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.

中文翻译：

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q-rung orthopair模糊环境下线积分的基本理论及其应用

在决策（DM）过程中，为了更自由地表达 DM 专家对成员和非成员等级的信念，Yager 提出了 q-rung orthopair 模糊集（q-ROFS）作为模糊集的扩展。最近，为了聚合离散和连续的 q-ROF 信息，许多学者提供了各种聚合算子。然而，仍然有一些连续的 q-ROF 信息不能通过现有的聚合算子聚合。在本文中，我们提出了 q-rung orthopair 模糊环境 (q-ROFE) 下的两种新型线积分，它们为涉及离散或连续 q-ROF 信息的多属性决策 (MADM) 提供了更广泛的方法选择范围. 我们首先构造 q-ROFE 下的第一种线积分形式并研究它们的性质，我们证明了特定条件均值定理，但否定了它的一般条件。此外，还提供了一些关于它的不等式。之后，我们给出了q-ROFE下第二种线积分形式的两种表征形式，研究了两种线积分之间的关​​系。更重要的是，我们展示了q-ROFE下的Green公式，它连接了现有的q-rung orthopair模糊积分理论。之后，我们分别在两条线积分的基础上提供了两种聚合算子。作为它们的应用，我们给出了基于两种线积分聚合算子的两种新颖的 MADM 方法。并给出了一些例子来演示线积分算子的聚合过程。