Two kinds of orthopair soft sets as novel approaches to granular computing based on parametrization

Document Type : Original Article


1 Islamabad Model College for Boys G-11/1, Islamabad, Pakistan

2 Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan

3 Department of Applied Mathematics, Xi an University of Posts and Telecommunications, Xi an, China


This paper aims at introducing two types of orthopair soft sets, which might serve as novel approaches to granular computing based on parametrization. These generalized soft sets emerge naturally when linguistic parameters are employed to convey uncertainty attached to elements of certain sets. Concepts of uncertainty measures attached to the parameters and the whole orthopair soft sets are presented as well. The proposed uncertainty measures are useful for classifying elements of the set of parameters. Different types of granularity measures associated with parameters are presented and are extended to orthopair soft sets. Collective wisdom is helpful in decision making based on consensus. A numerical example is given to demonstrate how orthopair soft sets can be employed in this regard.


[1] M. Agarwal, K.K. Biswas, M. Hanmandlu, Generalized intuitionistic fuzzy soft sets with ap-
plications in decision-making, Applied Soft Computing, 13 (2013), 3552{3566.
[2] M.I. Ali, Another view on q-rung orthopair fuzzy sets, International Journal of Intelligent
Systems, 33 (2018), 2139{2153.
[3] A. Ali, M.I. Ali, N. Rehman, Soft dominance based rough sets with applications in information
systems, International Journal of Approximate Reasoning, 113 (2019), 171{195.
[4] M.I. Ali, B. Davvaz, M. Shabir, Generalized fuzzy S-acts and their characterization by soft
S-acts, Neural Computing and Applications, 21(1) (2012), 9{17.
[5] M.I. Ali, F. Feng, X.Y. Liu, W.K. Min, M. Shabir, On some new operations in soft set theory,
Computers and Mathematics with Applications, 57 (2009), 1547{1553.
[6] M.I. Ali, N. Mehmood, J.M. Zhan, N. Shah, Soft linear programming: An application of soft
vector spaces, Journal of Information and Optimization Sciences, 41(3) (2020), 679{704.
[7] M.I. Ali, M. Shabir, M. Naz, Algebraic structures of soft sets associated with new operations,
Computers and Mathematics with Applications, 61 (2011), 2647{2654.
[8] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87{96.
[9] K. Atanassov, Intuitionistic fuzzy sets, Springer, Heidelberg, 1999.
[10] D. Ciucci, Orthopairs: A simple and widely used way to model uncertainty, Fundamenta
Informaticae, 108 (2011), 287{304.
[11] D. Dubois, H. Prade, Two-fold fuzzy sets and rough sets{some issues in knowledge represen-
tation, Fuzzy Sets and Systems, 23 (1987), 3{18.
[12] D. Dubois, H. Prade, An introduction to bipolar representation of information and preference,
International Journal of Intelligent Systems, 23 (2008), 866{877.
[13] F. Feng, H. Fujita, M.I. Ali, R.R. Yager, X. Liu, Another view on generalized intuitionistic
fuzzy soft sets and related multiattribute decision making methods, IEEE Transactions on
Fuzzy Systems, 27(3) (2019), 474{488.
[14] F. Feng, Z. Xu, H. Fujita, M. Liang, Enhancing PROMETHEE method with intuitionistic
fuzzy soft sets, International Journal of Intelligent Systems, 35 (2020), 1071{1104.
[15] F. Feng, Y. Zheng, B. Sun, M. Akram, Novel score functions of generalized orthopair fuzzy
membership grades with application to multiple attribute decision making, Granular Comput-
ing, (2021),
[16] M.Y. Gentilhomme, Les ensembles  ous en linguistique, Cahiers de Linguistique Theoretique
et Applique, Bucarest, 47 (1968), 47{65.
[17] P.K. Maji, R. Biswas, A.R. Roy, Intuitionistic fuzzy soft sets, Journal of Fuzzy Mathematics,
9(3) (2001), 677{692.
[18] D. Molodtsov, Soft set theory{ rst results, Computers and Mathematics with Applications,
37 (1999), 19{31.
[19] M. Naz, M. Shabir, On fuzzy bipolar soft sets, their algebraic structures and applications,
Journal of Intelligent and Fuzzy Systems, 26(4) (2014), 1645{1656.
[20] Z. Pawlak, Rough sets, International Journal of Computer and Information Sciences, 11 (1982),
[21] Z. Pawlak, Rough sets{theoretical aspects of reasoning about data, Kluwer Academic Publisher,
[22] W. Pedrycz, Shadowed sets: Representing and processing fuzzy sets, IEEE Transaction on
Systems, Man and Cybernetics{PART B: Cybernetics, 28(1) (1998), 103{109.
[23] W. Pedrycz, Shadowed sets: Bridging fuzzy and rough sets, in: Rough Fuzzy Hybridization
(S. Pal, A. Skowron, Eds.), Springer{Verlag, Singapore, (1999), 179{199.
[24] W. Pedrycz, G. Vukovich, Granular computing with shadowed sets, International Journal of
Intelligent Systems, 17 (2002), 173{197.
[25] N. Rehman, A. Ali, M.I. Ali, C. Park, SDMGRS: Soft dominance based multi granulation
rough sets and their applications in con ict analysis problems, IEEE Access, 6 (2018), 31399{
[26] R.R. Yager, Pythagorean membership grades in multi-criteria decision making, Technical Re-
port MII-3301. New Rochelle, NY: Machine Intelligence Institute, Iona College, 2013.
[27] R.R. Yager, Pythagorean membership grades in multi-criteria decision making, IEEE Trans-
actions on Fuzzy Systems, 22 (2014), 958{965.
[28] R.R. Yager, Generalized orthopair fuzzy sets, IEEE Transactions on Fuzzy Systems, 25 (2017),
[29] R.R. Yager, A. M. Abbasov, Pythagorean membership grades, complex numbers, and decision
making, International Journal of Intelligent Systems, 28 (2013), 436{452.
[30] Y.Y. Yao, X. Li, Comparison of rough-set and interval-set models for uncertain reasoning,
Fundamenta Informaticae, 27 (1997), 289{298.
[31] L.A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338{353.
[32] W.R. Zhang, Bipolar fuzzy sets and relations: A computational framework for cognitive mod-
eling and multiagent decision analysis, Proceedings of the First International Joint Conference
of The North American Fuzzy Information Processing Society Biannual Conference, (1994),