Arithmetic operations and ranking of hesitant fuzzy numbers by extension principle

Document Type : Research Paper


Department of Applied Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran


A hesitant fuzzy number (HFN) is important as a generalization of the fuzzy number for hesitant fuzzy analysis and takes some applications that were discussed in recent literature. In this paper, we develop the hesitant fuzzy arithmetic, which is based on the extension principle for hesitant fuzzy sets. Employing this principle, standard arithmetic operations on fuzzy numbers are extended to HFNs and we show that the outcome of these operations on two HFNs are an HFN.
Also we use the extension principle in HFSs for the ranking of HFNs, which may be an interesting topic.
In this paper, we show that the HFNs can be ordered in a natural way. To introduce a meaningful ordering of HFNs, we use a new lattice operation on HFNs based upon extension principle and  defining the Hamming distance on them.
Finally, the applications of them are explained on optimization and decision-making problems.


[1] J. C. R. Alcantud, V. Torra, Decomposition theorems and extension principles for hesitant fuzzy sets, Information Fusion, 41 (2018), 48-56.
[2] A. I. Ban, L. Coroianu, Simplifying the search for effective ranking of fuzzy numbers, IEEE Transactions on Fuzzy Systems, 23 (2015), 327-339.
[3] S. Bass, H. Kwakernaak, Rating and ranking of multiple-aspect alternatives using fuzzy sets, Automatica, 13 (1977), 47-58.
[4] C. R. Bector, S. Chandra, Fuzzy mathematical programming and fuzzy matrix games, Springer, Berlin, 2005.
[5] B. Bede, Mathematics of fuzzy sets and fuzzy logic, Springer, Heidelberg, 2013.
[6] P. P. Bonissone, A fuzzy sets based linguistic approach: Theory and applications, in: M.M. Gupta, E. Sanchez (Eds.), Approximate Reasoning in Decision Analysis, North-Holland, Amsterdam, (1982), 329-339.
[7] F. J. Cabrerizo, M. A. Martinez, M. J. Cobo, S. Alonso, E. Herrera-Viedma, Hesitant fuzzy sets: A bibliometric study, 5th International Conference on CoDIT, (2018), 659-664.
[8] N. Chen, Z. Xua, M. Xia, Correlation coefficients of hesitant fuzzy sets and their applications to clustering analysis, Applied Mathematical Modelling, 37(4) (2013), 2197-2211.
[9] C. H. Cheng, A new approach for ranking fuzzy numbers by distance method, Fuzzy Sets and Systems, 95 (1998), 307-317.
[10] T. C. Chu, C. T. Tsao, Ranking fuzzy numbers with an area between the centroid point and original point, Computers and Mathematics with Applications, 43(1-2) (2002), 111-117.
[11] R. Chutia, B. Chutia, A new method of ranking parametric form of fuzzy numbers using value and ambiguity, Applied Soft Computing, 52 (2017), 1154-1168.
[12] D. Dubios, H. Prade, Operations on fuzzy numbers, International Journal of Systems Science, 9 (1978), 613-626.
[13] B. Farhadinia, A series of score functions for hesitant fuzzy sets, Information Sciences, 277 (2014), 102-110.
[14] B. Farhadinia, F. Javier Cabrerizo, E. Herrera Viedma, Horizontal representation of a hesitant fuzzy set and its application to multiple attribute decision making, Iranian Journal of Fuzzy Systems, 16(5) (2019), 1-13.
[15] X. Gou, X. Wang, Z. Xu, F. Herrera, Consensus based on multiplicative consistent double hierarchy linguistic preferences: Venture capital in real estate market, International Journal of Strategic Property Management, 24 (2019), 1-23.
[16] X. Gou, Z. Xu, Double hierarchy linguistic term set and its extensions: The state of the art survey, International Journal of Intelligent Systems, 36(2) (2021), 832-865.
[17] X. Gou, Z. Xu, H. Liao, Group decision making with compatibility measures of hesitant fuzzy linguistic preference relations, Soft Computing, 23 (2019), 1511-1527.
[18] X. Gou, Z. Xu, W. Zhou, Interval consistency repairing method for double hierarchy hesitant fuzzy linguistic preference relation and application in the diagnosis of lung cancer, Economic Research-Ekonomska Istrazivanja, 34(1) (2021), 1-20.
[19] P. Grzegrorzewski, The Hamming distance between intuitionistic fuzzy sets, in: The Proceeding of the IFSA 2003 World Congress, ISTANBUL, 2003.
[20] X. Gu, Y. Wang, B. Yang, A method for hesitant fuzzy multiple attribute decision making and its application to risk investment, Journal of Convergence Information Technology, 6(6) (2011), 282-287.
[21] M. Hanss, Applied fuzzy arithmetic, Springer, Berlin, Germany, 2005.
[22] R. Jain, A procedure for multiple-aspect decision making using fuzzy sets, International Journal of Systems Science, 8(l) (1977), 1-7.
[23] A. Kaufmann, M. M. Gupta, Introduction to fuzzy arithmetic: Theory and applications, Van Nostrand Reinhold, New York, NY, USA, 1991.
