A novel bidirectional projection measures of circular intuitionistic fuzzy sets and its application to multiple attribute group decision-making problems

Hu Wang; Hu Wang

doi:10.3934/math.2025468

AIMS Mathematics

2025, Volume 10, Issue 5: 10283-10307. doi: 10.3934/math.2025468

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A novel bidirectional projection measures of circular intuitionistic fuzzy sets and its application to multiple attribute group decision-making problems

Hu Wang ^,

College of Mathematics and Physics, Mianyang Teachers' College, Mianyang 621000, China

Received: 12 February 2025 Revised: 26 April 2025 Accepted: 29 April 2025 Published: 06 May 2025
MSC : 28E10, 90B50

Atanassov recently proposed a new circular intuitionistic fuzzy set (CIFS) as an extension of intuitionistic fuzzy sets to express uncertain information by a circle with centered membership, non-membership, and radius r. Circular intuitionistic fuzzy sets can express uncertain information more flexibly than the intuitionistic fuzzy set. In this paper, we first propose a new method for calculating the radius r of CIFSs by ordinary least squares (OLS). We introduce some notions, such as modules of the circular intuitionistic fuzzy set and the cosine of the included angle between membership and non-membership vectors of the circular intuitionistic fuzzy set. Then, we define a new bidirectional projection measure of circular intuitionistic fuzzy sets, which takes into account the difference between different CIFSs in terms of membership degree and non-membership degree and radius r. The proposed bidirectional projection measures show superiority compared with some recent research works through numerical examples. Finally, the method is applied to a multi-attribute decision-making problem with group expert decision-making to prove the effectiveness and accuracy of the method.
- intuitionistic fuzzy sets,
- circular intuitionistic fuzzy sets,
- bidirectional projection measures,
- multi-attribute group decision making
Citation: Hu Wang. A novel bidirectional projection measures of circular intuitionistic fuzzy sets and its application to multiple attribute group decision-making problems[J]. AIMS Mathematics, 2025, 10(5): 10283-10307. doi: 10.3934/math.2025468

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Abstract

Atanassov recently proposed a new circular intuitionistic fuzzy set (CIFS) as an extension of intuitionistic fuzzy sets to express uncertain information by a circle with centered membership, non-membership, and radius r. Circular intuitionistic fuzzy sets can express uncertain information more flexibly than the intuitionistic fuzzy set. In this paper, we first propose a new method for calculating the radius r of CIFSs by ordinary least squares (OLS). We introduce some notions, such as modules of the circular intuitionistic fuzzy set and the cosine of the included angle between membership and non-membership vectors of the circular intuitionistic fuzzy set. Then, we define a new bidirectional projection measure of circular intuitionistic fuzzy sets, which takes into account the difference between different CIFSs in terms of membership degree and non-membership degree and radius r. The proposed bidirectional projection measures show superiority compared with some recent research works through numerical examples. Finally, the method is applied to a multi-attribute decision-making problem with group expert decision-making to prove the effectiveness and accuracy of the method.

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