Multi-attribute decision-making method based on utility three-way decision in Pythagorean fuzzy environments

doi:10.11772/j.issn.1001-9081.2025091111

Journal of Computer Applications ›› 2026, Vol. 46 ›› Issue (10): 3280-3287.DOI: 10.11772/j.issn.1001-9081.2025091111

• Advanced computing • Previous Articles

Multi-attribute decision-making method based on utility three-way decision in Pythagorean fuzzy environments

Fuyue TAO¹^,², Lei WANG¹^,²(), Min LI¹^,²

^1.School of Information Engineering，Jiangxi University of Water Resources and Electric Power，Nanchang Jiangxi 330099，China
^2.Jiangxi Province Key Laboratory of Smart Water Conservancy （Jiangxi University of Water Resources and Electric Power），Nanchang Jiangxi 330099，China

Received:2025-09-23 Revised:2025-12-26 Accepted:2025-12-30 Online:2026-03-18 Published:2026-10-10
Contact: Lei WANG
About author:TAO Fuyue， born in 2001， M. S. candidate. Her research interests include granular computing， rough set.
LI Min， born in 1975， Ph. D.， professor. His research interests include data mining， knowledge discovery， machine learning.
Supported by:
National Natural Science Foundation of China(61562061)

毕达哥拉斯模糊环境下基于效用三支决策的多属性决策方法

陶富悦¹^,², 王磊¹^,²(), 黎敏¹^,²

^1.江西水利电力大学信息工程学院，南昌 330099
^2.智慧水利江西省重点实验室（江西水利电力大学），南昌 330099

通讯作者: 王磊
作者简介:陶富悦（2001—），女，安徽六安人，硕士研究生，主要研究方向：粒计算、粗糙集
黎敏（1975—），男，江西南昌人，教授，博士，主要研究方向：数据挖掘、知识发现、机器学习。
基金资助:
国家自然科学基金资助项目(61562061);国家自然科学基金资助项目(62566041)

Abstract

Abstract:

The traditional Multi-Attribute Decision-Making （MADM） methods are based on two-way decision， which involve only acceptance or rejection. Three-Way Decision （3WD） reduces decision risks by introducing a deferred decision. In the current complex and dynamic decision-making environments， applying 3WD to provide solutions to MADM problems has emerged as a hot research topic. Therefore， a utility three-way multi-attribute decision-making model based on Pythagorean Fuzzy Numbers （PFNs） similarity was proposed. First， the relative utility function was extended to the Pythagorean fuzzy environment， and a similarity measure of PFNs was integrated， to establish a similarity-based relative utility function. Then， the ideal superior and inferior reference sequences were obtained using the score functions， and grey relational coefficients were computed on the basis of distance measures between PFNs， thus proposing a method for calculating conditional probabilities based on Grey Relational Analysis （GRA）. Based on this， a ranking strategy was introduced via expected utility， thereby obtaining a three-way multi-attribute decision-making model based on PFN similarity. The feasibility of the proposed method was validated through a site selection assessment case for new municipal solid waste treatment plants. Experimental results demonstrate that the proposed method can categorize alternatives precisely and reduce decision risks. Compared to the traditional MADM methods， it incorporates objective weights with 3WD classification to overcome the limitation of only ranking； compared to the three-way MADM methods， its GRA optimized conditional probabilities are more consistent with the classification logic of 3WD.

Key words: Pythagorean Fuzzy Set (PFS), Multi-Attribute Decision-Making (MADM), Three-Way Decision (3WD), utility function, similarity measure, three-way multi-attribute decision-making

