¹上海理工大学理学院，上海

²上海大学管理学院，上海

收稿日期：2022年9月26日；录用日期：2022年10月19日；发布日期：2022年10月28日

摘要

针对区间毕达哥拉斯犹豫模糊环境下的变量隶属度、非隶属度和属性的关联性融合问题，本文创新地将Heronian信息集成算子与区间毕达哥拉斯犹豫模糊数结合，定义了区间毕达哥拉斯犹豫模糊Heronian加权平均算子(IVPHFHWM)和区间毕达哥拉斯犹豫模糊Heronian几何加权算子(IVPHFGHWM)，研究了IVPHFHWM算子和IVPHFGHWM算子的置换不变性、单调性、有界性。然后，针对属性值为区间毕达哥拉斯犹豫模糊数的多属性问题，建立了基于区间毕达哥拉斯犹豫模糊Heronian加权算子的多属性决策模型，给出了一种综合的区间毕达哥拉斯犹豫模糊数的多属性决策方法。最后，通过多属性决策实例说明了Heronian信息集成算子在区间毕达哥拉斯犹豫模糊环境中的可行性和优越性。

关键词

区间毕达哥拉斯犹豫模糊数，Heronian算子，多属性决策

Interval Pythagorean Hesitating Fuzzy Heronian Information Integration Operator and Their Application to Multi-Attribute Decision Making

Yang Zhang¹, Ying Ji², Zixin Zhou¹, Ruijuan Geng¹

¹College of Science, University of Shanghai for Science and Technology, Shanghai

²School of Management, Shanghai University, Shanghai

Received: Sep. 26^th, 2022; accepted: Oct. 19^th, 2022; published: Oct. 28^th, 2022

ABSTRACT

For the correlation fusion of variable membership, non-membership and attribute in the interval Pythagorean hesitant fuzzy environment, this paper innovatively combined the Heronian information integration operator with the interval Pythagorean hesitant fuzzy number. Interval Pythagorean hesitant Fuzzy Heronian weighted mean operators (IVPHFHWM) and interval Pythagorean hesitant Fuzzy Heronian geometric weighted mean operators (IVPHFGHWM) are defined, and study the permutation invariance, monotonicity and boundedness of these operators. Then, establish a multi-attribute decision making model based on interval Pythagorean hesitant fuzzy Heronian weighting operator for multi-attribute problems with interval Pythagorean hesitant fuzzy number, and give a comprehensive multi-attribute decision making method. Finally, an example of multi-attribute decision making is given to illustrate the feasibility and rationality of the method.

Keywords:Interval Pythagorean Hesitant Fuzzy Number, Heronian Operator, Multi Attribute Decision Making Problem

Copyright © 2022 by author(s) and Hans Publishers Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

1. 引言

1986年Atanassov在经典模糊集理论 [1] 的基础上提出了直觉模糊集 [2]，随后，Atanassov等又将其推广并提出区间直觉模糊集理论 [3]，使用隶属度 $\mu$ 和非隶属度 $\nu$ 来描绘信息的不确定性，并满足 $\mu +\nu <1$。但在实际决策问题中可能出现隶属度和非隶属度之和大于1的情况，显然此时使用直觉模糊理论分析容易出现偏差。因此，Yager [4] [5] 引入了毕达哥拉斯模糊集来解决该问题，其满足隶属度 $\mu$ 和非隶属度 $\nu$ 的平方和不大于1，即 ${\mu }^2+{\nu }^2\le 1$。此后，Torra提出了犹豫模糊集 [6] 的概念，使用几个模糊数的集合来表示一个元素属于一个集合的隶属度，用来反映多个决策者对其不同的决策判断。近年来一些学者为了更加准确地描述决策信息的模糊性和不确定性，将毕达哥拉斯模糊集和犹豫模糊集结合起来提出了毕达哥拉斯犹豫模糊集。但是不同的学者对其有不同的构造方法，刘卫峰 [7] 和Khan [8] 认为一个毕达哥拉斯犹豫模糊数的隶属度和非隶属度分别属于两个犹豫模糊集合，Wei [9] 认为多个毕达哥拉斯模糊数组成的集合为一个毕达哥拉斯犹豫模糊数。Zhang [10] 对毕达哥拉斯犹豫模糊集进行推广，用区间数代替原有的实数的表示形式。提出了区间毕达哥拉斯犹豫模糊集，可以更加准确地描述实际问题中的模糊信息，具有更强的模糊信息处理和表达能力。

