Some frank aggregation operators based on the interval-valued intuitionistic fuzzy numbers

Document Type : Research Paper


1 Department of Mathematics, Payame Noor University, Tehran, Iran

2 Department of Mathematics, Semnan University, Semnan, Iran


In this article, we introduce the interval-valued intuitionistic fuzzy set (\textbf{IVIFS}), which are generalized forms of intuitionistic fuzzy set (\textbf{IFS}) and fuzzy set, this is because in intuitionistic fuzzy sets the non-membership function also applies to evaluations, and these sets are useful for modelling ambiguous concepts that abound in real problems. Here we try to look for new methods for more practical solutions in optimization problems for various sciences such as computer science, mathematics, engineering, medicine, psychology, climate and etc. First, with the introduction of t-norm Frank, an action we construct some Frank aggregation operators on interval-valued intuitionistic fuzzy numbers (\textbf{IVIFN}s), including the Frank weighted averaging operator, Frank-ordered weighted averaging operator, Frank hybrid weighted averaging operator, Frank geometric weighted averaging operator, Frank geometric-ordered weighted averaging operator, and Frank geometric hybrid weighted averaging operator. Also, examine some of the characteristics of these operators. In the following, we introduce two multiple attribute group decision-making methods (\textbf{MAGDM}) based on such operators. Finally, we provide illustrative examples of these methods.