[24] G. J. Klir, B. Yuan, Fuzzy sets and fuzzy logic, theory and applications, Prentice Hall PTR, 1995.
[25] D. F. Li, A ratio ranking method of triangular intuitionistic fuzzy numbers and its application to MADM problems, Computers and Mathematics with Applications, 60 (2010), 1557-1570.
[26] C. Li, H. Zhao, Z. Xu, Hesitant fuzzy psychological distance measure, International Journal of Machine Learning and Cybernetics, 11 (2020), 2089-2100.
[27] H. Liao, Z. Xu, Extended hesitant fuzzy hybrid weighted aggregation operators and their application in decision making, Soft Computing, 19(9) (2015), 2551-2564.
[28] H. Liu, L. Jiang, Optimizing consistency and consensus improvement process for hesitant fuzzy linguistic preference relations and the application in group decision making, Information Fusion, 56 (2020), 114-127.
[29] A. Mahmodi Nejad, M. Mashinchi, Ranking fuzzy numbers based on the areas on the left and the right sides of fuzzy number, Computers and Mathematics with Applications, 61(2) (2011), 431-442.
[30] H. B. Mitchell, Ranking intuitionistic fuzzy numbers, International Journal of Uncertainty, Fuzziness and Knowledge Based Systems, 12 (2004), 377-386.
[31] J. A. Morente-Molinera, F. J. Cabrerizo, J. Mezei, C. Carlsson, E. Herrera-Viedma, A dynamic group decision making process for high number of alternatives using hesitant fuzzy ontologies and sentiment analysis, KnowledgeBased Systems, 195 (2020), 105657.
[32] M. Ranjbar, S. Effati, Symmetric and right-hand-side hesitant fuzzy linear programming, IEEE Transactions on Fuzzy Systems, 28(2) (2020), 215-227.
[33] M. Ranjbar, A. V. Kamyad, S. Effati, Fuzzy classification as a decision making problem in hesitant environments, International Journal of Information and Decision Sciences, 11(1) (2019), 22-35.
[34] M. Ranjbar, S. M. Miri, S. Effati, Hesitant fuzzy numbers with (α, k)-cuts in compact intervals and applications, Expert Systems with Applications, 151 (2020), 113363.
[35] Z. Ren, Z. Xu, H. Wang, Normal wiggly hesitant fuzzy sets and their application to environmental quality evaluation, Knowledge-Based Systems, 159 (2018), 286-297.
[36] K. Rezaei, H. Rezaei, New distance and similarity measures for hesitant fuzzy soft sets, Iranian Journal of Fuzzy Systems, 16(6) (2019), 159-176.
[37] S. Rezvani, Ranking generalized exponential trapezoidal fuzzy numbers based on variance, Applied Mathematics and Computation, 262 (2015), 191-198.
[38] R. M. Rodriguez, Z. S. Xu, L. Martinez, Hesitant fuzzy information for information fusion in decision making, Information Fusion, 42 (2018), 62-63.
[39] M. Sakawa, Fuzzy sets and interactive multiobjective optimization, Plenum Press, New York, 1993.
[40] G. Sun, X. Guan, X. Yi, Z. Zhou, Grey relational analysis between hesitant fuzzy sets with applications to pattern recognition, Expert Systems with Applications, 92 (2018), 521-532.
[41] M. Tang, H. Liao, Managing information measures for hesitant fuzzy linguistic term sets and their applications in designing clustering algorithms, Information Fusion, 50 (2019), 30-42.
[42] V. Torra, Hesitant fuzzy sets, International Journal of Intelligent Systems, 25 (2010), 529-539.
[43] J. Wang, X. Ma, Z. Xu, J. Zhan, Three-way multi-attribute decision making under hesitant fuzzy environments, Information Sciences, 552 (2021), 328-351.
[44] M. Xia, Z. Xu, Hesitant fuzzy information aggregation in decision making, International Journal of Approximate Reasoning, 52(3) (2011), 395-407.
[45] Y. W. Xu, F. Y. Meng, N. Wang, Correlation coefficients of linguistic interval hesitant fuzzy sets and their application, Iranian Journal of Fuzzy Systems, 16(4) (2019), 65-81.
[46] Z. Xu, M. Xia, Distance and similarity measures for hesitant fuzzy sets, Information Sciences, 181(11) (2011), 2128-2138.
[47] R. R. Yager, On choosing between fuzzy subsets, Kybernetes, 9 (1980), 151-154.
[48] R. R. Yager, A procedure for ordering fuzzy subests of the unit interval, Information Sciences, 24 (1981), 143-161.
[49] W. Zeng, R. Ma, Q. Yin, Z. Xu, Similarity measure of hesitant fuzzy sets based on implication function and clustering analysis, IEEE Access, 8 (2020), 119995-120008.
[50] F. Zhang, J. Ignatius, C. P. Lim, Y. Zhao, A new method for ranking fuzzy numbers and its application to group decision making, Applied Mathematical Modelling, 38(4) (2014), 1563-1582.
[51] X. Zhang, Z. Xu, Novel distance and similarity measures on hesitant fuzzy sets with applications to clustering analysis, Journal of Intelligent and Fuzzy System, 28(5) (2015), 2279-2296.
[52] B. Zhu, Z. Xu, M. Xia, Hesitant fuzzy geometric Bonferroni means, Information Sciences, 205 (2012), 72-85.
[53] H. J. Zimmermann, Fuzzy set theory and its applications, Kluwer Academic Publishers, 1996.