In some multi-attribute decision-making (MADM) models studying attributes’ interactive phenomena is very important for the minimizing decision risks. Usually, the Choquet integral type aggregations are considered in such problems. However, the Choquet integral aggregations do not consider all attributes’ interactions; therefore, in many cases, when these interactions are revealed in less degree, they do not perceive these interactions and their utility in MADM problems is less useful. For the decision of this problem, we create the Choquet integral-based new aggregation operators’ family which considers all pair interactions between attributes. The problem under the discrimination q-rung picture linguistic and q-rung orthopair fuzzy environments is considered. Construction of a 2-order additive fuzzy measure (TOAFM) involves pair interaction indices and importance values of attributes of a MADM model. Based on the attributes’ pair interactions for the identification of associated probabilities of a 2-order additive fuzzy measure, the Shapley entropy maximum principle is used. The associated probabilities q-rung picture linguistic weighted averaging (APs-q-RPLWA) and the associated probabilities q-rung picture linguistic weighted geometric (APs-q-RPLWG) aggregation operators are constructed with respect to TOAFM. For an uncertainty pole of experts’ evaluations on attributes regarding the possible alternatives, the associated probabilities of a fuzzy measure are used. The second pole of experts’ evaluations as arguments of the aggregation operators by discrimination q-rung picture linguistic values is presented. Discrimination q-rung picture linguistic evaluations specify the attribute’s dominant, neutral and non-dominant impacts on the selection of concrete alternative from all alternatives. Constructed operators consider the all relatedness between attributes in any consonant attribute structure. Main properties on the rightness of extensions are showed: APs-q-RPLWA and APs-q-RPLWG operators match with q-rung picture linguistic Choquet integral averaging and geometric operators for the lower and upper capacities of order two. The conjugation among the constructed operators is also considered. Connections between the new operators and the compositions of dual triangular norms $\left(T_p,S_p^q\right)$ and $(T_{\min },S_{\max })$ are also constructed. Constructed operators are used in evaluation of a selection reliability index (SRI) of candidate service centers in the facility location selection problem, when small degree interactions are observed between attributes. In example MADM, the difference in optimal solutions is observed between the Choquet integral aggregation operators and their new extensions. The difference, however, is due to the need to use indices of all interactions between attributes.

中文翻译：

---

判别q-Rung图片语言环境下交互式MADM的相关概率

在一些多属性决策（MADM）模型中，研究属性的交互现象对于最小化决策风险非常重要。通常，此类问题会考虑 Choquet 整数类型聚合。但是，Choquet 积分聚合并未考虑所有属性的交互；因此，在许多情况下，当这些交互作用被较少地揭示时，他们不会察觉到这些交互作用，并且它们在 MADM 问题中的效用就没有那么大了。为了解决这个问题，我们创建了基于 Choquet 积分的新聚合算子家族，它考虑了属性之间的所有对交互。考虑了区分q-rung图片语言和q-rung orthopair模糊环境下的问题。2 阶加性模糊测度 (TOAFM) 的构建涉及 MADM 模型属性的配对交互指数和重要性值。基于用于识别二阶加性模糊测度的关联概率的属性对相互作用，使用了Shapley 熵最大值原理。关联概率 q-rung 图片语言加权平均 (APs-q-RPLWA) 和关联概率 q-rung 图片语言加权几何 (APs-q-RPLWG) 聚合算子是相对于 TOAFM 构建的。对于专家对可能的替代方案的属性评估的不确定性极点，使用了模糊度量的相关概率。提出了专家评估的第二个极点，作为通过区分 q 级图片语言值的聚合算子的参数。辨别 q-rung 图片语言评估指定属性对从所有备选方案中选择具体备选方案的显性、中性和非显性影响。构造运算符考虑任何辅音属性结构中属性之间的所有相关性。扩展正确性的主要性质显示：APs-q-RPLWA 和APs-q-RPLWG 算子与q-rung 图片语言Choquet 积分平均和二阶上下容量的几何算子匹配。还考虑了构造运算符之间的共轭。新算子与对偶三角范数的组合之间的联系 区分 q 级图片语言评估指定属性对从所有备选方案中选择具体备选方案的显性、中性和非显性影响。构造运算符考虑任何辅音属性结构中属性之间的所有相关性。扩展正确性的主要性质显示：APs-q-RPLWA 和APs-q-RPLWG 算子与q-rung 图片语言Choquet 积分平均和二阶上下容量的几何算子匹配。还考虑了构造运算符之间的共轭。新算子与对偶三角范数的组合之间的联系 辨别 q-rung 图片语言评估指定属性对从所有备选方案中选择具体备选方案的显性、中性和非显性影响。构造运算符考虑任何辅音属性结构中属性之间的所有相关性。扩展正确性的主要性质显示：APs-q-RPLWA 和APs-q-RPLWG 算子与q-rung 图片语言Choquet 积分平均和二阶上下容量的几何算子匹配。还考虑了构造运算符之间的共轭。新算子与对偶三角范数的组合之间的联系 构造运算符考虑任何辅音属性结构中属性之间的所有相关性。扩展正确性的主要性质显示：APs-q-RPLWA 和APs-q-RPLWG 算子与q-rung 图片语言Choquet 积分平均和二阶上下容量的几何算子匹配。还考虑了构造运算符之间的共轭。新算子与对偶三角范数的组合之间的联系 构造运算符考虑任何辅音属性结构中属性之间的所有相关性。扩展正确性的主要性质显示：APs-q-RPLWA 和APs-q-RPLWG 算子与q-rung 图片语言Choquet 积分平均和二阶上下容量的几何算子匹配。还考虑了构造运算符之间的共轭。新算子与对偶三角范数的组合之间的联系 还考虑了构造运算符之间的共轭。新算子与对偶三角范数的组合之间的联系 还考虑了构造运算符之间的共轭。新算子与对偶三角范数的组合之间的联系$\left({吨}_{磷},{秒}_{磷}^q\right)$ 和 $({吨}_{\mathrm{分钟}},{秒}_{\mathrm{最大限度}})$也在建设中。当在属性之间观察到小程度的交互时，构造算子用于评估设施选址问题中候选服务中心的选择可靠性指数 (SRI)。在示例 MADM 中，在 Choquet 积分聚合运算符及其新扩展之间观察到最佳解决方案的差异。然而，不同之处在于需要使用属性之间所有交互的索引。