Yue et al. [1] presented some theories of triangular intuitionistic fuzzy numbers (TIFNs) and triangular intuitionistic fuzzy sets (TIFSs) in Alexandria Engineering Journal. These theories have not been proved in the above reference; hence they are proved in this paper. In addition, this paper also provides some novel theorems and related proofs on TIFNs and TIFSs. Moreover, the presented theories are applied in the field of two-sided matching decision-making. A numerical case is adopted for demonstrating the effectiveness of the proposed theorems and method. This work will effectively complement the theories of TIFNs and TIFSs, and popularize their application scope.

Aiming at the multi-attribute bilateral matching problem with unknown attribute weights under a linguistic intuitionistic fuzzy environment, a decision method based on TODIM considering satisfaction and fairness degrees is proposed. First, the theories of linguistic intuitionistic fuzzy sets and bilateral matching are given, and the multi-attribute bilateral matching problem under a linguistic intuitionistic fuzzy environment is described. To solve this problem, according to linguistic intuitionistic fuzzy preference matrices, the overall attribute dominances are calculated based on TODIM; considering group consensus, a new method is proposed to calculate attribute weights based on linguistic intuitionistic fuzzy induced ordered weighted averaging (LIFIOWA) operator; then, the overall dominances of bilateral subjects are obtained by aggregating the overall attribute dominances and attribute weights. Furthermore, the overall dominances are standardized to calculate the satisfaction degrees of bilateral subjects; the fairness degrees of bilateral subjects are calculated considering the loss attenuation coefficient. Based on satisfaction degree matrices, fairness degree matrices and bilateral matching matrices, multiple bilateral matching models are established and then solved to obtain the optimal bilateral matching scheme. Finally, an example shows the effectiveness, reliability and accuracy of the proposed method. The research results indicate the following main characteristics of the proposed method: (1) A new method for calculating the unknown attribute weights using LIFIOWA operator is proposed. (2) According to the TODIM idea, a calculation method for fairness degree considering the loss attenuation coefficient is proposed. (3) Considering satisfaction and fairness degrees, multiple bilateral matching models under a linguistic intuitionistic fuzzy environment are established.

In view of the lack of research on graduate students and supervisors matching (GSSM) decision-making, this study proposes a novel many-to-one matching decision-making method for graduate students and supervisors (GSS) considering conformity psychology and graduate students’ preferences from a stability-based fairness perspective. First, the many-to-one matching problem for GSS is described. To tackle this problem, linguistic term matrices (LTMs) provided by bilateral subjects are transformed into Pythagorean fuzzy matrices (PFMs). On the one hand, according to the transference relation of graduate students, a conformity coefficient matrix is built. Then a PFM considering conformity psychology is established. Attribute weight vectors adjusted by conformity coefficients are formulated based on comparison information of graduate students and the conformity coefficient matrix. On this basis, a comprehensive PFM of graduate students is constructed. On the other hand, a preference coefficient matrix of graduate students is built by using TODIM (a Portuguese acronym for interactive and multicriteria decision making). And a PFM considering graduate students’ preferences is constructed. Attribute weight vectors of supervisors are determined based on attribute comparison information. On this basis, a comprehensive PFM of supervisors is set up. Furthermore, satisfaction matrices of GSS are constructed by using TOPSIS (technique for order preference by similarity to the ideal solution) and grey correlation degrees. A many-to-one GSSM model considering stability-based fairness is established by introducing matching matrix and stable matching matrix. The many-to-one matching model is then transformed into a one-to-one matching model by introducing virtual supervisor subjects; the optimal matching scheme between GSS is obtained by solving the above model. Finally, the feasibility, effectiveness and innovation of the proposed method are verified by an example analysis. The key findings of this study are listed as follows: (1) A new score of Pythagorean fuzzy numbers (PFNs) is proposed. (2) A method for converting linguistic term sets (LTSs) into PFNs is developed. (3) A weight calculation that combines conformity psychology with BWM is improved. (4) A novel method for satisfaction calculation considering conformity psychology and graduate students’ preferences is introduced. (5) A GSSM model considering stability-based fairness is built.