Pythagorean fuzzy set (PFS) is applied to the problems of multi-attribute decision-making, and a similarity measure of grey relational analysis (GRA) and PFS based novel method for MADM is presented, and the corresponding algorithm is designed. Firstly, a new similarity measurement method is proposed, and some important properties are proved. Furthermore, the ranking of decision options is achieved by comparing the grey relational similarity of each option with the optimal and worst ideal sequences of the PFS when the attribute values are considered as the Pythagorean fuzzy number (PFN). Finally, a detailed example and comparative experiments are illustrated to prove the effectiveness and correctness of the proposed method.

In this paper, a comprehensive evaluation method integrating the rough sets with the Intuitionistic Fuzzy Analytic Hierarchy Process (IFAHP) is proposed, in order to overcome the IFAHP not to accurately ascertain the weight of each attribute in terms of real numbers. This comprehensive evaluation method integrating with the advantages of objective and subjective weighting methods, and it makes up for the shortcomings of these two methods to a certain extent. First, the objective weight of each attribute is calculated using the knowledge granulation of the rough sets; second, a new evaluation function is proposed to improve the IFAHP by transferring the intuitionistic fuzzy number obtained into a real one and to obtain the subjective weight of attributes through normalization; third, the subjective and objective weight values are combined by a weighting coefficient to figure out the combined weight of attributes; finally, the values are fused to calculate the integrated evaluation value, thereby obtaining the ranking result. A numerical example shows the feasibility and validity of the ranking result obtained, which has been proved practical to solve the multi-attribute decision-making problems.

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