Policy decision making based on some averaging aggregation operators of t-spherical fuzzy sets; a multi-attribute decision making approach

Document Type : Original Article


Department of Mathematics & Statistics, International Islamic University Islamabad, 44000 Islamabad, Pakistan.


Multi-attribute decision making (MADM) is a hot research area in fuzzy mathematics and to deal with that, the averaging and geometric aggregation operators (AOs) are the widely used tools. The aim of this manuscript is to propose the notion of averaging and geometric AOs in the environment of T-spherical fuzzy sets (TSFSs). TSFS enables the selection of grades of memberships from considerably a larger domain and hence overcome the drawbacks of the existing fuzzy frameworks. In this paper, we develop some novel operations for TSFSs including algebraic sum, product etc. Based on new operations some averaging AOs including T-spherical fuzzy weighted averaging (TSFWA) and T-spherical fuzzy weighted geometric (TSFWG) operators are developed. The monotonicity, idempotency and boundedness of the defined operators are investigated, and their fitness is validated using induction method. With the help of an illustrative example, the problem of policy decision making using a MADM algorithm is solved. The new proposed work and the existing literature is compared numerically and the advantages of the TSFWA and TSFWG operators are investigated over existing work.


Zadeh, L.A., Information and control. Fuzzy sets, 1965. 8(3): p. 338-353.
Zadeh, L. A. (1975). The concept of a linguistic variable and its application to approximate reasoning—I. Information sciences8(3), 199-249.‏
Atanassov, K. T. (1999). Intuitionistic fuzzy sets. In Intuitionistic fuzzy sets (pp. 1-137). Physica, Heidelberg.‏
Atanassov, K. T. (1989). More on intuitionistic fuzzy sets. Fuzzy sets and systems, 33(1), 37-45.‏
Atanassov, K. and G. Gargov, Interval valued intuitionistic fuzzy sets. Fuzzy sets and systems, 1989. 31(3): p. 343-349.
Torra, V. (2010). Hesitant fuzzy sets. International Journal of Intelligent Systems, 25(6), 529-539.‏
Torra, V., & Narukawa, Y. (2009, August). On hesitant fuzzy sets and decision. In 2009 IEEE International Conference on Fuzzy Systems (pp. 1378-1382). IEEE.‏
Yager, R. R. (2013, June). Pythagorean fuzzy subsets. In 2013 joint IFSA world congress and NAFIPS annual meeting (IFSA/NAFIPS) (pp. 57-61). IEEE.‏
Rahman, K., Khan, M. A., Ullah, M., & Fahmi, A. (2017). Multiple attribute group decision making for plant location selection with Pythagorean fuzzy weighted geometric aggregation operator. The Nucleus54(1), 66-74.‏
Cuong, B. Picture fuzzy sets-First results. Part 1. in Seminar Neuro-Fuzzy Systems with Applications. 2013.
Cuong, B. C., & Kreinovich, V. (2013, December). Picture Fuzzy Sets-a new concept for computational intelligence problems. In 2013 third world congress on information and communication technologies (WICT 2013) (pp. 1-6). IEEE.‏
Cuong, B. C., & Van Hai, P. (2015, October). Some fuzzy logic operators for picture fuzzy sets. In 2015 seventh international conference on knowledge and systems engineering (KSE) (pp. 132-137). IEEE.‏
Garg, H. (2017). Some picture fuzzy aggregation operators and their applications to multicriteria decision-making. Arabian Journal for Science and Engineering, 42(12), 5275-5290.‏
Wei, G. (2016). Picture fuzzy cross-entropy for multiple attribute decision making problems. Journal of Business Economics and Management17(4), 491-502.‏
Wei, G. (2017). Picture fuzzy aggregation operators and their application to multiple attribute decision making. Journal of Intelligent & Fuzzy Systems, 33(2), 713-724.‏
Wei, G. (2017). Some cosine similarity measures for picture fuzzy sets and their applications to strategic decision making. Informatica28(3), 547-564.‏
Wei, G. (2018). Some similarity measures for picture fuzzy sets and their applications. Iranian Journal of Fuzzy Systems15(1), 77-89.‏
Mahmood, T., Ullah, K., Khan, Q. and Jan, N., 2019. An approach toward decision-making and medical diagnosis problems using the concept of spherical fuzzy sets. Neural Computing and Applications, 31(11), pp.7041-7053..
