Differential calculus of interval-valued q-rung orthopair fuzzy functions and their applications

The interval-valued q-rung orthopair fuzzy set (IVq-ROFS) provides an extension of Yager's q-rung orthopair fuzzy set (q-ROFS), where membership and nonmembership degrees are subsets of the closed interval [0,1]. In such a situation, it is more superior for decision makers to provide their judgments by intervals instead of crisp numbers due to the uncertainty and vagueness in real life. In this paper, we study the calculus theories of IVq-ROFS from the microscopic. In particular, we first introduce the elementary arithmetic of interval-valued q-rung orthopair fuzzy values (IVq-ROFVs), including addition, multiplication, and their inverse. They are the basis for analysis and calculation throughout the work. In addition, we discuss and prove in detail the operation properties and aggregation operators of IVq-ROFVs. Then, we introduce the concept of interval-valued q-rung orthopair fuzzy functions (IVq-ROFFs), which is the main research object of this paper. After that, we further discuss the continuity, derivatives and differentials of IVq-ROFFs. We also find that the derivatives of IVq-ROFFs are closely related to elasticity, which is an important concept in economics. Finally, we provide some application examples to verify the feasibility and effectiveness of the derived results.