[1] S. Abbasbandy and T. Allahviranloo, Numerical solution of fuzzy differential equation by Runge-Kutta method,
11(1) (2004) 117–129.
[2] S. Abbasbandy, T. Allahviranloo, O. L´opez-Pouso and J. J. Nieto, ´ Numerical methods for fuzzy differential
inclusions, Comput. Math. Appl. 48 (2004) 1633–1641.
[3] T. Allahviranloo and M. Barkhordari Ahmadi, Fuzzy Laplace transforms, Soft Comput. 14, Article number: 235
[4] T. Allahviranloo, S. Abbasbandy, N. Ahmady and E. Ahmady, Improved predictor-corrector method for solving
fuzzy initial value problems, Inf. Sci. 179(7) (2009) 945–955.
[5] T. Allahviranloo, S. Abbasbandy, O. Sedaghgatfar and P. Darabi, A new method for solving fuzzy integrodifferential equation under generalized differentiability, Neural Comput. Appl. 21 (2012) 191—196.
[6] T. Allahviranloo, Z. Gouyandeh, A. Armand and A. Hasanoglu, On fuzzy solutions for heat equation based on
generalized Hukuhara differentiability, Fuzzy Sets Syst. 265 (2015) 1–23.
[7] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst. 20 (1986) 87–96.
[8] K. T. Atanassov, More on intuitionistic fuzzy sets, Fuzzy Sets Syst. 33 (1989) 37–46.
[9] K. T. Atanassov and G. Gargov, Interval-valued intuitionistic fuzzy sets, Fuzzy Sets Syst. 3 (1989) 343–349.
[10] K. T. Atanassov, Operators over interval-valued intuitionistic fuzzy sets, Fuzzy Sets Syst. 64 (1994) 159–174.
[11] G. Beliakov, A. Pradera and T. Calvo, Aggregation Functions: A Guide For Practitioners, Heidelberg, Germany:
Springer, 2007.
[12] G. Beliakov, H. Bustince, S. James, T. Calvo and J. Fernandez, Aggregation for Atanassov’s intuitionistic and
interval valued fuzzy sets: The median operator, IEEE Trans. Fuzzy Syst. 20(3) (2012) 487–498.
[13] Y. J. Cho, Th. M. Rassias and R. Saadat, Fuzzy Operator Theory in Mathematical Analysis, Springer, 2018.
[14] G. Deschrijver and E. E Kerre, A generalization of operators on intuitionistic fuzzy sets using triangular norms
and conorms, Notes Intuitionistic Fuzzy Sets, 8(1) (2002) 19–27.
[15] G. Deschrijver, C. Cornelis and E. E. Kerre, On the representation of intuitionistic fuzzy t-norms and t-conorms,
IEEE Trans. Fuzzy Syst. 12(1) (2004) 45–61.
[16] R. Ezzatia, T. Allahviranloo, S. Khezerloo and M. Khezerloo, An approach for ranking of fuzzy numbers, Expert
Syst. Appl. 39(1) (2012) 690–695.
[17] V. L. Gomathi Nayagam, S. Muralikrishnan and G. Sivaraman, Multicriteria decision making method based on
interval-valued intuitionistic fuzzy sets, Expert Syst. Appl. 38(3) (2011) 1464–1467.
[18] M. Goudarzi, S. M. Vaezpour and R. Saadati, On the intuitionistic fuzzy inner product spaces, Chaos, Sol. Fract.
41(3) (2009) 1105–1112.
[19] Z. Gouyandeh, T. Allahviranloo, S. Abbasbandy and A. Armand, A fuzzy solution of heat equation under generalized Hukuhara differentiability by fuzzy Fourier transform, Fuzzy Sets Syst. 309 (2017) 81–97.
[20] M. Grabisch, J. L. Marichal, R. Mesiar and E. Pap, Aggregation functions: Means, Info. Sci. 181(1) (2011) 1–22.
[21] S. B. Hosseini, R. Saadati and M. Amini, Alexandroff theorem in fuzzy metric spaces, Math. Sci. Res. J. 2004.
[22] S. B. Hosseini, D. Oregan and R. Saadati, Some results on intuitionistic fuzzy spaces, Iran. J. Fuzzy Syst. 4(1)
(2007) 53–64.
[23] P. E. Klement and R. Mesiar, Triangular norms, Tatra Mt. Math. Pupl. 13 (1997) 169–193.
[24] P. Liu, Some Hamacher aggregation operators based on the interval-valued intuitionistic fuzzy numbers and their
application to group, Decision Making, 22(1) (2014) 83–97.
[25] X. D. Liu, S. H. Zheng and F. I. Xiong, Entropy and subsethood for general interval-valued intuitionistic fuzzy
sets, in Fuzzy Systems and Knowledge Discovery, Fuzzy Syst. Knowledge Disc. 3613 (2005) 42–52.
[26] A. Mahmoodirad, T. Allahviranloo and S. Niroomand, A new effective solution method for fully intuitionistic
fuzzy transportation problem, Soft Comput. 23 (2019) 4521—4530.
[27] P. Melin, G. E. Martinez and R. Svetkov, Choquet and Sugeno integrals and intuitionistic fuzzy integrals as
aggregation operators, Notes Intuit. Fuzzy Set. 23(1) (2017) 95–99.
[28] E. Pap, Ch. Park and R. Saadati, Additive σ−Random operator inequality and rhom-derivations in fuzzy Banach
algebras, U. P. B. Sci. Bull., Series A, 82(2) (2020) 3–14.
[29] J. H. Park, Y. B. Park and R. Saadati, Some results in intuitionistic fuzzy metric spaces, J. Comput. Anal. Appl.
10(4) (2008) 441–451.
[30] J. Qin and X. Liu, Frank aggregation operators for triangular interval type-2 fuzzy set and its application in
multiple attribute group, J. Appl. Math. 2014 (2014), Article ID 923213, 24 pages.
[31] R. Saadati and M. Vaezpour, Some results on fuzzy Banach spaces, J. Appl. Math. Comput. 17(1) (2005) 475–484.
[32] R. Saadati and J. H. Park, Intuitionistic fuzzy Euclidean normed spaces, Commun. Math. Anal. 1(2) (2006) 85–90.
[33] R. Saadati and J. H. Park, On the intuitionistic fuzzy topological spaces, Chaos, Solit. Fract. 27(2) (2006) 331–344.[34] R. Saadati, Common fixed point theorem in intuitionistic fuzzy metric spaces, Albanian J. Math. 2(4) (2008)
[35] R. Saadati, M. Vaezpour and Y. J. Cho, Quicksort algorithm: Application of a fixed point theorem in intuitionistic
fuzzy quasi-metric spaces at a domain of words, J. Comput. Appl. Math. 228(1) (2009) 219–225.
[36] R. Saadati, On the topology of fuzzy metric type spaces, Filomat, 29(1) (2015) 133–141.
[37] S. Salahshour and T. Allahviranloo, Application of fuzzy differential transform method for solving fuzzy Volterra
integral equations, Appl. Math. Model. 37(3) (2013) 1016–1027.
[38] C. Tan and Q. Zhang, Fuzzy multiple attribute decision-making based on interval valued intuitionistic fuzzy sets,
Proc. IEEE Int. Conf. Syst, Man, Cybern, vol. 2, Taipei, Taiwan, 2006, pp. 1404-1407.
[39] J. Q. Wang, K. J. Li and H. Y. Zhang, Interval-valued intuitionistic fuzzy multi-criteria decision-making approach
based on prospect score function, Knowl-Based Syst. 27 (2012) 119–125.
[40] W. Wang and X. Liu, Intuitionistic fuzzy information aggregation using Einstein operations, IEEE Trans. Fuzzy
Syst. 20(5) (2012) 923–938.
[41] W. Z. Wang and X. W. Liu, Intuitionistic fuzzy geometric aggregation operators based on Einstein operations,
Int. J. Intell. Syst. 26 (2011) 1049–1075.
[42] Z. Wang and J. Xu, A fractional programming method for interval-valued intuitionistic fuzzy multi-attribute
decision making, Proc. 49th IEEE Int. Conf. Decision Control, 2010, pp. 636-641.
[43] G. W. Wei and W. D. Yi, Induced interval-valued intuitionistic fuzzy OWG operator, Proc. 5th Int. Conf. Fuzzy
Syst. Knowl. Discovery, 2008, pp. 605-609.
[44] M. M. Xia, Z. S. Xu and B. Zhu, Some issues on intuitionistic fuzzy aggregation operators based on Archimedean
t-conorm and t-norm, Knowl. Based Syst. 31 (2012) 78–88.
[45] Z. S. Xu and R. R. Yager, Intuitionistic and interval-valued intutionistic fuzzy preference relations and their
measures of similarity for the evaluation of agreement within a group, Fuzzy Optim. Decis. Mak. 8 (2009) 123–
[46] L. A. Zadeh, Fuzzy sets, Inf. Cont. 8 (1965) 338–356.
Volume 12, Special Issue
December 2021
Pages 325-342
  • Receive Date: 14 February 2021
  • Revise Date: 19 April 2021
  • Accept Date: 25 May 2021