Ullah, K., Mahmood, T., & Jan, N. (2018). Similarity measures for T-spherical fuzzy sets with applications in pattern recognition. Symmetry10(6), 193.‏
Ullah, K., Hassan, N., Mahmood, T., Jan, N., & Hassan, M. (2019). Evaluation of investment policy based on multi-attribute decision-making using interval valued T-spherical fuzzy aggregation operators. Symmetry11(3), 357.‏
Ullah, K., Garg, H., Mahmood, T., Jan, N., & Ali, Z. (2020). Correlation coefficients for T-spherical fuzzy sets and their applications in clustering and multi-attribute decision making. Soft Computing24(3), 1647-1659.‏
Garg, H., Munir, M., Ullah, K., Mahmood, T., & Jan, N. (2018). Algorithm for T-spherical fuzzy multi-attribute decision making based on improved interactive aggregation operators. Symmetry10(12), 670.‏
Liu, P., Khan, Q., Mahmood, T., & Hassan, N. (2019). T-spherical fuzzy power Muirhead mean operator based on novel operational laws and their application in multi-attribute group decision making. Ieee Access7, 22613-22632.‏
Adlassnig, K. P. (1986). Fuzzy set theory in medical diagnosis. IEEE Transactions on Systems, Man, and Cybernetics16(2), 260-265.‏
Bellman, R. E., & Zadeh, L. A. (1970). Decision-making in a fuzzy environment. Management science17(4), B-141.‏
Xu, Z. (2007). Some similarity measures of intuitionistic fuzzy sets and their applications to multiple attribute decision making. Fuzzy Optimization and Decision Making6(2), 109-121.‏
Pei, Z., & Zheng, L. (2012). A novel approach to multi-attribute decision making based on intuitionistic fuzzy sets. Expert Systems with Applications39(3), 2560-2566.‏
Liu, H. W., & Wang, G. J. (2007). Multi-criteria decision-making methods based on intuitionistic fuzzy sets. European Journal of Operational Research179(1), 220-233.‏
Ye, J. (2009). Multicriteria fuzzy decision-making method based on a novel accuracy function under interval-valued intuitionistic fuzzy environment. Expert systems with Applications36(3), 6899-6902.‏
Xu, Z. (2011). Approaches to multiple attribute group decision making based on intuitionistic fuzzy power aggregation operators. Knowledge-Based Systems24(6), 749-760.‏
Xu, Z. (2007). Intuitionistic fuzzy aggregation operators. IEEE Transactions on fuzzy systems15(6), 1179-1187.‏
Garg, H., & Rani, D. (2019). Novel aggregation operators and ranking method for complex intuitionistic fuzzy sets and their applications to decision-making process. Artificial Intelligence Review, 1-26.‏
Edalatpanah, S. A. (2020). Neutrosophic structured element. Expert systems37(5), e12542.‏
Garg, H. (2016). A new generalized Pythagorean fuzzy information aggregation using Einstein operations and its application to decision making. International Journal of Intelligent Systems31(9), 886-920.‏
Garg, H., Rani, M., Sharma, S. P., & Vishwakarma, Y. (2014). Intuitionistic fuzzy optimization technique for solving multi-objective reliability optimization problems in interval environment. Expert Systems with Applications41(7), 3157-3167.‏
Garg, H. (2016). A novel accuracy function under interval-valued Pythagorean fuzzy environment for solving multicriteria decision making problem. Journal of Intelligent & Fuzzy Systems31(1), 529-540.‏
Garg, H. (2016). Generalized intuitionistic fuzzy interactive geometric interaction operators using Einstein t-norm and t-conorm and their application to decision making. Computers & Industrial Engineering101, 53-69.‏