Zheng [11] 构造了基础的区间毕达哥拉斯犹豫模糊集的相关算子，并将其运用于多属性决策环境中。李龙妹 [12] 等在不确定环境下对区间毕达哥拉斯犹豫模糊集进行研究，给出度量不确定信息的模糊因子和直觉因子。黄月 [13] 将区间值毕达哥拉斯犹豫模糊信息融入到粗糙集进行分析，增强了处理不确定信息的鲁棒性。李进军 [14] 将Heronian算子引入到区间值毕达哥拉斯模糊集中，给出区间毕达哥拉斯模糊Heronian平均(IVPFHM)算子的定义及基本性质。李龙妹 [15] 等基于香农熵将模糊因子和直觉因子结合，提出了一个区间毕达哥拉斯犹豫模糊熵公式和相似性度量，并成功运用于决策问题中。到目前为止关于区间值毕达哥拉斯犹豫模糊集的研究还很少，而且还没有学者研究Heronian平均算子在区间毕达哥拉斯犹豫模糊集中的应用。

在现实决策问题中属性之间都存在相互依赖和影响性，而大多数的信息集成算子都无法考虑到属性之间的相互关联性。Heronian平均算子从属性变量之间的影响性出发，综合考虑变量之间的关联性，使得决策结果更加准确和客观。基于此，本文将Heronian平均算子推广至区间毕达哥拉斯犹豫模糊集中，首先定义IVPHFHWM算子和IVPHFGHWM算子，介绍并证明了IVPHFHWM算子和IVPHFGHWM算子的置换不变性、有界性和单调性。最后基于IVPHFHWM算子和IVPHFGHWM算子提出了一种基于优势理论的比率分析和全乘法形式的区间毕达哥拉斯犹豫模糊多属性决策排名方法(IVPHF-RAFMDT)，并通过实例验证和对比分析说明了该方法在区间毕达哥拉斯犹豫模糊环境中的有效性和优越性。

2. 预备知识

区间毕达哥拉斯犹豫模糊集

定义1 [10] 设X是论域，称P为X上一个区间毕达哥拉斯犹豫模糊集(IVPHFS)定义为：

$P=\left\{\langle x,h_P\left(x\right)\rangle |x\in X\right\}$

其中 $\begin{array}{l}{h}_P\left(x\right)=\{\langle {\Gamma }_P\left(x\right),{\Psi }_P\left(x\right)\rangle |{\Gamma }_P\left(x\right)=\left[{\mu }_P{\left(x\right)}^L,{\mu }_P{\left(x\right)}^U\right]\subseteq \left[0,1\right],\\ {\Psi }_P\left(x\right)=\left[{\nu }_P{\left(x\right)}^L,{\nu }_P{\left(x\right)}^U\right]\subseteq \left[0,1\right],{\left({\mu }_P{\left(x\right)}^U\right)}^2+{\left({\nu }_P{\left(x\right)}^U\right)}^2\le 1\}\end{array}$

其中 ${\Gamma }_P\left(x\right)$ 和 ${\Psi }_P\left(x\right)$ 分别表示X中的元素x属于集合P的可能隶属度区间和可能非隶属度区间。

称

$\begin{array}{l}{\pi }_P\left(x\right)=\{\underset{\begin{array}{l}{\mu }_P{\left(x\right)}^U,{\mu }_P{\left(x\right)}^L\in {\Gamma }_P\left(x\right)\\ {\nu }_P{\left(x\right)}^U,{\nu }_P{\left(x\right)}^L\in {\Psi }_P\left(x\right)\end{array}}{\cup }\left[{\pi }_P{\left(x\right)}^L,{\pi }_P{\left(x\right)}^U\right]|{\pi }_P{\left(x\right)}^L=\sqrt{1-{\left({\mu }_P{\left(x\right)}^U\right)}^2-{\left({\nu }_P{\left(x\right)}^U\right)}^2},\\ \begin{array}{c}\\ \\ \end{array}{\pi }_P{\left(x\right)}^U=\sqrt{1-{\left({\mu }_P{\left(x\right)}^L\right)}^2-{\left({\nu }_P{\left(x\right)}^L\right)}^2}\}\end{array}$

