Abstract
Normal intuitionistic fuzzy numbers (NIFNs), which combine the normal fuzzy number (NFN) with intuitionistic number, can easily express the stochastic fuzzy information existing in real decision making, and power-average (PA) operator can consider the relationships of different attributes by assigned weighting vectors which depend upon the input arguments. In this paper, we extended PA operator to process the NIFNs. Firstly, we defined some basic operational rules of NIFNs by considering the interaction operations of intuitionistic fuzzy sets (IFSs), established the distance between two NIFNs, and introduced the comparison method of NIFNs. Then, we proposed some new aggregation operators, including normal intuitionistic fuzzy weighted interaction averaging (NIFWIA) operator, normal intuitionistic fuzzy power interaction averaging (NIFPIA) operator, normal intuitionistic fuzzy weighted power interaction averaging (NIFWPIA) operator, normal intuitionistic fuzzy generalized power interaction averaging (NIFGPIA) operator, and normal intuitionistic fuzzy generalized weighted power interaction averaging (NIFGWPIA) operator, and studied some properties and some special cases of them. Based on these operators, we developed a decision approach for multiple attribute decision-making (MADM) problems with NIFNs. The significant characteristics of the proposed method are that: (1) it is easier to describe the uncertain information than the existing fuzzy sets and stochastic variables; (2) it used the interaction operations in part of IFSs which could overcome the existing weaknesses in operational rules of NIFNs; (3) it adopted PA operator which could relieve the influence of unreasonable data given by biased decision makers; and (4) it made the decision-making results more flexible and reliable because it was with generalized parameter which could be regard as the risk attitude value of decision makers. Finally, an illustrative example is given to verify its feasibility, and to compare with the existing methods.
Highlights
Since Zadeh [1] proposed fuzzy set (FS), the research and applications based on FS have made many achievements, especially the interval numbers, triangular fuzzy numbers (TFNs) and trapezoidal fuzzy numbers (TrFNs) have become the important tools for expressing the fuzzy information
Normal intuitionistic fuzzy numbers (NIFNs); (3) it adopted power average (PA) operator which could relieve the influence of unreasonable data given by biased decision makers; and (4) it made the decision-making results more flexible and reliable because it was with generalized parameter which could be regard as the risk attitude value of decision makers
Comparing with the method proposed by Wang and Li [15], the advantages of the developed method in this paper are that it can give the comprehensive value of each alternative based on the power interaction averaging operators of NIFNs by considering the relationships of different attributes and the interaction between the membership degree (MD) and non-membership degree (NMD), and the advantages of the method proposed by
Summary
Since Zadeh [1] proposed fuzzy set (FS), the research and applications based on FS have made many achievements, especially the interval numbers, triangular fuzzy numbers (TFNs) and trapezoidal fuzzy numbers (TrFNs) have become the important tools for expressing the fuzzy information. Wang and Li [14] proposed some operational laws, the score function and comparison method for NIFNs, developed some induced intuitionistic normal fuzzy related aggregation operators and applied them to multiple attribute decision making (MADM). NIFNs can better express the stochastic and fuzzy information, and PA operator can better deal with the relationship between the fused values which can relieve the influence of unreasonable data given by biased decision makers, at the same time, the interaction operational laws for IFNs can take into the interactions between MD and NMD account and overcome the weaknesses in existing operational rules.
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