Additive Integrals of $q$ -Rung Orthopair Fuzzy Functions

The <inline-formula> <tex-math notation="LaTeX">${q}$ </tex-math></inline-formula>-rung orthopair fuzzy set (<inline-formula> <tex-math notation="LaTeX">${q}$ </tex-math></inline-formula>-ROFS) is a powerful tool to deal with uncertainty and ambiguity in real life. The theoretical basis for processing the continuous <inline-formula> <tex-math notation="LaTeX">${q}$ </tex-math></inline-formula>-rung orthopair fuzzy information is <inline-formula> <tex-math notation="LaTeX">${q}$ </tex-math></inline-formula>-rung orthopair fuzzy calculus (<inline-formula> <tex-math notation="LaTeX">${q}$ </tex-math></inline-formula>-ROFC) and the main object is <inline-formula> <tex-math notation="LaTeX">${q}$ </tex-math></inline-formula>-rung orthopair fuzzy functions (<inline-formula> <tex-math notation="LaTeX">${q}$ </tex-math></inline-formula>-ROFFs). Recently, the authors proposed derivatives and differentials of <inline-formula> <tex-math notation="LaTeX">${q}$ </tex-math></inline-formula>-ROFFs in the framework of <inline-formula> <tex-math notation="LaTeX">${q}$ </tex-math></inline-formula>-ROFC. In this paper, we aim to further study the <inline-formula> <tex-math notation="LaTeX">${q}$ </tex-math></inline-formula>-rung orthopair fuzzy integral (<inline-formula> <tex-math notation="LaTeX">${q}$ </tex-math></inline-formula>-ROFI). It is the most important and fundamental part of the <inline-formula> <tex-math notation="LaTeX">${q}$ </tex-math></inline-formula>-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 <inline-formula> <tex-math notation="LaTeX">${q}$ </tex-math></inline-formula>-ROFFs. In particular, we begin with the indefinite integral of <inline-formula> <tex-math notation="LaTeX">${q}$ </tex-math></inline-formula>-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 <inline-formula> <tex-math notation="LaTeX">${q}$ </tex-math></inline-formula>-ROFFs under additive operations, and obtain the explicit integral formula. Some properties of <inline-formula> <tex-math notation="LaTeX">${q}$ </tex-math></inline-formula>-ROFIs, such as comparison, algebraic operations, and mean value theorem are analyzed. Finally, we generalize the <inline-formula> <tex-math notation="LaTeX">${q}$ </tex-math></inline-formula>-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 <inline-formula> <tex-math notation="LaTeX">${q}$ </tex-math></inline-formula>-rung orthopair fuzzy data with the <inline-formula> <tex-math notation="LaTeX">${q}$ </tex-math></inline-formula>-ROFIs.