Garg, H. (2016). A novel correlation coefficients between Pythagorean fuzzy sets and its applications to decisionā€making processes. International Journal of Intelligent Systems31(12), 1234-1252.‏
Garg, H. (2016). Some series of intuitionistic fuzzy interactive averaging aggregation operators. SpringerPlus5(1), 1-27.‏
Liao, H., & Xu, Z. (2014). Some new hybrid weighted aggregation operators under hesitant fuzzy multi-criteria decision making environment. Journal of Intelligent & Fuzzy Systems26(4), 1601-1617.‏
Xia, M., Xu, Z., & Chen, N. (2013). Some hesitant fuzzy aggregation operators with their application in group decision making. Group Decision and Negotiation22(2), 259-279.‏
Ali, Z., & Mahmood, T. (2020). Maclaurin symmetric mean operators and their applications in the environment of complex q-rung orthopair fuzzy sets. Computational and Applied Mathematics39, 1-27.‏
Peng, X., & Yang, Y. (2015). Some results for Pythagorean fuzzy sets. International Journal of Intelligent Systems30(11), 1133-1160.‏
Garg, H. (2020). Exponential operational laws and new aggregation operators for intuitionistic multiplicative set in multiple-attribute group decision making process. Information Sciences538, 245-272.‏
Gupta, S., Garg, H., & Chaudhary, S. (2020). Parameter estimation and optimization of multi-objective capacitated stochastic transportation problem for gamma distribution. Complex & Intelligent Systems6(3), 651-667.‏
Garg, H. (2020). New ranking method for normal intuitionistic sets under crisp, interval environments and its applications to multiple attribute decision making process. Complex & Intelligent Systems6, 559-571.‏
Munir, M., Kalsoom, H., Ullah, K., Mahmood, T., & Chu, Y. M. (2020). T-spherical fuzzy Einstein hybrid aggregation operators and their applications in multi-attribute decision making problems. Symmetry12(3), 365.‏
Liu, P., Mahmood, T., & Ali, Z. (2020). Complex q-rung orthopair fuzzy aggregation operators and their applications in multi-attribute group decision making. Information11(1), 5.‏
Ullah, K., Mahmood, T., Ali, Z., & Jan, N. (2020). On some distance measures of complex Pythagorean fuzzy sets and their applications in pattern recognition. Complex & Intelligent Systems6(1), 15-27.‏
Ullah, K., Mahmood, T., & Garg, H. (2020). Evaluation of the performance of search and rescue robots using T-spherical fuzzy hamacher aggregation operators. International Journal of Fuzzy Systems22(2), 570-582.‏
Mahmood, T., Ullah, K., & Khan, Q. (2018). Some aggregation operators for bipolar-valued hesitant fuzzy information. Infinite Study.‏
Mao, X., Guoxi, Z., Fallah, M., & Edalatpanah, S. A. (2020). A Neutrosophic-Based Approach in Data Envelopment Analysis with Undesirable Outputs. Mathematical Problems in Engineering2020.‏
Ali, Z., Mahmood, T., & Yang, M. S. (2020). TOPSIS Method Based on Complex Spherical Fuzzy Sets with Bonferroni Mean Operators. Mathematics8(10), 1739.‏
Ali, Z., Mahmood, T., & Yang, M. S. (2020). Complex T-spherical fuzzy aggregation operators with application to multi-attribute decision making. Symmetry12(8), 1311.‏
Kumar, R., Edalatpanah, S. A., Jha, S., & Singh, R. (2019). A Pythagorean fuzzy approach to the transportation problem. Complex & intelligent systems5(2), 255-263.‏
Yang, W., Cai, L., Edalatpanah, S. A., & Smarandache, F. (2020). Triangular single valued neutrosophic data envelopment analysis: application to hospital performance measurement. Symmetry12(4), 588.‏