Isomorphic Multiplicative Transitivity for Intuitionistic and Interval-Valued Fuzzy Preference Relations and Its Application in Deriving Their Priority Vectors

Abstract

Intuitionistic fuzzy preference relations (IFPRs) are used to deal with hesitation while interval-valued fuzzy preference relations (IVFPRs) are for uncertainty in multi-criteria decision making (MCDM). This article 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 (FPRs). 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-divisiontransitivity. A multiplicative consistency based multi-objective 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 existing methods where crisp priority vectors were derived as a consequence of modelling transitivity just for the intuitionistic membership function and not for the intuitionistic non-membership function. Finally, a numerical example concerning green supply selection is given to validate the efficiency and practicality of the proposed multiplicative consistency MOP model.

On interval‐valued fuzzy decision‐making using soft likelihood functions

“…For the multiattribute decision-making problem in the interval-valued fuzzy environment, the assessment information of alternatives is generally expressed by IVFSs. To select the optimal alternative, some score functions [40][41][42] and representative values 43 are defined to measure a scalar value of IVFSs, and the bigger the value, the more preferred its corresponding IVFS. A sophisticated representative value of Atanassov-type intuitionistic membership grades called golden rule was proposed by Yager 43…”

“…So we obtain the index function σ k ( ) can be obtained by Equation (42) and the results are shown in Table 8.…”

Large Group Decision-Making Approach Based on Stochastic MULTIMOORA: An Application of Doctor Evaluation in Healthcare Service

“…He (2015a, 2015b) researched the theoretical and practical problems of the consensus strategy based on the minimum cost and maximum return with interval preference opinions. Stochastic multiattribute decision theory is applied to uncertainty mathematical theories [4][5][6][7][8][9][10][11][12][13][17][18][19][20], Logic-Mathematical theory [11], rough set theory [14], game theory [17], consensus decision [18][19][20], matrix games [4], fuzzy information [5,6], and so on.…”

Additively Consistent Interval-Valued Intuitionistic Fuzzy Preference Relations and Their Application to Group Decision Making

“…Later, Atanassov and Gargov [5] used the IFS to propose an interval-valued intuitionistic fuzzy set (IVIFS). Consequently, the interval-valued fuzzy preference relation (IVFPR) [6][7][8][9] and the interval-valued intuitionistic fuzzy (IVIF) preference relations (IVIFPR) [10][11][12][13][14] appeared in succession.…”