摘要：

传统的多属性决策（MADM）方法基于二支决策，即接受和拒绝，而三支决策（3WD）通过增加一个延迟决策降低决策风险。在当前复杂多变的决策环境下，利用3WD为MADM提供解决方案已经成为一个热门研究。本文提出一种基于毕达哥拉斯模糊数（PFN）相似度的效用三支多属性决策模型。首先，将相对效用函数拓展到毕达哥拉斯模糊环境下，结合PFN的相似度度量，建立一个基于相似度的相对效用函数；其次，利用记分函数得到理想优、劣母序列，结合PFN的距离度量计算灰色关联系数，进而提出一种基于灰色关联分析（GRA）的条件概率计算方法；在此基础上，利用期望效用引入排序策略得到基于PFN相似度的效用三支多属性决策模型。通过开发新的生活垃圾处理厂中的选址评估案例，验证该方法的可行性。实验结果表明，该方法可以精准划分方案分类，降低决策风险的可行性，相较于传统MADM方法，以客观权重结合3WD分类，突破了仅排序局限；相较于三支MADM方法，经灰色关联优化条件概率，更贴合3WD的分类逻辑。

关键词: 毕达哥拉斯模糊集, 多属性决策, 三支决策, 效用函数, 相似度度量, 三支多属性决策

CLC Number:

TP391

Fuyue TAO, Lei WANG, Min LI. Multi-attribute decision-making method based on utility three-way decision in Pythagorean fuzzy environments[J]. Journal of Computer Applications, 2026, 46(10): 3280-3287.

陶富悦, 王磊, 黎敏. 毕达哥拉斯模糊环境下基于效用三支决策的多属性决策方法[J]. 《计算机应用》唯一官方网站, 2026, 46(10): 3280-3287.

Figures/Tables 14

Tab. 1 Relative utility function

行动	$C$	$¬ C$
$a P$	$u P P$	$u P N = 0$
$a B$	$u B P = ξ u P P$	$u B N = ξ u N N$
$a N$	$u N P = 0$	$u N N$

Tab. 1 Relative utility function

行动	$C$	$¬ C$
$a P$	$u P P$	$u P N = 0$
$a B$	$u B P = ξ u P P$	$u B N = ξ u N N$
$a N$	$u N P = 0$	$u N N$

Tab. 2 Relative utility function under single attribute

行动	$X$	$¬ X$
$b P$	$u P P i j = s (z i j, z j +)$	$u P N i j = 0$
$b B$	$u B P i j = ξ u P P i j = ξ s (z i j, z j +)$	$u B N i j = ξ u N N i j = ξ s (z i j, z j -)$
$b N$	$u N P i j = 0$	$u N N i j = s (z i j, z j -)$

Tab. 2 Relative utility function under single attribute

行动	$X$	$¬ X$
$b P$	$u P P i j = s (z i j, z j +)$	$u P N i j = 0$
$b B$	$u B P i j = ξ u P P i j = ξ s (z i j, z j +)$	$u B N i j = ξ u N N i j = ξ s (z i j, z j -)$
$b N$	$u N P i j = 0$	$u N N i j = s (z i j, z j -)$

Tab. 3 Comprehensive relative utility function based on similarity

行动	$X$	$¬ X$
$b P$	$u P P i = ∑ j = 1 n w j u P P i j$	$u P N i = 0$
$b B$	$u B P i = ξ ∑ j = 1 n w j u B P i j$	$u B N i = ξ ∑ j = 1 n w j u B N i j$
$b N$	$u N P i = 0$	$u N N i = ∑ j = 1 n w j u N N i j$

Tab. 3 Comprehensive relative utility function based on similarity

行动	$X$	$¬ X$
$b P$	$u P P i = ∑ j = 1 n w j u P P i j$	$u P N i = 0$
$b B$	$u B P i = ξ ∑ j = 1 n w j u B P i j$	$u B N i = ξ ∑ j = 1 n w j u B N i j$
$b N$	$u N P i = 0$	$u N N i = ∑ j = 1 n w j u N N i j$

