Abstract
Yager's q -rung orthopair fuzzy set is a generalization of fuzzy sets, whose prominent feature is that the q th 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.