毕达哥拉斯犹豫模糊集多属性决策研究

doi:10.12305/j.issn.1001-506X.2024.03.24

摘要/Abstract

摘要：

针对属性间相互关联, 评价信息为毕达哥拉斯犹豫模糊信息的多属性决策问题, 首先通过研究犹豫度对决策结果的影响, 提出一种新的毕达哥拉斯犹豫模糊集得分函数, 解决了现有得分函数中存在的不足。其次, 提出一种最小公倍数规范化原则, 解决了现有方法容易引入误差的缺陷。最后, 针对属性关联的多属性决策问题, 基于λ-模糊测度与Choquet积分, 提出了一种拓展交互式多准则决策(interative multi-criteria decision-making, TODIM)方法, 既解决了属性关联的问题, 又通过前景理论反映了决策者的心理行为特征。实例分析与敏感性分析验证了所提算法的正确性与有效性。

关键词: 毕达哥拉斯犹豫模糊集, 得分函数, 最小公倍数规范化原则, 属性关联, 交互式多准则决策方法

Abstract:

In view of the multi-attribute decision-making problem in which the attributes are interrelated and the evaluation information is Pythagorean hesitation fuzzy information, first of all, by studying the influence of hesitation on the decision-making results, a new Pythagorean hesitation fuzzy set score function is proposed to solve the shortcomings of the existing score functions. Secondly, a least common multiple standardization principle is proposed to solve the defect that the existing methods are easy to introduce errors. Finally, for the multi-attribute decision-making problem of attribute association, based on λ-fuzzy measure and Choquet integral, an extended interative multi-criteria decision-making (TODIM) method is put forward, which not only solves the problem of criteria interaction, but also reflects the psychological behavior characteristics of decision makers through prospect theory. The correctness and effectiveness of the proposed algorithm are verified through case analysis and sensitivity analysis.

Key words: Pythagorean hesitation fuzzy set, score function, least common multiple normalization principle, attribute association, interative multi-criteria decision-making (TODIM) method

中图分类号:

C934

关欣, 刘赢. 毕达哥拉斯犹豫模糊集多属性决策研究[J]. 系统工程与电子技术, 2024, 46(3): 982-991.

Xin GUAN, Ying LIU. Research on multi-attribute decision-making for Pythagorean hesitation fuzzy sets[J]. Systems Engineering and Electronics, 2024, 46(3): 982-991.

