Ranking method of Pythagorean fuzzy numbers characterized by curved trapezoidal area

doi:10.3785/j.issn.1008-9497.2022.04.001

Journal of Zhejiang University (Science Edition)

2022, Vol. 49

Issue (4): 391-397 DOI: 10.3785/j.issn.1008-9497.2022.04.001

Mathematics and Computer Science

Ranking method of Pythagorean fuzzy numbers characterized by curved trapezoidal area

Yujie TAO¹,Chunfeng SUO²(

)

^1.School of Mathematics，Tonghua Normal University, Tonghua 134002 Jilin Province China
^1.School of Mathematics and Statistics，Beihua University，, Jilin 134002 Jilin Province China

Download:

HTML( 7 )

PDF(974KB)
Export: BibTeX | EndNote (RIS)

Abstract

Pythagorean fuzzy set (PFS) is an extension of traditional intuitionistic fuzzy set (IFS). It can deal with decision-making problems with multi-attribute information in a wider area. In this paper, we first point out errors in the ranking criterion of Pythagorean fuzzy number (PFN) proposed in a paper, and analyze the reasons that cause these errors through the derivation of reliable information (accuracy function). Then, we propose a new score function through the curved trapezoidal area (CTA) corresponding to the reliable information, which provides a ranking criterion of PFN.The basic properties of the score function are discussed. Finally, we show an example indicating the effectiveness and advantage of the new ranking method.

Key words： Pythagorean fuzzy number (PFN) amount of reliable information curved trapezoidal area (CTA) score function ranking method

Received: 21 February 2022 Published: 13 July 2022

CLC:

O 159

Corresponding Authors: Chunfeng SUO E-mail: 1242362420@qq.com

	Service
	E-mail this article
	Add to my bookshelf
	Add to citation manager
	E-mail Alert
	RSS
	Articles by authors
	Yujie TAO
	Chunfeng SUO

Cite this article:

Yujie TAO,Chunfeng SUO. Ranking method of Pythagorean fuzzy numbers characterized by curved trapezoidal area. Journal of Zhejiang University (Science Edition), 2022, 49(4): 391-397.

URL:

https://www.zjujournals.com/sci/EN/Y2022/V49/I4/391

基于曲边梯形面积刻画毕达哥拉斯模糊数的排序方法

毕达哥拉斯模糊集（Pythagorean fuzzy set，PFS）是传统直觉模糊集（intuitionistic fuzzy set，IFS）的扩展，能在更广泛区域处理多属性信息决策问题。首先，针对某文献毕达哥拉斯模糊数（Pythagorean fuzzy number，PFN）排序方法存在的错误，分析了其产生原因。其次，在毕达哥拉斯模糊环境下，基于可靠信息量所对应的曲边梯形面积（curved trapezoidal area，CTA）提出了新的得分函数公式，进而给出了PFN的排序准则，并讨论了该得分函数的基本性质。最后，用实例说明给出的排序方法克服了其他方法的某些缺陷，具有一定优势。

关键词： 毕达哥拉斯模糊数（PFN）, 可靠信息量, 曲边梯形面积（CTA）, 得分函数, 排序方法

Table 1 Comparison of four ranking function formulas and their ranking criteria

序号	给定PFN $α ? i$ 和 $β ? i$	文献［10-12］方法		本文方法
序号	给定PFN $α ? i$ 和 $β ? i$	能否比较 $α ? i$ 和 $β ? i$	存在的缺陷	能否比较 $α ? i$ 和 $β ? i$	S_CTA（ $α$ ）	S_CTA（ $β$ ）	结果
1	$α ? 1 = (0.05,0.62) β ? 1 = (0.08,0.64)$	文献［10-12］均能比较	文献［10-12］与IFNs的结果矛盾	能	0.011 7	0.018 1	$β 1 ? α 1$
2	$α ? 2 = (0.5,0.5) β ? 2 = (0.6,0.6)$	文献［11-12］不能比较，文献［10］能比较	文献［11］得分函数值均为0.5，文献［12］得分函数值均为0.0，不能比较只能视为等价	能	0.152 2	0.157 6	$β ? 2 ? α 2$
3	$α ? 3 = (0.4,0.2) β ? 3 = (0.3,0.1)$	文献［10-12］均能比较	文献［11］与IFNs的结果矛盾	能	0.171 7	0.139 7	$α ? 3 ? β ? 3$
4	$α 4 = (0.4,0.1) β ? 4 = (0.5,0.4)$	文献［10-12］均能比较	文献［10］与IFNs的结果矛盾	能	0.175 0	0.190 7	$β 4 ? α ? 4$

Table 2 Comparison of the proposed method and the methods in references ［10-12］

Fig.1 Geometric diagram of information reliability derived from reference ［9］

Fig.2 Geometric diagram of reliable information area and hesitant information area of PFN

