The theory of fuzzy sets considers that everything exhibits some elasticity and is a matter of degree. When fuzzy numbers are used to evaluate the judgements of decision makers (DMs) in pairwise comparisons of alternatives following the analytic hierarchy process, the flexibility experienced by DMs has been exhibited. In order to capture this aspect of flexibility, it is important to know how to realize the flexibility degree of fuzzy numbers and further present a method of realizing its quantification. In this paper, a definition of the flexibility degree of fuzzy numbers is proposed. Some formulas are proposed to quantify the flexibility and rigidity degrees of interval numbers, triangular fuzzy numbers, and trapezoidal fuzzy numbers. A group decision making (GDM) model is developed under the consideration of the flexibility of DMs. By considering the effects of the applied scale and the reciprocal relation, the flexibility degree of interval multiplicative reciprocal comparison matrices is further defined, which is used to evaluate the flexibility degree of the DM involved in the decision process. An RD-IOWGA operator is proposed to aggregate individual interval multiplicative reciprocal matrices by associating more importance to that with less flexibility. A new algorithm is shown to solve GDM problems with interval multiplicative reciprocal preference relations. Numerical studies are carried out to illustrate the new definitions and offer some comparative analysis. The observations reveal that the developed consensus method can be used to model the GDM with a dominant position.

中文翻译：

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模糊数的柔性度及其对群体决策模型的启示


模糊集理论认为一切事物都表现出一定的弹性，并且只是程度的问题。当模糊数用于评估决策者（DM）在遵循层次分析法的备选方案的成对比较中的判断时，DM 所体验到的灵活性就得到了体现。为了捕捉这一方面的灵活性，重要的是要知道如何实现模糊数的灵活性程度，并进一步提出实现其量化的方法。本文提出了模糊数柔性度的定义。提出了一些公式来量化区间数、三角模糊数和梯形模糊数的灵活性和刚性程度。考虑到DM的灵活性，提出了群体决策（GDM）模型。考虑应用尺度和倒数关系的影响，进一步定义了区间乘性倒数比较矩阵的灵活性程度，用于评价决策过程中DM的灵活性程度。提出了 RD-IOWGA 算子，通过将更重要的内容与灵活性较低的内容相关联来聚合各个区间乘法倒数矩阵。提出了一种新的算法来解决具有区间乘性互惠偏好关系的 GDM 问题。进行了数值研究来说明新的定义并提供一些比较分析。观察结果表明，所开发的共识方法可用于对具有主导地位的 GDM 进行建模。