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参考文献 29

1	ZADEH L A . Fuzzy sets[J]. Information & Control, 1965, 8 (3): 338- 353.
2	ATANASSOV K T . Intuitionistic fuzzy sets[J]. Fuzzy Sets & Systems, 1986, 20 (1): 87- 96.
3	YAGER R R , ABBASOV A M . Pythagorean membership grades, complex numbers and decision making[J]. International Journal of Intelligent Systems, 2013, 28 (5): 436- 452. doi: 10.1002/int.21584
4	YAGER R R . Pythagorean membership grades in multicriteria decision making[J]. IEEE Trans. on Fuzzy Systems, 2014, 22 (4): 958- 965. doi: 10.1109/TFUZZ.2013.2278989
5	TORRA V . Hesitant fuzzy sets[J]. International Journal of Intelligent Systems, 2010, 25 (6): 529- 539.
6	刘卫锋, 何霞. 毕达哥拉斯犹豫模糊集[J]. 模糊系统与数学, 2016, 30 (4): 107- 115.
	LIU W F , HE X . Pythagorean hesitant fuzzy set[J]. Fuzzy Systems and Mathematics, 2016, 30 (4): 107- 115.
7	刘卫锋, 何霞, 常娟. 毕达哥拉斯犹豫模糊集的相关测度[J]. 控制与决策, 2019, 34 (5): 1018- 1024.
	LIU W F , HE X , CHANG J . Correlation measures of Pythagorean hesitant fuzzy set[J]. Control and Decision, 2019, 34 (5): 1018- 1024.
8	WEI G W , MAO L . Dual hesitant Pythagorean fuzzy hamacher aggregation operators in multiple attribute decision making[J]. Archives of Control Sciences, 2017, 27 (3): 365- 395. doi: 10.1515/acsc-2017-0024
9	WEI G W , WANG J , WEI C , et al. Dual hesitant Pythagorean fuzzy Hamy mean operators in multiple attribute decision making[J]. IEEE Access, 2019, 7, 86697- 86716. doi: 10.1109/ACCESS.2019.2924974
10	TANG X Y , WEI G W . Dual hesitant Pythagorean fuzzy Bonferroni mean operators in multi-attribute decision making[J]. Archives of Control Sciences, 2019, 29 (2): 339- 386.
11	LIANG D C , XU Z S . The new extension of TOPSIS method for multiple criteria decision making with hesitant Pythagorean fuzzy sets[J]. Applied Soft Computing Journal, 2017, 60 (11): 167- 179.
12	常娟, 杜迎雪, 刘卫锋. 广义毕达哥拉斯犹豫模糊集混合加权距离测度及决策应用[J]. 浙江大学学报(理学版), 2021, 48 (3): 304- 313.
	CHANG J , DU Y X , LIU W F . Generalized Pythagorean hesitant fuzzy set hybrid weighted distance measure and its application to decision making[J]. Journal of Zhejiang University (Science Edition), 2021, 48 (3): 304- 313.
13	MUHAMMAD S A , SALEEM A , ASAD A , et al. Pythagorean hesitant fuzzy sets and their application to group decision making with incomplete weight information[J]. Journal of Intelligent & Fuzzy Systems, 2017, 33 (6): 3971- 3985.
14	WU Q , LIN W H , ZHOU L G , et al. Enhancing multiple attribute group decision making flexibility based on information fusion technique and hesitant Pythagorean fuzzy sets[J]. Computers & Industrial Engineering, 2019, 127 (1): 954- 970.
15	AKRAM M , LUQMAN A , KAHRAMAN C . Hesitant Pythagorean fuzzy ELECTRE-Ⅱ method for multi-criteria decision-making problems[J]. Applied Soft Computing, 2021, 108 (3): 107479.
16	AKRAM M , LUQMAN A , ALCANTUD J . An integrated ELECTRE-Ⅰ approach for risk evaluation with hesitant Pytha-gorean fuzzy information[J]. Expert Systems with Application, 2022, 200 (8): 116945.
17	林萍萍, 李登峰, 江彬倩, 等. 属性关联的双极容度多属性决策VIKOR方法[J]. 系统工程理论与实践, 2021, 41 (8): 2147- 2156.
	LIN P P , LI D F , JIANG B Q , et al. Bipolarcapacities multi-attribute decision making[J]. Systems Engineering-Theory & Practice, 2021, 41 (8): 2147- 2156.
18	方冰, 韩冰, 闻传花. 基于新型距离测度的概率犹豫模糊多属性群决策方法[J]. 控制与决策, 2022, 37 (3): 729- 736.
	FANG B , HAN B , WEN C H . Probabilistic hesitant fuzzy group decision-making based on new distance measure[J]. Control and Decision, 2022, 37 (3): 729- 736.
19	尹东亮, 崔国恒, 黄晓颖, 等. 基于改进得分函数和前景理论的区间毕达哥拉斯模糊多属性决策[J]. 系统工程与电子技术, 2022, 44 (11): 3463- 3469. doi: 10.12305/j.issn.1001-506X.2022.11.21
	YIN D L , CUI G H , HUANG X Y , et al. Interval Pythagoras fuzzy multi-attribute decision making based on improved score function and prospect theory[J]. Systems Engineering and Electronics, 2022, 44 (11): 3463- 3469. doi: 10.12305/j.issn.1001-506X.2022.11.21
20	李美娟, 卢锦呈. 基于一种新得分函数和累积前景理论的毕达哥拉斯模糊TOPSIS法[J]. 控制与决策, 2022, 37 (2): 483- 492.
	LI M J , LU J C . Pythagorean fuzzy TOPSIS based on novel score function and cumulative prospect theory[J]. Control and Decision, 2022, 37 (2): 483- 492.
21	谭春桥, 贾媛. 基于证据理论和前景理论的犹豫-直觉模糊语言多准则决策方法[J]. 控制与决策, 2017, 32 (2): 333- 339.
	TAN C Q , JIA Y . Multi-criteria decision-making method based on evidence theory and prospect theory for hesitant-intuitionistic fuzzy linguistic[J]. Control and Decision, 2017, 37 (2): 333- 339.
22	LOURENZUTTI R , KROHLING R A , REFORMAT M Z . Choquet based TOPSIS and TODIM for dynamic and heterogeneous decision making with criteria interaction[J]. Information Sciences, 2017, 408 (10): 41- 69.
23	TAN C Q , CHEN X H . Interval-valued intuitionistic fuzzy multicriteria group decision making based on VIKOR and choquet integral[J]. Journal of Applied Mathematics, 2013, (9): 656879.
24	WANG J Q , WANG J , CHEN Q H , et al. An outranking approach for multi-criteria decision-making with hesitant fuzzy linguistic term sets[J]. Information Sciences, 2014, 280 (10): 338- 351.
25	TIAN X L , NIU M L , ZHANG W K , et al. A novel TODIM based on prospect theory to select green supplier with q-rung orthopair fuzzy set[J]. Technological and Economic Development of Economy, 2020, 27 (2): 284- 310. doi: 10.3846/tede.2020.12736
26	WU P , ZHOU L G , CHEN H Y , et al. Additive consistency of hesitant fuzzy linguistic preference relation with a new expansion principle for hesitant fuzzy linguistic term sets[J]. IEEE Trans. on Fuzzy Systems, 2018, 27 (4): 716- 730.
27	WEI G , ALSAADI F E , HAYAT T , et al. A linear assignment method for multiple criteria decision analysis with hesitant fuzzy sets based on fuzzy measure[J]. International Journal of Fuzzy Systems, 2017, 19 (3): 607- 614. doi: 10.1007/s40815-016-0177-x
28	NIE R X , TIAN Z P , WANG J Q , et al. Pythagorean fuzzy multiple criteria decision analysis based on Shapley fuzzy measures and partitioned normalized weighted Bonferroni mean operator[J]. International Journal of Intelligent Systems, 2019, 34 (2): 297- 324. doi: 10.1002/int.22051
29	CHEN C Y , HUANG J J . Integration of genetic algorithms and neural networks for the formation of the classifier of the hierarchical Choquet integral[J]. Information Sciences, 2020, 537 (10): 46- 61.