Fig. 1 Model framework

Tab. 4 Pythagorean fuzzy information system

备选方案	$v 1$	$v 2$	$v 3$	$v 4$	$v 5$	$v 6$
$a 1$	（0.38， 0.60）	（0.81， 0.38）	（0.75， 0.35）	（0.77， 0.19）	（0.72， 0.18）	（0.70， 0.43）
$a 2$	（0.70， 0.22）	（0.34， 0.40）	（0.31， 0.74）	（0.48， 0.33）	（0.30， 0.68）	（0.24， 0.76）
$a 3$	（0.64， 0.15）	（0.66， 0.37）	（0.83， 0.32）	（0.39， 0.63）	（0.82， 0.47）	（0.87， 0.20）
$a 4$	（0.81， 0.33）	（0.45， 0.33）	（0.60， 0.26）	（0.78， 0.43）	（0.37， 0.53）	（0.21， 0.73）
$a 5$	（0.14， 0.62）	（0.16， 0.61）	（0.21， 0.55）	（0.81， 0.35）	（0.15， 0.53）	（0.75， 0.13）

Tab. 4 Pythagorean fuzzy information system

备选方案	$v 1$	$v 2$	$v 3$	$v 4$	$v 5$	$v 6$
$a 1$	（0.38， 0.60）	（0.81， 0.38）	（0.75， 0.35）	（0.77， 0.19）	（0.72， 0.18）	（0.70， 0.43）
$a 2$	（0.70， 0.22）	（0.34， 0.40）	（0.31， 0.74）	（0.48， 0.33）	（0.30， 0.68）	（0.24， 0.76）
$a 3$	（0.64， 0.15）	（0.66， 0.37）	（0.83， 0.32）	（0.39， 0.63）	（0.82， 0.47）	（0.87， 0.20）
$a 4$	（0.81， 0.33）	（0.45， 0.33）	（0.60， 0.26）	（0.78， 0.43）	（0.37， 0.53）	（0.21， 0.73）
$a 5$	（0.14， 0.62）	（0.16， 0.61）	（0.21， 0.55）	（0.81， 0.35）	（0.15， 0.53）	（0.75， 0.13）

Tab. 5 Comprehensive relative utility function based on similarity for instance

效用函数	$a 1$	$a 2$	$a 3$	$a 4$	$a 5$
$u P P i$	0.876 0	0.584 7	0.813 3	0.687 0	0.593 4
$u B P i$	0.613 2	0.409 3	0.569 3	0.480 9	0.415 4
$u B N i$	0.420 2	0.597 7	0.402 6	0.499 1	0.535 9
$u N N i$	0.600 3	0.853 9	0.575 1	0.713 0	0.765 6

Tab. 5 Comprehensive relative utility function based on similarity for instance

效用函数	$a 1$	$a 2$	$a 3$	$a 4$	$a 5$
$u P P i$	0.876 0	0.584 7	0.813 3	0.687 0	0.593 4
$u B P i$	0.613 2	0.409 3	0.569 3	0.480 9	0.415 4
$u B N i$	0.420 2	0.597 7	0.402 6	0.499 1	0.535 9
$u N N i$	0.600 3	0.853 9	0.575 1	0.713 0	0.765 6

Tab. 6 Decision thresholds

备选方案	$α i$	$β i$	$P r (X [a i])$
$a 1$	0.615 2	0.227 0	0.657 6
$a 2$	0.773 1	0.384 9	0.350 7
$a 3$	0.622 6	0.232 6	0.600 8
$a 4$	0.707 7	0.307 9	0.498 4
$a 5$	0.750 7	0.356 1	0.406 4

Tab. 6 Decision thresholds

备选方案	$α i$	$β i$	$P r (X [a i])$
$a 1$	0.615 2	0.227 0	0.657 6
$a 2$	0.773 1	0.384 9	0.350 7
$a 3$	0.622 6	0.232 6	0.600 8
$a 4$	0.707 7	0.307 9	0.498 4
$a 5$	0.750 7	0.356 1	0.406 4

Tab. 7 Impact of parameter ξ on classification results

$ξ$	POS	BND	NEG
0.1	$a 1, a 2, a 3, a 4, a 5$	$∅$	$∅$
0.3	$a 1, a 3, a 4, a 5$	$∅$	$a 2$
0.5	$a 1, a 3$	$∅$	$a 2, a 4, a 5$
0.7	$a 1$	$a 3, a 4, a 5$	$a 2$
0.9	$∅$	$a 1, a 2, a 3, a 4, a 5$	$∅$

