拓展区间Fermatean模糊前景理论综合评价方法

doi:10.16381/j.cnki.issn1003-207x.2022.0663

摘要/Abstract

摘要：

针对现有聚合综合评价信息视角单一、忽略指标相关性以及评价者非理性心理给出的异常偏好信息等问题，本文基于区间Fermatean模糊评价信息，建立加权幂Heronian平均算子和拓展前景理论的综合评价方法。首先，采用区间Fermatean模糊集（IVFFS）表征群组专家的评价信息，更大范围内囊括偏好信息。为了解决输入指标的交互关系及整体均衡性问题，提出广义区间Fermatean模糊加权幂Heronian平均（GIVFFWPHM）算子，并讨论了算子的重要性质。随后，根据群组成员偏好确立了专家前景权重，定义两个区间Fermatean模糊数间的相关系数以确定指标的客观权重。此外，基于拓展前景理论建立了一种考虑群组专家有限理性的多指标综合评价流程框架。最后，选取5个区域就乡村振兴赋能脱贫攻坚成效评价进行算例分析，并进行稳健性检验和有效对比分析，验证了本文方法的可行性和优越性。

关键词: 区间Fermatean模糊集, 广义区间Fermatean模糊加权幂Heronian平均算子, 前景理论, 相关系数, 乡村振兴群组综合评价

Abstract:

In view of the existing problems of aggregative comprehensive evaluation information， such as single perspective， ignoring the correlation of indicators and abnormal preference information given by the irrational psychology of the evaluator， etc. Therefore， based on the interval-valued Fermatean fuzzy evaluation information， a comprehensive evaluation method of weighted power Heronian average operator and extended prospect theory is established in this paper. First， interval-valued Fermatean fuzzy sets （IVFFS） are used to represent the evaluation information of group experts， and the preference information is included in a larger range. In order to solve the interaction of input indices and the global equilibrium problem， a generalized interval-valued Fermatean fuzzy weighted power Heronian average （GIVFFWPHM） operator is proposed， and the important properties of the operator are discussed. Then， according to the different preferences of each evaluator in the group， the expert prospect weight is established， and the correlation coefficient between two interval-valued Fermatean fuzzy numbers is defined to determine the objective weight of the index. In addition， based on the extended prospect theory， a multi-index comprehensive evaluation process framework considering the bounded rationality of group experts is established. Finally， the feasibility and superiority of the proposed method are verified by robustness test and effective comparative analysis in five regions selected to evaluate the effectiveness of rural revitalization enabling poverty alleviation.

Key words: interval-valued Fermatean fuzzy set, generalized interval-valued Fermatean fuzzy weighted power Heronian mean operator, prospect theory, correlation coefficient, group comprehensive evaluation of rural vitalization

中图分类号:

C934

罗世华, 刘俊. 拓展区间Fermatean模糊前景理论综合评价方法[J]. 中国管理科学, 2025, 33(6): 129-139.

Shihua Luo, Jun Liu. A Comprehensive Evaluation Method Based on Extended Interval-Valued Fermatean Fuzzy Prospect Theory[J]. Chinese Journal of Management Science, 2025, 33(6): 129-139.

图/表 12

图1

表1

表2

表3

表4

表5

表6

表7

表8

算子聚合运算参数变化下的排序"

