Cubic q-Fractional Fuzzy Sets and Their Applications

Recently, many studies have been conducted on developing cubic sets, like cubic intuitionistic fuzzy sets, cubic Pythagorean fuzzy sets, and cubic q-rung orthopair fuzzy sets. However, the cubic set combines interval-valued and fuzzy sets. But both parts of these fuzzy sets cannot attain the maximum value (equal to 1) due to the restriction at the sum of memberships and non-membership grades. For example if one has data of the form Ξ ={|x∈ X}, then clearly these data cannot be handled through cubic q-rung orthopair fuzzy sets. To cover this situation, we introduce the notions of cubic q-fractional fuzzy sets ( Cqf_rFs ), combining the interval-valued q-fractional fuzzy sets ( IVqf_rFs ) and q-fractional fuzzy sets ( qf_rFs ) and allowing them to attain the maximum value by introducing a new parameter q≥ 2. We first introduce the concept of interval-valued q-fractional fuzzy sets ( IVqf_rFs ) with elemental properties. Then we propose the novel idea of cubic q-fractional fuzzy sets ( Cqf_rFs ) and discuss their sensitivity analysis. We also provide the fundamental arithmetic operations of cubic q-fractional fuzzy sets ( Cqf_rFs ) and properties. In the end, we propose the correlation coefficients to measure the relationship between cubic q-fractional fuzzy sets ( Cqf_rFs ). Finally, we presented a numerical example of the evaluation of using a digital library in the education department by considering its advantages and disadvantages using the developed correlation coefficients for user X . Thus, by knowing a user’s priorities, the digital library database can be updated.