α

方法	$α ? 1 = (0.7,0.5) β ? 1 = (0.5,0.1)$		$α ? 2 = (0.4,0.4) β ? 2 = (0.6,0.6)$		$α ? 3 = (0.02,0.8) β ? 3 = (0.1,0.84)$
方法	得分值	排序结果	得分值	排序结果	得分值	排序结果
文献［10］	$S p x d (α ? 1) ? = ? 1.000 ? 9 S p x d (β ? 1) ? = ? 0.730 ? 6$	$α ? 1 ? β 1$	$S p x d (α ? 2) ? = ? 0.595 ? 2 S p x d (β ? 2) ? = ? 0.781 ? 3$	$β ? 2 ? α 2$	$S p x d (α ? 3) ? = ? 0.387 ? 9 S p x d (β ? 3) ? = ? 0.388 ? 3$	$β ? 3 ? α 3$
文献［11］	$V ? (α ? 1) ?? = ?? 0.590 ? 4 V ? ? (β ? 1) ? ? = ? 0.690 ? 9$	$β ? 1 ? α 1$	$V ? (α ? 2) ?? = ?? 0.500 ? 0 V ? ? (β ? 2) ? ? = ? 0.500 ? 0$	$α 2 ～ β ? 2$	$V ? (α ? 3) ?? = ?? 0.112 ? 6 V ? ? (β ? 3) ? ? = ? 0.140 ? 8$	$β ? 3 ? α 3$
文献［12］	$S p e n g (α ? 1) ? = ? 0.255 ? 5 S p e n g (β ? 1) ? = ? 0.284 ? 2$	$β ? 1 ? α 1$	$S p e n g (α ? 2) ? = ? 0.000 ? 0 S p e n g (β ? 2) ? = ? 0.000 ? 0$	$α 2 ～ β ? 2$	$S p e n g (α ? 3) ? = ? - 0.695 ? 2 S p e n g (β ? 3) ? = ? - 0.743 ? 2$	$α 3 ? β ? 3$
本文	$S C T A (α ? 1) ?? = ? 0.228 ? 3 S C T A ? (β ? 1) ? ? = ? 0.246 ? 2$	$β ? 1 ? α 1$	$S C T A ? (α ? 2) ?? = ?? 0.136 ? 3 S C T A ? (β ? 2) ? ? = ? 0.157 ? 6$	$β ? 2 ? α 2$	$S C T A (α ? 3) ?? = ?? 0.002 ? 9 S C T A (β ? 3) ? ? = ? 0.012 ? 3$	$β ? 3 ? α 3$

Table 3 A comparison of the proposed method and the ranking method in references ［10-12］


[1]	ATANASSOV K. Intuitionistic fuzzy sets ［J］. Fuzzy Sets and Systems，1986， 20（1）： 87-96. DOI：10. 1016/S0165-0114（86）80034-3 doi: 10. 1016/S0165-0114（86）80034-3

[2]	CHEN S M， TAN J M. Handling multicriteria fuzzy decision-making problems based on vague set theory ［J］. Fuzzy Sets and Systems， 1994， 67（2）：163-172. DOI：10.1016/0165-0114（94）90084-1 doi: 10.1016/0165-0114（94）90084-1

[3]	HONG D H， CHOI C H. Multicriteria fuzzy decision- making problems based on vague set theory ［C］// Fuzzy Sets and Systems.Amsterdam：Elsevier， 2000， 114： 103-113. DOI：10.1016/S0165-0114（98）00271-1 doi: 10.1016/S0165-0114（98）00271-1

[4]	YAGER R R， ABBASOV A M. Pythagorean membership grades， complex numbers and decision making［J］. International Journal of Intelligent Systems， 2013， 28： 436-452. DOI：10.1002/int. 21584 doi: 10.1002/int. 21584

[5]	YAGER R R. Pythagorean membership grades in multicriteria decision making［J］. IEEE Transactions on Fuzzy Systems， 2014， 22（4）： 958-965. DOI：10.1109/TFUZZ.2013.2278989 doi: 10.1109/TFUZZ.2013.2278989

[6]	ZHANG X L， XU Z S. Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets［J］. International Journal of Intelligent Systems， 2014， 29（12）：1061-1078．DOI：10.1002/int.21676 doi: 10.1002/int.21676

[7]	PENG X D， YANG Y. Some results for Pythagorean fuzzy sets［J］. International Journal of Intelligent Systems， 2015， 30（11）： 1133-1160． DOI：10. 1002/int.21738 doi: 10. 1002/int.21738

[8]	ZHANG X L. Multicriteria Pythagorean fuzzy decision analysis： A hierarchical QUALIFLEX approach with the closeness index-based ranking methods ［J］. Information Sciences， 2016， 33（1）：104-124. DOI：10.1016/j.ins.2015.10.012 doi: 10.1016/j.ins.2015.10.012

[9]	WAN S P， JIN Z， WANG F. A new ranking method for Pythagorean fuzzy numbers［C］// 12 International Conference on Intelligent Systems and Knowledge Engineering（ISKE）. Nanjing：IEEE， 2017. DOI： 10. 1109/ISKE.2017.8258763 doi: 10. 1109/ISKE.2017.8258763

[10]	PENG X D， DAI J. Approaches to Pythagorean fuzzy stochastic multi-criteria decision making based on prospect theory and regret theory with new distance measure and score function［J］. International Journal of Intelligent Systems， 2017， 32（1）：1187-1214. DOI：10.1002/int.21896 doi: 10.1002/int.21896

[11]	LI D Q， ZENG W Y. Distance measure of Pythagorean fuzzy sets［J］. International Journal of Intelligent Systems， 2018， 33（2）：348-361. DOI：10. 1002/int.21934 doi: 10. 1002/int.21934

[1]	Dan LI,Guijun WANG. The proof and generalization of Hamming distance measure in the Pythagorean fuzzy environment[J]. Journal of Zhejiang University (Science Edition), 2023, 50(4): 402-408.

[2]	Dan Li,Guijun WANG. Complete separability and approximation of the intuitionistic polygonal fuzzy number space.[J]. Journal of Zhejiang University (Science Edition), 2022, 49(5): 532-539.

[3]	. [J]. Journal of Zhejiang University (Science Edition), 2022, 49(4): 398-407.

[4]	LI Yanhong. The subadditivity and autocontinuity of the generalized quasi Sugeno integral[J]. Journal of Zhejiang University (Science Edition), 2020, 47(1): 67-71.

Viewed

Full text

Abstract

Cited

Shared

Discussed