Distance measures of picture fuzzy sets and interval-valued picture fuzzy sets with their applications

Sijia Zhu; Zhe Liu; Sijia Zhu; Zhe Liu

doi:10.3934/math.20231525

AIMS Mathematics

2023, Volume 8, Issue 12: 29817-29848. doi: 10.3934/math.20231525

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Research article

Distance measures of picture fuzzy sets and interval-valued picture fuzzy sets with their applications

Sijia Zhu ^{1
,
,},
Zhe Liu ^{2
,
,}

1.
Cw Chu College, Jiangsu Normal University, Xuzhou, 221116, China
2.
School of Computer Sciences, Universiti Sains Malaysia, Penang, 11800, Malaysia

Received: 28 September 2023 Revised: 28 October 2023 Accepted: 30 October 2023 Published: 02 November 2023
MSC : 03E72, 28E10, 90B50

Picture fuzzy sets (PFSs) are a versatile generalization of fuzzy sets and intuitionistic fuzzy sets (IFSs), providing a robust framework for modeling imprecise, uncertain, and inconsistent information across various fields. As an advanced extension of PFSs, interval-valued picture fuzzy sets (IvPFSs) offer superior capabilities for handling incomplete and indeterminate information in various practical applications. Distance measures have always been an important topic in fuzzy sets and their variants. Some existing distance measures for PFSs have shown limitations and may yield counterintuitive results under certain conditions. Furthermore, there are currently few studies on distance measures for IvPFSs. To solve these problems, in this paper we devised a series of novel distance measures between PFSs and IvPFSs inspired by the Hellinger distance. Specifically, all the distance measures were divided into two parts: One considered the positive membership degree, neutral membership degree and negative membership degree, and the other added the refusal membership degree. Moreover, the proposed distance measures met some important properties, including boundedness, non-degeneracy, symmetry, and consistency, but also showed superiority compared to the existing measures, as confirmed through numerical comparisons. Finally, the proposed distance measures were validated in pattern recognition and medical diagnosis applications, indicating that the proposed distance measures can deliver credible, reasonable results, particularly in similar cases.
- picture fuzzy sets,
- interval-valued picture fuzzy sets,
- Hellinger distance,
- pattern recognition,
- medical diagnosis
Citation: Sijia Zhu, Zhe Liu. Distance measures of picture fuzzy sets and interval-valued picture fuzzy sets with their applications[J]. AIMS Mathematics, 2023, 8(12): 29817-29848. doi: 10.3934/math.20231525

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Abstract

Picture fuzzy sets (PFSs) are a versatile generalization of fuzzy sets and intuitionistic fuzzy sets (IFSs), providing a robust framework for modeling imprecise, uncertain, and inconsistent information across various fields. As an advanced extension of PFSs, interval-valued picture fuzzy sets (IvPFSs) offer superior capabilities for handling incomplete and indeterminate information in various practical applications. Distance measures have always been an important topic in fuzzy sets and their variants. Some existing distance measures for PFSs have shown limitations and may yield counterintuitive results under certain conditions. Furthermore, there are currently few studies on distance measures for IvPFSs. To solve these problems, in this paper we devised a series of novel distance measures between PFSs and IvPFSs inspired by the Hellinger distance. Specifically, all the distance measures were divided into two parts: One considered the positive membership degree, neutral membership degree and negative membership degree, and the other added the refusal membership degree. Moreover, the proposed distance measures met some important properties, including boundedness, non-degeneracy, symmetry, and consistency, but also showed superiority compared to the existing measures, as confirmed through numerical comparisons. Finally, the proposed distance measures were validated in pattern recognition and medical diagnosis applications, indicating that the proposed distance measures can deliver credible, reasonable results, particularly in similar cases.

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