为区间值毕达哥拉斯犹豫模糊集的区间犹豫度集合。

为了方便，称 $\tilde{P}=h_{\tilde{P}}\left(x\right)$ 为一个区间毕达哥拉斯犹豫模糊数，其中 $\tilde{P}=\left\{\langle \tilde{\Gamma },\tilde{\Psi }\rangle |\tilde{\Gamma }=\left[{\mu }^L,{\mu }^U\right],\tilde{\Psi }=\left[{\nu }^L,{\nu }^U\right],{\left({\mu }^U\right)}^2+{\left({\nu }^U\right)}^2\le 1\right\}$。当 $\tilde{P}$ 中只含有一个元素时，则区间值毕达哥拉斯犹豫模糊集转化为区间值毕达哥拉斯模糊集。当 $\tilde{P}$ 中所有元素都满足 ${\mu }^L={\mu }^U$，${\nu }^L={\nu }^U$ 时，则区间值毕达哥拉斯犹豫模糊集转化为毕达哥拉斯犹豫模糊集。当 $\tilde{P}$ 中所有元素都满足 ${\mu }^U+{\nu }^U<1$ 时，则区间值毕达哥拉斯犹豫模糊集可以视为区间值直觉犹豫模糊集。

定义2 [10] 设 $\tilde{P}$，${\tilde{P}}_1$ 和 ${\tilde{P}}_2$ 为三个区间毕达哥拉斯犹豫模糊数，其中 $\tilde{P}=\left\{\langle \tilde{\Gamma },\tilde{\Psi }\rangle |\tilde{\Gamma }=\left[{\mu }^L,{\mu }^U\right],\tilde{\Psi }=\left[{\nu }^L,{\nu }^U\right]\right\}$，${\tilde{P}}_1=\left\{\langle {\tilde{\Gamma }}_1,{\tilde{\Psi }}_1\rangle |\tilde{\Gamma }=\left[{\mu }_1^L,{\mu }_1^U\right],\tilde{\Psi }=\left[{\nu }_1^L,{\nu }_1^U\right]\right\}$，${\tilde{P}}_2=\left\{\langle {\tilde{\Gamma }}_2,{\tilde{\Psi }}_2\rangle |\tilde{\Gamma }=\left[{\mu }_2^L,{\mu }_2^U\right],\tilde{\Psi }=\left[{\nu }_2^L,{\nu }_2^U\right]\right\}$，$\lambda >0$ 则

$\lambda \tilde{P}=\left\{\langle \left[\sqrt{1-{\left(1-{\left({\mu }^L\right)}^2\right)}^{\lambda }},\sqrt{1-{\left(1-{\left({\mu }^U\right)}^2\right)}^{\lambda }}\right],\left[{\left({\nu }^L\right)}^{\lambda },{\left({\nu }^U\right)}^{\lambda }\right]\rangle |\langle \tilde{\Gamma },\tilde{\Psi }\rangle \in \tilde{P}\right\}$

${\tilde{P}}^{\lambda }=\left\{\langle \left[{\left({\mu }^L\right)}^{\lambda },{\left({\mu }^U\right)}^{\lambda }\right],\left[\sqrt{1-{\left(1-{\left({\nu }^L\right)}^2\right)}^{\lambda }},\sqrt{1-{\left(1-{\left({\nu }^U\right)}^2\right)}^{\lambda }}\right]\rangle |\langle \tilde{\Gamma },\tilde{\Psi }\rangle \in \tilde{P}\right\}$

${\tilde{P}}^C=\left\{\langle \left[{\nu }^L,{\nu }^U\right],\left[{\mu }^L,{\mu }^U\right]\rangle |\langle \tilde{\Gamma },\tilde{\Psi }\rangle \in \tilde{P}\right\}$

