Yager's q-rung orthopair fuzzy set is a generalization of fuzzy sets, whose prominent feature is that the qth power sum of the membership and the nonmembership degrees is equal to or less than one, and we call its core, an ordered pair, q-rung orthopair fuzzy number (q-ROFN). More recently, the scholars have constructed the q-rung orthopair fuzzy calculus (q-ROFC), which can effectively deal with continuous q-rung orthopair fuzzy information. Nevertheless, the q-ROFC is only based on the basic operational laws of the q-ROFNs, in fuzzy theory, Archimedean t-norms and t-conorms (ATTs) are a significant class of continuous triangular norms and conorms, which are the generalizations of the intersection and union related to fuzzy sets. Thus, in order to extend the q-ROFC to a wider area, in this article, we systematically discuss the q-rung orthopair fuzzy double integrals (q-ROFDIs) in the frame of ATTs. First, we construct the q-ROFDI in the frame of Archimedean t-conorms in detail, and then provide its concrete value. In addition, we reveal the relationships with respect to two types of q-rung orthopair fuzzy spaces. Based on which, we can easily obtain another types of q-ROFDI. After that, we investigate their fundamental properties in detail so as to comprehend these kinds of q-ROFDIs in-depth. Finally, we point out the essences of these kinds of q-ROFDIs, on the basis of which we provide a practical application to show their effectiveness and elasticity via comparing with the existing methods.

The q-rung orthopair fuzzy set ( q-ROFS) that the sum of the qth power of the membership degree and the qth power of the nonmembership degree is restricted to one is a generalization of fuzzy set (FS). Recently, many researchers have given a series of aggregation operators to fuse q-rung orthopair fuzzy discrete information. Subsequently, although some scholars have also focused on studying q-rung orthopair fuzzy continuous information and give its continuity, derivative, differential, and integral, those studies are only considered from the perspective of multivariable fuzzy functions. Thus, the main aim of the paper is to study the q-rung orthopair fuzzy continuous single variable information. In this paper, we first define the concept of q-rung orthopair single variable fuzzy function ( q-ROSVFF) to describe the fuzzy continuous information, and give its domain to make sure that this kind of function is meaningful. Afterward, we propose the limits, continuities, and infinitesimal of q-ROSVFFs, and offer the relationship between the limit of q-ROSVFF and that of q-ROSVFF infinitesimal. On the basis of the definition of derivative in mathematical analysis, we define the subtraction and division derivatives and basic operational rules, and offer the simpler proofs for the derivatives of q-ROSVFFs. What is more, we propose the subtraction and division differential invariances, and give the approximate calculation formulas of q-ROSVFFs when the value of independent variable is changed small enough. In the real situation, fundamental functions cannot be used to express more complicated functions, thus we define the compound q-ROSVFFs and give their chain rules of subtraction and division derivatives. Finally, we use numerical examples by simulation to verify the feasibility and veracity of the approximate calculation on q-ROSVFFs.

Yager's q-rung orthopair fuzzy sets (q-ROFSs), which extend Zadeh's fuzzy sets, use the membership and nonmembership functions to describe things' vague characteristics, and the sum of the qth-power for the membership and nonmembership functions is less than or equal to 1. More recently, some scholars have proposed a series of aggregation operators to fuse q-rung orthopair fuzzy discrete information. However, so far, there is no research on aggregating q-rung orthopair fuzzy continuous information. Thus, we proposed q-rung orthopair fuzzy definite integrals (q-ROFDIs) to fill this vacancy. First, we further study the operations of q-rung orthopair fuzzy numbers (q-ROFNs) that are the core of q-ROFSs. We also introduce the limit of a q-ROFN sequence. Subsequently, we construct the q-ROFDIs step-by-step, give their concrete values, and discuss their integrability criteria from two perspectives. From the perspectives of modern analysis and the operational laws of q-ROFNs, we investigate the q-ROFDIs in detail, which are concise and considerably different from the investigative techniques of the previous research on aggregating continuous information. Finally, a practical example is provided to show the effectiveness, elasticity, and superiority of the q-ROFDIs via comparing them with the existing methods.