序号	$p, q$ 参数	综合前景感知价值	综合排序结果
1	$p = 0.5, q = 1$	$ψ 1 = 0.4899, ψ 2 = 0.5417, ψ 3 = 1.7847, ψ 4 = 0.6914, ψ 5 = 1.4665$	$A 3 ≻ A 5 ≻ A 4 ≻ A 2 ≻ A 1$
2	$p = 1 . q = 0.5$	$ψ 1 = 0.4899, ψ 2 = 0.5417, ψ 3 = 1.7847, ψ 4 = 0.6914, ψ 5 = 1.4665$	$A 3 ≻ A 5 ≻ A 4 ≻ A 2 ≻ A 1$
3	$p = 1, q = 2$	$ψ 1 = 0.4689, ψ 2 = 0.5418, ψ 3 = 1.7848, ψ 4 = 0.6911, ψ 5 = 1.4718$	$A 3 ≻ A 5 ≻ A 4 ≻ A 2 ≻ A 1$
4	$p = 2, q = 1$	$ψ 1 = 0.4689, ψ 2 = 0.5418, ψ 3 = 1.7848, ψ 4 = 0.6911, ψ 5 = 1.4718$	$A 3 ≻ A 5 ≻ A 4 ≻ A 2 ≻ A 1$
5	$p = 2, q = 5$	$ψ 1 = 0.4600, ψ 2 = 0.5415, ψ 3 = 1.7859, ψ 4 = 0.6910, ψ 5 = 1.4748$	$A 3 ≻ A 5 ≻ A 4 ≻ A 2 ≻ A 1$
6	$p = 5, q = 2$	$ψ 1 = 0.4600, ψ 2 = 0.5415, ψ 3 = 1.7859, ψ 4 = 0.6910, ψ 5 = 1.4748$	$A 3 ≻ A 5 ≻ A 4 ≻ A 2 ≻ A 1$
7	$p = 5, q = 10$	$ψ 1 = 0.4615, ψ 2 = 0.5412, ψ 3 = 1.7868, ψ 4 = 0.6912, ψ 5 = 1.4745$	$A 3 ≻ A 5 ≻ A 4 ≻ A 2 ≻ A 1$
8	$p = 10, q = 5$	$ψ 1 = 0.4615, ψ 2 = 0.5412, ψ 3 = 1.7868, ψ 4 = 0.6912, ψ 5 = 1.4745$	$A 3 ≻ A 5 ≻ A 4 ≻ A 2 ≻ A 1$

表8

图2

表9

不同综合评价方法排序结果"

评价方法	计算结果	综合排序结果
PWA	$s c 1 = 0.1014, s c 2 = 0.0701, s c 3 = 0.2519, s c 4 = 0.0136, s c 5 = 0.0860$	$A 3 ≻ A 1 ≻ A 5 ≻ A 2 ≻ A 4$
PWG	$s c 1 = 0.1081, s c 2 = 0.0768, s c 3 = 0.2537, s c 4 = 0.0148, s c 5 = 0.0973$	$A 3 ≻ A 1 ≻ A 5 ≻ A 2 ≻ A 4$
TOPSIS	$C I 1 = 0.4035, C I 2 = 0.4235, C I 3 = 0.5981, C I 4 = 0.4744, C I 5 = 0.5746$	$A 3 ≻ A 5 ≻ A 4 ≻ A 2 ≻ A 1$
VIKOR	$Q 1 = 1.0000, Q 2 = 0.4204, Q 3 = 0.0000, Q 4 = 0.5865, Q 5 = 0.7006$	$A 3 ≻ A 2 ≻ A 4 ≻ A 5 ≻ A 1$
PT	$V 1 = 0.2611, V 2 = 0.5010, V 3 = 0.7303, V 4 = 0.3852, V 5 = 0.5126$	$A 3 ≻ A 5 ≻ A 2 ≻ A 4 ≻ A 1$
IVFFS-PT	$ψ 1 = 0.4794, ψ 2 = 0.5418, ψ 3 = 1.7846, ψ 4 = 0.6912, ψ 5 = 1.4690$	$A 3 ≻ A 5 ≻ A 4 ≻ A 2 ≻ A 1$

表9

图3

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区域	C₁	C₂	C₃	C₄	C₅
A₁	{[0.51,0.60],[0.38,0.70]}	{[0.37,0.50],[0.62,0.69]}	{[0.24,0.53],[0.51,0.62]}	{[0.17,0.36],[0.57,0.72]}	{[0.11,0.25],[0.80,0.92]}
A₂	{[0.52,0.74],[0.22,0.35]}	{[0.77,0.92],[0.31,0.43]}	{[0.47,0.61],[0.60,0.85]}	{[0.36,0.42],[0.64,0.82]}	{[0.18,0.32],[0.33,0.65]}
A₃	{[0.63,0.82],[0.50,0.62]}	{[0.54,0.64],[0.65,0.73]}	{[0.53,0.66],[0.11,0.25]}	{[0.63,0.87],[0.42,0.56]}	{[0.47,0.76],[0.34,0.54]}
A₄	{[0.33,0.57],[0.64,0.81]}	{[0.47,0.62],[0.54,0.69]}	{[0.42,0.70],[0.60,0.65]}	{[0.65,0.80],[0.55,0.78]}	{[0.22,0.55],[0.55,0.90]}
A₅	{[0.15,0.25],[0.60,0.85]}	{[0.30,0.54],[0.53,0.60]}	{[0.47,0.63],[0.50,0.65]}	{[0.50,0.65],[0.40,0.60]}	{[0.20,0.45],[0.35,0.66]}

