Intuitionistic fuzzy preference relations (IFPRs) are used to deal with hesitation, while interval-valued fuzzy preference relations (IVFPRs) are for uncertainty in multicriteria decision making (MCDM). This paper aims to explore the isomorphic multiplicative transitivity for IFPRs and IVFPRs, which builds the substantial relationship between hesitation and uncertainty in MCDM. To do that, the definition of the multiplicative transitivity property of IFPRs is established by combining the multiplication of intuitionistic fuzzy sets and Tanino's multiplicative transitivity property of fuzzy preference relations. It is proved to be isomorphic to the multiplicative transitivity of IVFPRs derived via Zadeh's extension principle. The use of the multiplicative transitivity isomorphism is twofold: 1) to discover the substantial relationship between IFPRs and IVFPRs, which will bridge the gap between hesitation and uncertainty in MCDM problems; and 2) to strengthen the soundness of the multiplicative transitivity property of IFPRs and IVFPRs by supporting each other with two different reliable sources, respectively. Furthermore, based on the existing isomorphism, the concept of multiplicative consistency for IFPRs is defined through a strict mathematical process, and it is proved to satisfy the following several desirable properties: weak transitivity, max–max transitivity, and center-division transitivity. A multiplicative consistency-based multiobjective programming (MOP) model is investigated to derive the priority vector from an IFPR. This model has the advantage of not losing information, as the priority vector representation coincides with that of the input information, which was not the case with the existing methods, where crisp priority vectors were derived as a consequence of the modeling transitivity just for the intuitionistic membership function and not for the intuitionistic nonmembership function. Finally, a numerical example concerning green supply selection is given to validate the efficiency and practicality of the proposed multiplicative consistency MOP model.

中文翻译：

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直觉和区间值模糊偏好关系的同构乘法传递性及其在推导优先向量中的应用

直觉模糊偏好关系 (IFPR) 用于处理犹豫，而区间值模糊偏好关系 (IVFPR) 用于多标准决策 (MCDM) 中的不确定性。本文旨在探索 IFPR 和 IVFPR 的同构乘法传递性，这在 MCDM 中建立了犹豫与不确定性之间的实质性关系。为此，结合直觉模糊集的乘法和模糊偏好关系的Tanino乘法传递性质，建立了IFPR的乘法传递性质的定义。它被证明与通过 Zadeh 扩展原理导出的 IVFPR 的乘法传递性是同构的。乘法传递同构的使用有两个方面：1）发现IFPRs和IVFPRs之间的实质关系，这将弥合MCDM问题中犹豫和不确定性之间的差距；2）通过分别用两个不同的可靠来源相互支持来加强 IFPR 和 IVFPR 的乘法传递性的合理性。此外，基于现有的同构，通过严格的数学过程定义了 IFPR 的乘法一致性概念，并证明它满足以下几个理想的性质：弱传递性、最大-最大传递性和中心分割传递性。研究了一种基于乘法一致性的多目标规划 (MOP) 模型，以从 IFPR 中导出优先级向量。这种模型的优点是不丢失信息，由于优先向量表示与输入信息的表示一致，而现有方法并非如此，其中清晰的优先向量是作为直觉隶属函数而不是直觉非隶属函数的建模传递性的结果得出的。最后，给出了一个关于绿色供应选择的数值例子，以验证所提出的乘法一致性 MOP 模型的效率和实用性。