Tab. 7 Impact of parameter ξ on classification results

$ξ$	POS	BND	NEG
0.1	$a 1, a 2, a 3, a 4, a 5$	$∅$	$∅$
0.3	$a 1, a 3, a 4, a 5$	$∅$	$a 2$
0.5	$a 1, a 3$	$∅$	$a 2, a 4, a 5$
0.7	$a 1$	$a 3, a 4, a 5$	$a 2$
0.9	$∅$	$a 1, a 2, a 3, a 4, a 5$	$∅$

Tab. 8 Ranking results of different MADM methods

方法	排序结果
本文方法	$x 1 > x 4 > x 2 > x 3 > x 6 > x 5$
文献［18］方法	$x 1 > x 4 > x 6 > x 3 > x 2 > x 5$
文献［20］方法	$x 1 > x 4 > x 6 > x 3 > x 2 > x 5$
文献［22］方法	$x 1 > x 4 > x 6 > x 3 > x 2 > x 5$

Tab. 8 Ranking results of different MADM methods

方法	排序结果
本文方法	$x 1 > x 4 > x 2 > x 3 > x 6 > x 5$
文献［18］方法	$x 1 > x 4 > x 6 > x 3 > x 2 > x 5$
文献［20］方法	$x 1 > x 4 > x 6 > x 3 > x 2 > x 5$
文献［22］方法	$x 1 > x 4 > x 6 > x 3 > x 2 > x 5$

Fig. 2 Classification results of different three-way MADM methods

Tab. 9 Ranking results of different three-way MADM methods

方法	排序结果
本文方法	$x 1 > x 3 > x 4 > x 2 > x 6 > x 5$
文献［25］方法	$x 1 > x 5 > x 4 > x 2 > x 6 > x 3$
文献［28］方法	$x 1 > x 4 > x 5 > x 2 > x 6 > x 3$

Tab. 9 Ranking results of different three-way MADM methods

方法	排序结果
本文方法	$x 1 > x 3 > x 4 > x 2 > x 6 > x 5$
文献［25］方法	$x 1 > x 5 > x 4 > x 2 > x 6 > x 3$
文献［28］方法	$x 1 > x 4 > x 5 > x 2 > x 6 > x 3$

Fig. 3 Spearman’s rank correlation coefficient of ranking results of three methods

