Basic theory of line integrals under the q‐rung orthopair fuzzy environment and their applications

In the process of decision making (DM), to have more freedom in expressing DM experts' belief about membership and nonmembership grade, Yager presented the q‐rung orthopair fuzzy sets (q‐ROFSs) as an extension of fuzzy sets. Recently, for aggregating discrete and continuous q‐ROF information, many scholars provided various aggregation operators. However, there are still some kinds of continuous q‐ROF information cannot be aggregated through existing aggregation operators. In this paper, we present two novel kinds of line integrals under the q‐rung orthopair fuzzy environment (q‐ROFE), which supply a wider range on method choice for multiple attribute decision making (MADM) concerning discrete or continuous q‐ROF information. We first construct the first form of line integral under the q‐ROFE and study their properties, we prove the specific condition mean valued theorem but negate the general condition for it. Besides, several inequalities with regard to it are also provided. After that, we give two characterization forms of the second form of line integral under the q‐ROFE, we study the relationship between two kinds of line integrals. What is more important, we show the Green formula under q‐ROFE, which connect the existing q‐rung orthopair fuzzy integration theory. Afterward, we provide two kinds of aggregation operators on the basis of two line integrals, respectively. As their applications, we give two novel MADM methods based on two kinds of line integral aggregation operators. And some examples are shown to demonstrate the aggregation process of line integral operators. We not only stress their availabilities and superiorities on aggregating continuous q‐ROF information, but also comparing with other existing aggregation methods for emphasizing the novel methods' abilities when deal with abnormal q‐ROF information.