doi:10.3785/j.issn.1008-9497.2022.04.002

Journal of Zhejiang University (Science Edition)

2022, Vol. 49

Issue (4): 398-407 DOI: 10.3785/j.issn.1008-9497.2022.04.002

Mathematics and Computer Science

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Abstract

Probabilistic hesitant fuzzy set (PHFS) is the extension of hesitant fuzzy set (HFS), that based on hesitant fuzzy set but adding a corresponding probability value for each membership degree. It is an effective tool to solve multi-attribute index decision-making problems, and can fully express the initial decision-making information given by experts. In this paper, we first introduce the basic definition and related operations of probability hesitant fuzzy number (PHFN), and present an improved method concerning the shortcomings of traditional PHFN score function and Hamming distance. Secondly, a new concept of Hamming distance and similarity of PHFN is proposed by appropriately supplementing the number of elements in membership set, and the similarity of probability hesitant fuzzy matrix (PHFM) is introduced according to the evaluation matrix given by experts. Finally, an interactive group evaluation method is given based on PHFM similarity in probabilistic hesitant fuzzy environment, and the effectiveness of the method is verified by an example.

Key words： probabilistic hesitant fuzzy number (PHFN) PHFN Hamming distance PHFN similarity interactive group evaluation

Received: 20 April 2021 Published: 13 July 2022

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

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https://www.zjujournals.com/sci/EN/Y2022/V49/I4/398

基于概率犹豫模糊相似度的交互式群体决策方法

概率犹豫模糊集（probabilistic hesitant fuzzy set，PHFS）是犹豫模糊集的推广，在犹豫模糊集基础上通过为每个隶属度添加与之相对应的概率，以全面表达专家赋予的初始决策信息，是处理多属性指标决策问题的一种有效工具。首先，介绍了概率犹豫模糊数（PHFN）的基本定义和相关运算，指出了传统PHFN的得分函数和汉明距离的不足，并给出了改进方法。其次，通过适当补充隶属度集合中的元素数量提出了PHFN的汉明距离和相似度概念，依据专家赋予的评价矩阵引入了概率犹豫模糊矩阵（PHFM）相似度。最后，在概率犹豫模糊环境下基于PHFM相似度提出了一种交互式群体评价算法，并用实例验证了算法的有效性。

关键词： 概率犹豫模糊数（PHFN）, PHFN汉明距离, PHFN相似度, 交互式群体评价

标准化PHFN	距离测度
标准化PHFN	按定义9计算	按本文定义10计算
$γ 1 = {0.4 / 0.3,0.6 / 0.7}, γ 2 = {0.3 / 0.4 ， 0.7 / 0.6}$	$d (γ 1, γ 2) = 0$	$D (γ 1, γ 2) = 0.042$
$γ 3 = {0.1 / ? 0.3,0.9 / ? 0.7}, γ 4 = {0.3 / ? 0.1 ， 0.7 / ? 0.9}$	$d (γ 3, γ 4) = 0$	$D (γ 3, γ 4) = 0.089$
$γ 5 = {0.1 / ? 0.4,0.9 / ? 0.6}, γ 6 = {0.4 / ? 0.1 ， 0.6 / ? 0.9}$	$d (γ 5, γ 6) = 0$	$D (γ 5, γ 6) = 0.129$

Table 1 Comparison of distance measures of three standardized PHFN

方案	$c 1$	$c 2$	$c 3$	$c 4$
Y₁	｛0.85/1｝	｛0.2/0.4，0.5/0.6｝	｛0.45/1｝	｛0.6/1｝
Y₂	｛0.7/1｝	｛0.1/0.3，0.6/0.7｝	｛0.57/1｝	｛0.31/1｝
Y₃	｛0.7/0.4，0.2/0.6｝	｛0.5/1｝	｛0.16/1｝	｛0.59/1｝
Y₄	｛0.3/0.7，0.56/0.1，0.7/0.2｝	｛0.3/1｝	｛0.18/1｝	｛0.4/1｝
Y₅	｛0.6/1｝	｛0.49/1｝	｛0.1/0.4，0.3/0.6｝	｛0.75/1｝