$\begin{array}{l}{\tilde{P}}_1\oplus {\tilde{P}}_2=\{\langle \left[\sqrt{{\left({\mu }_1^L\right)}^2+{\left({\mu }_2^L\right)}^2-{\left({\mu }_1^L\right)}^2{\left({\mu }_2^L\right)}^2},\sqrt{{\left({\mu }_1^U\right)}^2+{\left({\mu }_2^U\right)}^2-{\left({\mu }_1^U\right)}^2{\left({\mu }_2^U\right)}^2}\right],\\ \underset{}{\overset{}{}}\left[{\nu }_1^L{\nu }_2^L,{\nu }_1^U{\nu }_2^U\right]\rangle |\langle {\tilde{\Gamma }}_i,{\tilde{\Psi }}_i\rangle \in {\tilde{P}}_i,i=1,2\}\end{array}$

$\begin{array}{l}{\tilde{P}}_1\otimes {\tilde{P}}_2=\{\langle \left[{\mu }_1^L{\mu }_2^L,{\mu }_1^U{\mu }_2^U\right],[\sqrt{{\left({\nu }_1^L\right)}^2+{\left({\nu }_2^L\right)}^2-{\left({\nu }_1^L\right)}^2{\left({\nu }_2^L\right)}^2},\\ \sqrt{{\left({\nu }_1^U\right)}^2+{\left({\nu }_2^U\right)}^2-{\left({\nu }_1^U\right)}^2{\left({\nu }_2^U\right)}^2}]\rangle |\langle {\tilde{\Gamma }}_i,{\tilde{\Psi }}_i\rangle \in {\tilde{P}}_i,i=1,2\}\end{array}$

$\begin{array}{l}{\tilde{P}}_1\cup {\tilde{P}}_2=\{\langle \left[\max \left\{{\mu }_1^L,{\mu }_2^L\right\},\max \left\{{\mu }_1^U,{\mu }_2^U\right\}\right],[\min \left\{{\nu }_1^L,{\nu }_2^L\right\},\\ \min \left\{{\nu }_1^U,{\nu }_2^U\right\}]\rangle |\langle {\tilde{\Gamma }}_i,{\tilde{\Psi }}_i\rangle \in {\tilde{P}}_i,i=1,2\}\end{array}$

$\begin{array}{l}{\tilde{P}}_1\cap {\tilde{P}}_2=\{\langle \left[\min \left\{{\mu }_1^L,{\mu }_2^L\right\},\min \left\{{\mu }_1^U,{\mu }_2^U\right\}\right],[\max \left\{{\nu }_1^L,{\nu }_2^L\right\},\\ \max \left\{{\nu }_1^U,{\nu }_2^U\right\}]\rangle |\langle {\tilde{\Gamma }}_i,{\tilde{\Psi }}_i\rangle \in {\tilde{P}}_i,i=1,2\}\end{array}$

定义3 [10] 设 $\tilde{P}=\left\{\langle \tilde{\Gamma },\tilde{\Psi }\rangle |\tilde{\Gamma }=\left[{\mu }^L,{\mu }^U\right],\tilde{\Psi }=\left[{\nu }^L,{\nu }^U\right]\right\}$ 为区间毕达哥拉斯犹豫模糊数，其得分函数 $S\left(\tilde{P}\right)$ 为

$\begin{array}{c}S\left(\tilde{P}\right)=\frac{1}{2\#h}{\displaystyle \underset{\langle \tilde{\Gamma },\tilde{\Psi }\rangle \in \tilde{P}}{\sum }\left({\tilde{\Gamma }}^2-{\tilde{\Psi }}^2\right)}\\ =\left[\frac{1}{2\#h}{\displaystyle \underset{\langle \tilde{\Gamma },\tilde{\Psi }\rangle \in \tilde{P}}{\sum }\left({\left({\mu }^L\right)}^2-{\left({\nu }^U\right)}^2\right)},\frac{1}{2\#h}{\displaystyle \underset{\langle \tilde{\Gamma },\tilde{\Psi }\rangle \in \tilde{P}}{\sum }\left({\left({\mu }^U\right)}^2-{\left({\nu }^L\right)}^2\right)}\right]\end{array}$

