Yager's q-rung orthopair fuzzy set (q-ROFS) is a powerful tool to handle uncertainty and vagueness in real life. It expands the spatial scope of membership and non-membership, and therefore has a wider range of constraints and stronger modeling capabilities. However, to date, there is no investigation for q-Rung Orthopair fuzzy derivatives and differntials, which are very important for further developing q-rung orthopair fuzzy calculus (q-ROFC). The basic elements of a q-ROFS are q-rung orthopair fuzzy numbers (q-ROFNs), based on which we propose the q-rung orthopair fuzzy functions (q-ROFFs) and discuss their continuities in detail. Subsequently, we study the derivative of the q-ROFF, which reveals an accurate description on rate of change for continuous q-ROFF. Next, the differential operation of q-ROFF is established, thereby it provides an effective approximation on nonlinear problem in the q-ROFF environment. Finally, we present some numerical examples as explicit applications of q-ROFC.

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.

Probabilistic hesitant fuzzy sets add the probability value corresponding to each degree of membership on the basis of hesitant fuzzy sets, so as to express the initial decision information given by experts more accurately and comprehensively. In this paper, we mainly study how to integrate large-scare probabilistic hesitant fuzzy information more efficiently. We first discuss some basic operation laws of probabilistic hesitant fuzzy numbers, based on these the concepts of continuous probabilistic hesitant fuzzy sets and continuous probabilistic hesitant fuzzy functions are defined. They are the main research objects of our research. We further explore definite integrals of the continuous probabilistic hesitant fuzzy functions and their related properties. They have direct and powerful applications in continuous probabilistic hesitant fuzzy environments, and lay the foundation for subsequent theoretical analysis. Based on the weight density function, we finally get the weighted-integral operator of continuous probabilistic hesitant fuzzy information. Then some important properties of this integration operator are studied, including normalization, monotonicity and boundedness, etc. We are also devoted to revealing the inner connection between the continuous probabilistic hesitant fuzzy weighted-integral operator and the probabilistic hesitant fuzzy weighted averaging operator, the latter is usually used when dealing with discrete information. At last, we state why it is necessary to introduce a novel aggregation method based on definite integrals, and then provide an application of the proposed method to prove its validity and rationality.