Table 2 The initial evaluation matrix

R (1)

given by the expert

l 1

方案	$c 1$	$c 2$	$c 3$	$c 4$
Y₁	｛0.73/1｝	｛0.6/0.3，0.8/0.7｝	｛0.3/1｝	｛0.82/1｝
Y₂	｛0.68/1｝	｛0.5/1｝	｛0.3/0.4，0.7/0.6｝	｛0.5/1｝
Y₃	｛0.55/1｝	｛0.59/1｝	｛0.1/0.2，0.3/0.4，0.6/0.4｝	｛0.5/1｝
Y₄	｛0.7/0.2，0.8/0.8｝	｛0.1/1｝	｛0.43/1｝	｛0.16/1｝
Y₅	｛0.5/0.2，0.6/0.8｝	｛0.5/1｝	｛0.2/1｝	｛0.5/1｝

Table 3 The initial evaluation matrix

R (2)

given by the expert

l 2

方案	$c 1$	$c 2$	$c 3$	$c 4$
Y₁	｛0.6/1｝	｛0.34/1｝	｛0.3/0.2，0.4/0.8｝	｛0.5/1｝
Y₂	｛0.1/1｝	｛0.25/1｝	｛0.31/1｝	｛0.63/1｝
Y₃	｛0.2/0.2，0.3/0.8｝	｛0.43/1｝	｛0.12/1｝	｛0.4/1｝
Y₄	｛0.4/1｝	｛0.2/0.2，0.4/0.5，0.5/0.3｝	｛0.1/1｝	｛0.12/1｝
Y₅	｛0.53/1｝	｛0.1/0.3，0.4/0.7｝	｛0.3/1｝	｛0.2/1｝

Table 4 The initial evaluation matrix

R (3)

given by the expert

l 3

方案	$c 1$	$c 2$	$c 3$	$c 4$
Y₁	｛0.747/1｝	｛0.404/0.12，0.490/0.18， 0.527/0.28，0.596/0.42｝	｛0.354/0.2，0.386/0.8｝	｛0.669/1｝
Y₂	｛0.558/1｝	｛0.304/0.3，0.468/0.7｝	｛0.407/0.4，0.553/0.6｝	｛0.496/1｝
Y₃	｛0.339/0.12，0.368/0.18， 0.523/0.08，0.544/0.32｝	｛0.501/1｝	｛0.127/0.2，0.147/0.4， 0.334/0.4｝	｛0.502/1｝
Y₄	｛0.498/0.14，0.562/0.56， 0.570/0.02，0.622/0.04， 0.624/0.08，0.669/0.16｝	｛0.204/0.2，0.277/0.5， 0.319/0.3｝	｛0.251/1｝	｛0.237/1｝
Y₅	｛0.545/0.2，0.578/0.8｝	｛0.387/0.3，0.465/0.7｝	｛0.204/0.4，0.268/0.6｝	｛0.535/1｝

Table 5 Comprehensive evaluation matrix

R

after integration

方案	$c 1$	$c 2$	$c 3$	$c 4$
Y₁	｛0.760/1｝	｛0.402/0.12，0.500/0.18， 0.529/0.28，0.606/0.42｝	｛0.361/0.2，0.388/0.8｝	｛0.677/1｝
Y₂	｛0.585/1｝	｛0.301/0.3，0.487/0.7｝	｛0.421/0.4，0.567/0.6｝	｛0.480/1｝
Y₃	｛0.343/0.12，0.367/0.18， 0.548/0.08，0.565/0.32｝	｛0.516/1｝	｛0.129/0.2，0.201/0.4， 0.340/0.4｝	｛0.513/1｝
Y₄	｛0.499/0.14，0.564/0.56， 0.580/0.02，0.637/0.04， 0.634/0.08，0.684/0.16｝	｛0.209/0.2，0.269/0.5， 0.305/0.3｝	｛0.257/1｝	｛0.252/1｝
Y₅	｛0.549/0.2，0.582/0.8｝	｛0.408/0.3，0.470/0.7｝	｛0.194/0.4，0.267/0.6｝	｛0.563/1｝

Table 6 Comprehensive evaluation matrix

R'

of expert group after modifying weight

方法	得分函数					方案排序
方法	$γ ˉ 1$	$γ ˉ 2$	$γ ˉ 3$	$γ ˉ 4$	$γ ˉ 5$	方案排序
文献［5］方法	0.489	0.356	0.213	0.331	0.287	$Y 1 ? Y 2 ? Y 4 ? Y 5 ? Y 3$
文献［10］方法	0.425	0.391	0.285	0.343	0.253	$Y 1 ? Y 2 ? Y 4 ? Y 3 ? Y 5$
本文方法	0.287	0.161	0.138	0.105	0.154	$Y 1 ? Y 2 ? Y 5 ? Y 3 ? Y 4$

Table 7 Comparison of the score function and comprehensive ranking obtained by three methods


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