精确函数 $H\left(\tilde{P}\right)$ 为

$\begin{array}{c}H\left(\tilde{P}\right)=\frac{1}{2\#h}{\displaystyle \underset{\langle \tilde{\Gamma },\tilde{\Psi }\rangle \in \tilde{P}}{\sum }\left({\tilde{\Gamma }}^2+{\tilde{\Psi }}^2\right)}\\ =\left[\frac{1}{2\#h}{\displaystyle \underset{\langle \tilde{\Gamma },\tilde{\Psi }\rangle \in \tilde{P}}{\sum }\left({\left({\mu }^L\right)}^2+{\left({\nu }^L\right)}^2\right)},\frac{1}{2\#h}{\displaystyle \underset{\langle \tilde{\Gamma },\tilde{\Psi }\rangle \in \tilde{P}}{\sum }\left({\left({\mu }^U\right)}^2+{\left({\nu }^U\right)}^2\right)}\right]\end{array}$

其中 $\#h$ 表示 $\tilde{P}$ 中的元素个数。

定义4 [10] 设 ${\tilde{P}}_1$ 和 ${\tilde{P}}_2$ 为两个区间毕达哥拉斯犹豫模糊数，则有

如果 $P\left(S\left({\tilde{P}}_1\right)>S\left({\tilde{P}}_2\right)\right)>0.5$，则有 ${\tilde{P}}_1>{\tilde{P}}_2$。

如果 $P\left(S\left({\tilde{P}}_1\right)>S\left({\tilde{P}}_2\right)\right)=0.5$，那么

如果 $P\left(H\left({\tilde{P}}_1\right)>H\left({\tilde{P}}_2\right)\right)>0.5$，则有 ${\tilde{P}}_1>{\tilde{P}}_2$。

如果 $P\left(H\left({\tilde{P}}_1\right)>H\left({\tilde{P}}_2\right)\right)=0.5$，则有 ${\tilde{P}}_1={\tilde{P}}_2$。

Heronian平均算子

定义5 [16] 设 $p,q\ge 0$ 且不同时为0， ${\alpha }_i\ge 0\left(i=1,2,\cdots ,n\right)$，若HM满足：

$H{M}^{p,q}\left({\alpha }_1,{\alpha }_2,{\alpha }_3,\cdots ,{\alpha }_n\right)={\left(\frac{2}{n\left(n+1\right)}{\displaystyle \underset{i=1,j=i}{\overset{n}{\sum }}{\alpha }_i^p{\alpha }_j^q}\right)}^{\frac{1}{p+q}}$ (1)

则称HM为Heronian平均算子。

几何Heronian平均算子

定义6 [17] 设 $p,q\ge 0$ 且不同时为0， ${\alpha }_i\ge 0\left(i=1,2,\cdots ,n\right)$，若GHM满足：

${\text{GHM}}^{p,q}\left({\alpha }_1,{\alpha }_2,{\alpha }_3,\cdots ,{\alpha }_n\right)=\frac{1}{p+q}{\left({\displaystyle \underset{i=1,j=i}{\overset{n}{\prod }}\left(p{\alpha }_i+q{\alpha }_j\right)}\right)}^{\frac{2}{n\left(n+1\right)}}$ (2)

则称GHM为几何Heronian平均算子。

3. 区间毕达哥拉斯犹豫模糊Heronian加权平均(IVPHFHWM)算子

定义7设 $p,q\ge 0$ 且不同时为0， ${\tilde{P}}_i={\displaystyle \underset{\left[{\mu }_i^L,{\mu }_i^U\right]\in {\tilde{\Gamma }}_i,\left[{\nu }_i^L,{\nu }_i^U\right]\in {\tilde{\Psi }}_i}{\cup }\left\{\langle {\tilde{\Gamma }}_i,{\tilde{\Psi }}_i\rangle \right\}}\left(i=1,2,\cdots ,n\right)$ 为一组区间毕达哥拉斯犹豫模糊数，权重向量 $\omega ={\left({\omega }_1,{\omega }_2,{\omega }_3,\cdots ,{\omega }_n\right)}^{\text{T}}$ 且 ${\omega }_i\ge 0$，${\displaystyle \underset{i=1}{\overset{n}{\sum }}{\omega }_i}=1$，则IVPHFHWM算子为

${\text{IVPHFHWM}}^{p,q}\left({\tilde{P}}_1,{\tilde{P}}_2,{\tilde{P}}_3,\cdots ,{\tilde{P}}_n\right)={\left(\frac{2}{n\left(n+1\right)}\underset{i=1,j=i}{\overset{n}{\oplus }}\left({\left({\omega }_i{\tilde{P}}_i\right)}^p\otimes {\left({\omega }_j{\tilde{P}}_j\right)}^q\right)\right)}^{\frac{1}{p+q}}$ (3)