物流公司	C₁	C₂	C₃	C₄
A₁	〈{0.3, 0.8}, {0.4}〉	〈{0.6, 0.8}, {0.4, 0.5}〉	〈{0.7, 0.8}, {0.4, 0.5}〉	〈{0.5, 0.7, 0.8}, {0.4}〉
A₂	〈{0.7, 0.8}, {0.3, 0.4}〉	〈{0.3, 0.4, 0.5}, {0.7}〉	〈{0.3, 0.4}, {0.7, 0.8}〉	〈{0.6, 0.7}, {0.3, 0.4}〉
A₃	〈{0.4, 0.5}, {0.6}〉	〈{0.4, 0.5, 0.6}, {0.5}〉	〈{0.7, 0.8}, {0.4, 0.5}〉	〈{0.8, 0.9}, {0.3}〉
A₄	〈{0.5, 0.6}, {0.2, 0.3}〉	〈{0.7}, {0.4, 0.5}〉	〈{0.5, 0.6, 0.7}, {0.6}〉	〈{0.7, 0.8}, {0.2, 0.3}〉

属性组合	重要度	属性组合	重要度
ϕ	0	{C₂, C₃}	0.808 5
{C₁}	0.700 0	{C₂, C₄}	0.850 3
{C₂}	0.600 0	{C₃, C₄}	0.808 5
{C₃}	0.500 0	{C₁, C₂, C₃}	0.958 7
{C₄}	0.600 0	{C₁, C₂, C₄}	0.972 0
{C₁, C₂}	0.892 0	{C₁, C₃, C₄}	0.958 7
{C₁, C₃}	0.860 0	{C₂, C₃, C₄}	0.937 2
{C₁, C₄}	0.892 0	{C₁, C₂, C₃, C₄}	1

方法		全局比较值/贴近度	排序结果
传统TODIM法		ξ(x₁)=1, ξ(x₂)=0, ξ(x₃)=0.601 6, ξ(x₄)=0.743 7	A₁＞A₄＞A₃＞A₂
TOPSIS方法	D_PHFWA	δ(x₁)=0.549, δ(x₂)=0.281, δ(x₃)=0.654, δ(x₄)=0.592	A₃＞A₄＞A₁＞A₂
	D_PHFOWA	δ(x₁)=0.671, δ(x₂)=0.208, δ(x₃)=0.619, δ(x₄)=0.622	A₁＞A₄＞A₃＞A₂
	D_GPHFHWA	δ(x₁)=0.703, δ(x₂)=0.047, δ(x₃)=0.679, δ(x₄)=0.603	A₁＞A₃＞A₄＞A₂
本文算法		ξ(x₁)=1, ξ(x₂)=0, ξ(x₃)=0.801 6, ξ(x₄)=0.772 5	A₁＞A₃＞A₄＞A₂

[1]	尹东亮, 崔国恒, 黄晓颖, 张欢. 基于改进得分函数和前景理论的区间值毕达哥拉斯模糊多属性决策[J]. 系统工程与电子技术, 2022, 44(11): 3463-3469.
[2]	江文奇, 降晓璐. 面向属性关联的区间犹豫模糊型PROMETHEE决策方法[J]. 系统工程与电子技术, 2021, 43(11): 3250-3258.
[3]	尹胜, 杨桢, 陈思翼. 基于改进模糊熵的区间直觉模糊多属性决策[J]. 系统工程与电子技术, 2018, 40(5): 1079-1084.
[4]	张倩，陈桂明，颜宁，苏保忠. 直觉模糊和粗集的复合多属性保障力量优选[J]. Journal of Systems Engineering and Electronics, 2013, 35(8): 1697-1701.
[5]	王海鹏, 熊伟, 何友. 集中式多传感器广义相关算法[J]. Journal of Systems Engineering and Electronics, 2012, 34(10): 2045-2051.
[6]	司艳杰，魏法杰. 基于直觉模糊优选模型的混合型多属性决策[J]. Journal of Systems Engineering and Electronics, 2009, 31(12): 2893-2897.