区域	C₁	C₂	C₃	C₄	C₅
A₁	{[0.30,0.65],[0.25,0.50]}	{[0.30,0.70],[0.65,0.84]}	{[0.20,0.63],[0.50,0.75]}	{[0.15,0.28],[0.55,0.80]}	{[0.17,0.32],[0.70,0.90]}
A₂	{[0.45,0.75],[0.20,0.35]}	{[0.70,0.83],[0.50,0.75]}	{[0.35,0.65],[0.30,0.85]}	{[0.41,0.62],[0.80,0.91]}	{[0.10,0.35],[0.30,0.75]}
A₃	{[0.33,0.62],[0.50,0.65]}	{[0.35,0.76],[0.55,0.80]}	{[0.50,0.86],[0.20,0.30]}	{[0.55,0.83],[0.42,0.74]}	{[0.40,0.75],[0.30,0.55]}
A₄	{[0.50,0.65],[0.34,0.53]}	{[0.30,0.75],[0.55,0.80]}	{[0.22,0.45],[0.70,0.82]}	{[0.55,0.70],[0.65,0.85]}	{[0.32,0.65],[0.72,0.83]}
A₅	{[0.15,0.45],[0.55,0.83]}	{[0.32,0.65],[0.43,0.70]}	{[0.75,0.84],[0.44,0.65]}	{[0.60,0.75],[0.32,0.40]}	{[0.20,0.41],[0.30,0.65]}

区域	C₁	C₂	C₃	C₄	C₅
A₁	{[0.50,0.75],[0.22,0.55]}	{[0.37,0.68],[0.55,0.80]}	{[0.24,0.65],[0.44,0.77]}	{[0.16,0.35],[0.55,0.84]}	{[0.15,0.41],[0.65,0.82]}
A₂	{[0.32,0.74],[0.21,0.36]}	{[0.48,0.72],[0.40,0.76]}	{[0.23,0.66],[0.33,0.85]}	{[0.27,0.61],[0.72,0.87]}	{[0.13,0.42],[0.30,0.55]}
A₃	{[0.30,0.65],[0.43,0.69]}	{[0.48,0.71],[0.50,0.78]}	{[0.52,0.89],[0.25,0.45]}	{[0.42,0.73],[0.50,0.84]}	{[0.42,0.85],[0.34,0.65]}
A₄	{[0.35,0.58],[0.43,0.65]}	{[0.32,0.47],[0.59,0.93]}	{[0.20,0.45],[0.67,0.82]}	{[0.45,0.76],[0.60,0.82]}	{[0.52,0.75],[0.63,0.81]}
A₅	{[0.12,0.55],[0.56,0.85]}	{[0.30,0.75],[0.40,0.70]}	{[0.57,0.75],[0.53,0.67]}	{[0.65,0.87],[0.30,0.50]}	{[0.23,0.45],[0.35,0.75]}

区域	C₁	C₂	C₃	C₄	C₅
A₁	0.1986	0.2087	0.1982	0.2107	0.2151
A₂	0.1911	0.1758	0.2141	0.2092	0.2003
A₃	0.1933	0.2024	0.1877	0.1823	0.1805
A₄	0.2063	0.2039	0.2017	0.1961	0.2065
A₅	0.2107	0.2092	0.1983	0.2017	0.1976

区域	C₁	C₂	C₃	C₄	C₅
A₁	0.1976	0.2032	0.2031	0.2039	0.2114
A₂	0.1903	0.1891	0.2072	0.2123	0.2004
A₃	0.2017	0.2003	0.1921	0.1950	0.1817
A₄	0.1963	0.2012	0.2082	0.2033	0.2072
A₅	0.2141	0.2062	0.1894	0.1855	0.1993