Tab. 10 Theoretical analysis of three methods

方法

损失/效用

函数

Tab. 11 Running time under different data scales

数据集	方案数	属性数	平均运行时间/ms
D₁	5	6	12.3
D₂	10	6	38.7
D₃	30	10	425.1
D₄	100	15	8 324.9

References 28

[1]	Zarbakhshnia N， Wu Y， Govindan K， et al. A novel hybrid multiple attribute decision-making approach for outsourcing sustainable reverse logistics［J］. Journal of Cleaner Production， 2020， 242： No.118461.
[2]	Shen F， Huang Q， Su H， et al. An outranking approach for multi-attribute group decision-making with interval-valued hesitant fuzzy information［J］. Engineering Applications of Artificial Intelligence， 2024， 137（Pt A）： No.109120.
[3]	Mendonça G H M， Ferreira F G D C， Cardoso R T N， et al. Multi-attribute decision making applied to financial portfolio optimization problem［J］. Expert Systems with Applications， 2020， 158： No.113527.
[4]	Konneh K V， Adewuyi O B， Gamil M M， et al. A scenario-based multi-attribute decision making approach for optimal design of a hybrid off-grid system［J］. Energy， 2023， 265： No.125663.
[5]	Yao Y. Three-way decisions with probabilistic rough sets［J］. Information Sciences， 2010， 180（3）： 341-353.
[6]	Jia F， Liu P. A novel three-way decision model under multiple-criteria environment［J］. Information Sciences， 2019， 471： 29-51.
[7]	Zhang Q， Xie Q， Wang G. A novel three-way decision model with decision-theoretic rough sets using utility theory［J］. Knowledge-Based Systems， 2018， 159： 321-335.
[8]	Zhan J， Ye J， Ding W， et al. A novel three-way decision model based on utility theory in incomplete fuzzy decision systems［J］. IEEE Transactions on Fuzzy Systems， 2022， 30（7）： 2210-2226.
[9]	Qi Z， Li H， Zhang K， et al. An attribute fuzzy concept-oriented three-way utility decision model in multi-attribute environments［J］. Applied Soft Computing， 2023， 143： No.110353.
[10]	Zhang K， Dai J， Xu Z. The criterion-oriented three-way ranking and clustering strategies in fuzzy decision environments［J］. IEEE Transactions on Fuzzy Systems， 2022， 30（9）： 3841-3856.
[11]	Chen X Z， Zou L G. Three-way decision models based on ideal relations in multi-attribute decision-making［J］. Entropy， 2022， 24（7）： No.986.
[12]	Zhan J， Jiang H， Yao Y. Three-way multiattribute decision-making based on outranking relations［J］. IEEE Transactions on Fuzzy Systems， 2021， 29（10）： 2844-2858.
[13]	Ye J， Zhan J， Xu Z. A novel decision-making approach based on three-way decisions in fuzzy information systems［J］. Information Sciences， 2020， 541： 362-390.
[14]	Zadeh L. Fuzzy sets［J］. Information and Control， 1965， 8（3）： 338-353.
[15]	Atanassov K T. Intuitionistic fuzzy sets［J］. Fuzzy Sets and Systems， 1986， 20（1）： 87-96.
[16]	Yager R R. Pythagorean fuzzy subsets［C］// IFSA/NAFIPS 2013. Piscataway： IEEE， 2013： 57-61.
[17]	Ren P， Xu Z， Gou X. Pythagorean fuzzy TODIM approach to multi-criteria decision making［J］. Applied Soft Computing， 2016， 42： 246-259.
[18]	李德清，曾文艺，尹乾. 勾股模糊集的距离测度及其在多属性决策中的应用［J］. 控制与决策， 2017， 32（10）： 1817-1823.
	Li Deqing， Zeng Wenyi， Yin Qian. Distance measures of Pythagorean fuzzy sets and their applications in multi-attribute decision making［J］. Control and Decision， 2017， 32（10）： 1817-1823.
[19]	蔡香香，王磊，王冲，等. 一种结合勾股模糊相似度的多属性决策方法［J］. 南京大学学报（自然科学）， 2023， 59（1）： 35-42.
	Cai Xiangxiang， Wang Lei， Wang Chong， et al. A multi-attribute decision-making approach with Pythagorean fuzzy similarity［J］. Journal of Nanjing University （Natural Sciences）， 2023， 59（1）： 35-42.
[20]	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.
[21]	Wang H， He S， Li C， et al. Pythagorean uncertain linguistic variable Hamy mean operator and its application to multi-attribute group decision making［J］. IEEE/CAA Journal of Automatica Sinica， 2019， 6（2）： 527-539.
[22]	陈六新，罗南方. 基于前景理论的勾股模糊多属性决策［J］. 系统工程理论与实践， 2020， 40（3）： 726-735.
	Chen Liuxin， Luo Nanfang. Pythagorean fuzzy multi-criteria decision-making based on prospect theory［J］. Systems Engineering — Theory and Practice， 2020， 40（3）： 726-735.
[23]	Liu P， Wang Y， Jia F， et al. A multiple attribute decision making three-way model for intuitionistic fuzzy numbers［J］. International Journal of Approximate Reasoning， 2020， 119： 177-203.
[24]	Zhang H， Liu Y， Selang D， et al. A three-way decision with prospect-regret theory under Pythagorean fuzzy environments［J］. Applied Soft Computing， 2024， 167（Pt A）： No.112286.
[25]	Liang D， Xu Z， Liu D， et al. Method for three-way decisions using ideal TOPSIS solutions at Pythagorean fuzzy information［J］. Information Sciences， 2018， 435： 282-295.
[26]	Wang Y. Using the method of maximizing deviation to make decision for multi-indices［J］. Journal of Systems Engineering and Electronics， 1997， 8（3）： 21-26.
[27]	Zhang C， Hu Q， Zeng S， et al. IOWLAD-based MCDM model for the site assessment of a household waste processing plant under a Pythagorean fuzzy environment［J］. Environmental Impact Assessment Review， 2021， 89： No.106579.
[28]	Yi J H， Liu Y， Forrest J Y L， et al. A three-way decision approach with S-shaped utility function under Pythagorean fuzzy information［J］. Expert Systems with Applications， 2022， 210： No.118370.