定理1设 ${\tilde{P}}_i={\displaystyle \underset{\left[{\mu }_i^L,{\mu }_i^U\right]\in {\tilde{\Gamma }}_i,\left[{\nu }_i^L,{\nu }_i^U\right]\in {\tilde{\Psi }}_i}{\cup }\left\{\langle {\tilde{\Gamma }}_i,{\tilde{\Psi }}_i\rangle \right\}}\left(i=1,2,\cdots ,n\right)$ 为一组区间毕达哥拉斯犹豫模糊变量， $p,q\ge 0$ 且不同时为0，权重向量 $\omega ={\left({\omega }_1,{\omega }_2,{\omega }_3,\cdots ,{\omega }_n\right)}^{\text{T}}$ 且 ${\omega }_i\ge 0$，${\displaystyle \underset{i=1}{\overset{n}{\sum }}{\omega }_i}=1$，利用IVPHFHWM算子集成后的结果仍是区间毕达哥拉斯犹豫模糊变量，且

${\text{IVPHFHWM}}^{p,q}\left({\tilde{P}}_1,{\tilde{P}}_2,{\tilde{P}}_3,\cdots ,{\tilde{P}}_n\right)={\displaystyle \underset{\left[{\mu }_i^L,{\mu }_i^U\right]\in {\tilde{\Gamma }}_i,\left[{\nu }_i^L,{\nu }_i^U\right]\in {\tilde{\Psi }}_i,\left[{\mu }_j^L,{\mu }_j^U\right]\in {\tilde{\Gamma }}_j,\left[{\nu }_j^L,{\nu }_j^U\right]\in {\tilde{\Psi }}_j}{\cup }\left(\left[a,b\right],\left[c,d\right]\right)}$ (4)

其中

$\begin{array}{l}a={\left(\sqrt{1-{\displaystyle \underset{i=1,j=1}{\overset{n}{\prod }}{\left(1-{\left({\left(\sqrt{1-{\left(1-{\left({\mu }_i^L\right)}^2\right)}^{{\omega }_i}}\right)}^p{\left(\sqrt{1-{\left(1-{\left({\mu }_j^L\right)}^2\right)}^{{\omega }_j}}\right)}^q\right)}^2\right)}^{\frac{2}{n\left(n+1\right)}}}}\right)}^{\frac{1}{p+q}}\\ b={\left(\sqrt{1-{\displaystyle \underset{i=1,j=1}{\overset{n}{\prod }}{\left(1-{\left({\left(\sqrt{1-{\left(1-{\left({\mu }_i^U\right)}^2\right)}^{{\omega }_i}}\right)}^p{\left(\sqrt{1-{\left(1-{\left({\mu }_j^U\right)}^2\right)}^{{\omega }_j}}\right)}^q\right)}^2\right)}^{\frac{2}{n\left(n+1\right)}}}}\right)}^{\frac{1}{p+q}}\end{array}$

$\begin{array}{l}c=\sqrt{1-{\left(1-{\left({\displaystyle \underset{i=1,j=1}{\overset{n}{\prod }}\sqrt{1-{\left({\left(1-{\left({\mu }_i^L\right)}^{2{\omega }_i}\right)}^p{\left(1-{\left({\mu }_j^L\right)}^{2{\omega }_j}\right)}^q\right)}^{\frac{2}{n\left(n+1\right)}}}}\right)}^2\right)}^{\frac{1}{p+q}}}\\ d=\sqrt{1-{\left(1-{\left({\displaystyle \underset{i=1,j=1}{\overset{n}{\prod }}\sqrt{1-{\left({\left(1-{\left({\nu }_i^U\right)}^{2{\omega }_i}\right)}^p{\left(1-{\left({\nu }_j^U\right)}^{2{\omega }_j}\right)}^q\right)}^{\frac{2}{n\left(n+1\right)}}}}\right)}^2\right)}^{\frac{1}{p+q}}}\end{array}$

