Affiliations: [a] Department of Mathematics, Research Institute of Natural Sciences, Hanyang University, Seoul, Republic of Korea
| [b] Department of Mathematics & Statistics, Bacha Khan University Charsadda, Khyber Pakhtunkhwa, Pakistan
| [c] Department of Mathematics & Statistics, Riphah International University Hajj Complex I-14, Islamabad, Pakistan
Correspondence:
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Corresponding author. Noor Rehman, Department of Mathematics & Statistics, Bacha Khan University Charsadda, Khyber Pakhtunkhwa, Pakistan. E-mail: noorrehman@bkuc.edu.pk and Abbas Ali, Department of Mathematics & Statistics, Riphah International University Hajj Complex I-14, Islamabad, Pakistan. E-mail: abbasali5068@gmail.com.
Abstract: The q-rung orthopair fuzzy sets accommodate more uncertainties than the Pythagorean fuzzy sets and hence their applications are much extensive. Under the q-rung orthopair fuzzy set, the objective of this paper is to develop new types of q-rung orthopair fuzzy lower and upper approximations by applying the tolerance degree on the similarity between two objects. After employing tolerance degree based q-rung orthopair fuzzy rough set approach to it any times, we can get only the six different sets at most. That is to say, every rough set in a universe can be approximated by only six sets, where the lower and upper approximations of each set in the six sets are still lying among these six sets. The relationships among these six sets are established. Furthermore, we propose tolerance degree based multi granulation optimistic/pessimistic q-rung orthopair fuzzy rough sets and investigate some of their properties. Another main contribution of this paper is to disclose the ideas of different kinds of approximations called approximate precision, rough degree, approximate quality and their mutual relationship. Finally a technique is devloped to rank the alternatives in a q-rung orthopair fuzzy information system based on similarity relation. We find that the proposed method/technique is more efficient when compared with other existing techniques.