Multi-attribute decision-making method based on utility three-way decision in Pythagorean fuzzy environments

毕达哥拉斯模糊环境下基于效用三支决策的多属性决策方法

RichHTML

PDF

Knowledge

Abstract

Cite this article

share this article

Figures/Tables 14

References 28

Related Articles 15

Recommended Articles

Metrics

[1]	Kai XU, Qikai GAO, Ming YIN, Jingjing TAN. Trajectory similarity measurement algorithm based on three-dimensional space area division [J]. Journal of Computer Applications, 2024, 44(1): 318-323.
[2]	Lianpeng QIU, Chengyun SONG. Noise robust dynamic time warping algorithm [J]. Journal of Computer Applications, 2023, 43(6): 1855-1860.
[3]	Mingchao NING, Junbo ZHANG, Ge CHEN. Task scheduling algorithm for service-oriented architecture-based industrial software [J]. Journal of Computer Applications, 2023, 43(3): 885-893.
[4]	Yudan CHEN, Cuifang GAO, Wanqiang SHEN, Ping YIN. Iterative intuitionistic fuzzy K-modes algorithm [J]. Journal of Computer Applications, 2022, 42(2): 375-381.
[5]	ZHANG Shipeng, LI Yongzhong, DU Xiangtong. Intrusion detection model based on semi-supervised learning and three-way decision [J]. Journal of Computer Applications, 2021, 41(9): 2602-2608.
[6]	YANG Mengmeng, ZHANG Aihua. Fractal image compression based on gray-level co-occurrence matrix and simultaneous orthogonal matching pursuit [J]. Journal of Computer Applications, 2021, 41(5): 1445-1449.
[7]	HU Lihua, ZUO Weijian, NIE Yaoyao. Feature matching method based on weighted similarity measurement [J]. Journal of Computer Applications, 2021, 41(2): 511-516.
[8]	Yubin ZHOU, Hong XIAO, Tao WANG, Wenchao JIANG, Meng XIONG, Zhongtang HE. Health index construction and remaining useful life prediction of mechanical axis based on action cycle degradation similarity measurement [J]. Journal of Computer Applications, 2021, 41(11): 3192-3199.
[9]	Yike LYU, Kai XU, Zhenqiang HUANG. Trajectory similarity measurement method based on area division [J]. Journal of Computer Applications, 2020, 40(2): 578-583.
[10]	MA Weiping, LI Wenxin, SUN Jinchuan, CAO Pengxia. Binocular vision target positioning method based on coarse-fine stereo matching [J]. Journal of Computer Applications, 2020, 40(1): 227-232.
[11]	REN Jie, MIN Fan, WANG Min. Cost-sensitive active learning through farthest distance sum sampling [J]. Journal of Computer Applications, 2019, 39(9): 2499-2504.
[12]	FU Lidong, HAO Wei, LI Dan, LI Fan. Community dividing algorithm based on similarity of common neighbor nodes [J]. Journal of Computer Applications, 2019, 39(7): 2024-2029.
[13]	ZHANG Yonglai, ZHOU Yaojian. Review of clustering algorithms [J]. Journal of Computer Applications, 2019, 39(7): 1869-1882.
[14]	JIANG Yifan, YE Qing. Time series similarity measure based on Siamese neural network [J]. Journal of Computer Applications, 2019, 39(4): 1041-1045.
[15]	TENG Lei, LI Yuan, LI Zhixing, HU Feng. Cross-social network user alignment algorithm based on knowledge graph embedding [J]. Journal of Computer Applications, 2019, 39(11): 3198-3203.