91–109.