Additive Integrals of q -Rung Orthopair Fuzzy Functions.

The q -rung orthopair fuzzy set ( q -ROFS) is a powerful tool to deal with uncertainty and ambiguity in real life. The theoretical basis for processing the continuous q -rung orthopair fuzzy information is q -rung orthopair fuzzy calculus ( q -ROFC) and the main object is q -rung orthopair fuzzy functions ( q -ROFFs). Recently, the authors proposed derivatives and differentials of q -ROFFs in the framework of q -ROFC. In this paper, we aim to further study the q -rung orthopair fuzzy integral ( q -ROFI). It is the most important and fundamental part of the q -ROFC theoretical system with direct and powerful applications. Our contribution is the indefinite and definite integrals, and bridges the fuzzy calculus theoretical gap of the nonlinear q -ROFFs. In particular, we begin with the indefinite integral of q -ROFFs, which can be regarded as the anti-derivatives operations of our previous work. Some of their basic properties are discussed. Next, we give the accurate concept of definite integrals of q -ROFFs under additive operations, and obtain the explicit integral formula. Some properties of q -ROFIs, such as comparison, algebraic operations, and mean value theorem are analyzed. Finally, we generalize the q -ROFI to the case when membership and nonmembership functions are allowed to be correlated. After the theoretical results have been established, we present some numerical examples to demonstrate the rationality and effectiveness of integrating continuous q -rung orthopair fuzzy data with the q -ROFIs.