证明：由区间毕达哥拉斯犹豫模糊集的运算法则可得

${\omega }_i{\tilde{P}}_i={\displaystyle \underset{\begin{array}{l}\left[{\mu }_i^L,{\mu }_i^U\right]\in {\tilde{\Gamma }}_i\\ \left[{\nu }_i^L,{\nu }_i^U\right]\in {\tilde{\Psi }}_i\end{array}}{\cup }\left\{\langle \left[\sqrt{1-{\left(1-{\left({\mu }_i^L\right)}^2\right)}^{{\omega }_i}},\sqrt{1-{\left(1-{\left({\mu }_i^U\right)}^2\right)}^{{\omega }_i}}\right],\left[{\left({\nu }_i^L\right)}^{{\omega }_i},{\left({\nu }_i^U\right)}^{{\omega }_i}\right]\rangle \right\}}$

${\omega }_j{\tilde{P}}_j={\displaystyle \underset{\begin{array}{l}\left[{\mu }_j^L,{\mu }_j^U\right]\in {\tilde{\Gamma }}_j\\ \left[{\nu }_j^L,{\nu }_j^U\right]\in {\tilde{\Psi }}_j\end{array}}{\cup }\left\{\langle \left[\sqrt{1-{\left(1-{\left({\mu }_j^L\right)}^2\right)}^{{\omega }_j}},\sqrt{1-{\left(1-{\left({\mu }_j^U\right)}^2\right)}^{{\omega }_j}}\right],\left[{\left({\nu }_j^L\right)}^{{\omega }_j},{\left({\nu }_j^U\right)}^{{\omega }_j}\right]\rangle \right\}}$

则

$\begin{array}{l}{\left({\omega }_i{\tilde{P}}_i\right)}^p={\displaystyle \underset{\begin{array}{l}\left[{\mu }_i^L,{\mu }_i^U\right]\in {\tilde{\Gamma }}_i\\ \left[{\nu }_i^L,{\nu }_i^U\right]\in {\tilde{\Psi }}_i\end{array}}{\cup }\{\langle \left[{\left(\sqrt{1-{\left(1-{\left({\mu }_i^L\right)}^2\right)}^{{\omega }_i}}\right)}^p,{\left(\sqrt{1-{\left(1-{\left({\mu }_i^U\right)}^2\right)}^{{\omega }_i}}\right)}^p\right],}\\ \left[\sqrt{1-{\left(1-{\left({\nu }_i^L\right)}^{2{\omega }_i}\right)}^p},\sqrt{1-{\left(1-{\left({\nu }_i^U\right)}^{2{\omega }_i}\right)}^p}\right]\}\end{array}$

$\begin{array}{l}{\left({\omega }_j{\tilde{P}}_j\right)}^q={\displaystyle \underset{\begin{array}{l}\left[{\mu }_j^L,{\mu }_j^U\right]\in {\tilde{\Gamma }}_j\\ \left[{\nu }_j^L,{\nu }_j^U\right]\in {\tilde{\Psi }}_j\end{array}}{\cup }\{\langle \left[{\left(\sqrt{1-{\left(1-{\left({\mu }_j^L\right)}^2\right)}^{{\omega }_j}}\right)}^q,{\left(\sqrt{1-{\left(1-{\left({\mu }_j^U\right)}^2\right)}^{{\omega }_j}}\right)}^q\right],}\\ \left[\sqrt{1-{\left(1-{\left({\nu }_j^L\right)}^{2{\omega }_j}\right)}^q},\sqrt{1-{\left(1-{\left({\nu }_j^U\right)}^{2{\omega }_j}\right)}^q}\right]\rangle \}\end{array}$

