Abstract
The cubic uncertain fuzzy linguistic variable can easily express the fuzzy information, and the power average (PA) operator is a useful tool which provides more versatility in the information aggregation procedure. In this paper, we will combine the PA operator and Einstein operations to cubic uncertain linguistic environment and propose some new PA operators. Firstly, the definition and some basic operations of cubic uncertain linguistic number, power aggregation (PA) operator and Einstein operations are introduced. Then, we propose cubic uncertain linguistic fuzzy powered Einstein averaging operator, cubic uncertain linguistic fuzzy powered Einstein weighted (CULFPEWA) operator, cubic uncertain linguistic fuzzy Einstein geometric operator and cubic uncertain linguistic fuzzy Einstein weighted geometric (CULFPEWG) operator and discuss some properties of these in detail. Furthermore, we develop the decision-making methods for multi-attribute group decision-making problems with cubic uncertain linguistic information and give the detail decision steps. At last, an illustrate example is given to show the process of decision making and the effectiveness of the proposed method.
Highlights
Fuzzy set (FS) proposed by Zadeh [46] is a very valuable tool to develop the fuzzy information
In ‘‘Some cubic uncertain linguistic fuzzy powered Einstein operators’’ section, we propose the some cubic uncertain linguistic fuzzy powered Einstein operators, cubic uncertain linguistic fuzzy powered Einstein averaging operator, cubic uncertain linguistic fuzzy powered Einstein weighted operator, cubic uncertain linguistic fuzzy Einstein geometric (CULFPEG) operator and cubic uncertain linguistic fuzzy Einstein weighted geometric operator and introduce some properties and special cases of them. ‘‘The decisionmaking methods based on the CULFPEWA operator and CULFPEWG operator’’ section establishes the procedure of the decision-making method based on the CULFPEWA
The intuitionistic linguistic fuzzy number can be considered as a special case of cubic uncertain linguistic fuzzy numbers (CULFNs) when there is the element in membership and non-membership degrees
Summary
Fuzzy set (FS) proposed by Zadeh [46] is a very valuable tool to develop the fuzzy information. As FS has only a membership function, it is hard to term the more composite fuzzy information. Atanassov [3] further proposed the intuitionistic fuzzy set (IFS) which has a membership function and a non-membership function, so IFS has further advantages than FS on describing the Saleem Abdullah. Because the membership function and non-membership of IFS are crisp numbers which are hard to be acquired in real decision making, the choices of IFS are further extended [4]. Atanassov [5] proposed the interval-valued intuitionistic fuzzy set (IVIFS) which extended the membership and nonmembership to interval numbers. Zhang et al [47] gave the definition of the triangular intuitionistic fuzzy numbers
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