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：

---

直觉区间值模糊偏好关系的同构乘性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）用于处理多标准决策（MCDM）中的不确定性。本文旨在探讨 IFPR 和 IVFPR 的同构乘性及物性，从而建立 MCDM 中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Tanino的模糊偏好关系的乘性及物性，建立了IFPR的乘性及物性的定义。证明它与由Zadeh可拓原理导出的IVFPR的乘法传递性同构。乘法及物性同构的用途有两个：1）发现 IFPR 和 IVFPR 之间的实质性关系，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 乘性及物性的可靠性。此外，基于现有的同构，通过严格的数学过程定义了IFPR的乘性一致性概念，并证明其满足以下几个理想性质：弱传递性、最大-最大传递性和中心划分传递性。研究了基于乘法一致性的多目标规划 (MOP) 模型，以从 IFPR 导出优先级向量。 该模型具有不丢失信息的优点，因为优先级向量表示与输入信息的表示一致，而现有方法则不然，现有方法中，由于建模传递性的结果而得出清晰的优先级向量，只是为了直观隶属函数而不是直观的非隶属函数。最后，给出了一个关于绿色供应选择的数值例子，验证了所提出的乘性一致性MOP模型的效率和实用性。