所以

$\begin{array}{l}{\left({\omega }_i{\tilde{P}}_i\right)}^p\otimes {\left({\omega }_j{\tilde{P}}_j\right)}^q\\ ={\displaystyle \underset{\begin{array}{l}\left[{\mu }_i^L,{\mu }_i^U\right]\in {\tilde{\Gamma }}_i\\ \left[{\nu }_i^L,{\nu }_i^U\right]\in {\tilde{\Psi }}_i\end{array}}{\cup }\{\langle \left[{\left(\sqrt{1-{\left(1-{\left({\mu }_i^L\right)}^2\right)}^{{\omega }_i}}\right)}^p{\left(\sqrt{1-{\left(1-{\left({\mu }_j^L\right)}^2\right)}^{{\omega }_j}}\right)}^q,{\left(\sqrt{1-{\left(1-{\left({\mu }_i^U\right)}^2\right)}^{{\omega }_i}}\right)}^p{\left(\sqrt{1-{\left(1-{\left({\mu }_j^U\right)}^2\right)}^{{\omega }_j}}\right)}^q\right],}\\ \left[\sqrt{1-{\left(1-{\left({\nu }_i^L\right)}^{2{\omega }_i}\right)}^p{\left(1-{\left({\nu }_j^L\right)}^{2{\omega }_j}\right)}^q},\sqrt{1-{\left(1-{\left({\nu }_i^U\right)}^{2{\omega }_i}\right)}^p{\left(1-{\left({\nu }_j^U\right)}^{2{\omega }_j}\right)}^q}\right]\rangle \}\end{array}$

因此

由此可知，上述定理1得证。

定理2 (置换不变性)设 ${\tilde{P}}_i={\displaystyle \underset{\left[{\mu }_i^L,{\mu }_i^U\right]\in {\tilde{\Gamma }}_i,\left[{\nu }_i^L,{\nu }_i^U\right]\in {\tilde{\Psi }}_i}{\cup }\left\{\langle {\tilde{\Gamma }}_i,{\tilde{\Psi }}_i\rangle \right\}}\left(i=1,2,\cdots ,n\right)$ 为一组区间毕达哥拉斯犹豫模糊集， $\left(Q_1,Q_2,\cdots ,Q_n\right)$ 为 $\left({\tilde{P}}_1,{\tilde{P}}_2,{\tilde{P}}_3,\cdots ,{\tilde{P}}_n\right)$ 的任一置换，则

${\text{IVPHFHWM}}^{p,q}\left(Q_1,Q_2,\cdots ,Q_n\right)={\text{IVPHFHWM}}^{p,q}\left({\tilde{P}}_1,{\tilde{P}}_2,{\tilde{P}}_3,\cdots ,{\tilde{P}}_n\right)$

证明：因为 $\left(Q_1,Q_2,\cdots ,Q_n\right)$ 是 $\left({\tilde{P}}_1,{\tilde{P}}_2,{\tilde{P}}_3,\cdots ,{\tilde{P}}_n\right)$ 的任一置换，所以对于任意的 $Q_i$ 和 $Q_j$ 总存在唯一的 ${\tilde{P}}_x$ 和 ${\tilde{P}}_y$ 与之对应，所以

$\begin{array}{c}{\text{IVPHFHWM}}^{p,q}\left(Q_1,Q_2,\cdots ,Q_n\right)={\left(\frac{2}{n\left(n+1\right)}\underset{i=1,j=i}{\overset{n}{\oplus }}\left({\left({\omega }_i{Q}_i\right)}^p\otimes {\left({\omega }_j{Q}_j\right)}^q\right)\right)}^{\frac{1}{p+q}}\\ ={\left(\frac{2}{n\left(n+1\right)}\underset{x=1,y=x}{\overset{n}{\oplus }}\left({\left({\omega }_x{\tilde{P}}_x\right)}^p\otimes {\left({\omega }_y{\tilde{P}}_y\right)}^q\right)\right)}^{\frac{1}{p+q}}\\ ={\text{IVPHFHWM}}^{p,q}\left({\tilde{P}}_1,{\tilde{P}}_2,{\tilde{P}}_3,\cdots ,{\tilde{P}}_n\right)\end{array}$

定理3 (有界性) ${\tilde{P}}_i={\displaystyle \underset{\left[{\mu }_i^L,{\mu }_i^U\right]\in {\tilde{\Gamma }}_i,\left[{\nu }_i^L,{\nu }_i^U\right]\in {\tilde{\Psi }}_i}{\cup }\left\{\langle {\tilde{\Gamma }}_i,{\tilde{\Psi }}_i\rangle \right\}}\left(i=1,2,\cdots ,n\right)$ 为一组区间毕达哥拉斯犹豫模糊数，权重向量 $\omega ={\left({\omega }_1,{\omega }_2,{\omega }_3,\cdots ,{\omega }_n\right)}^{\text{T}}$ 且 ${\omega }_i\ge 0$，${\displaystyle \underset{i=1}{\overset{n}{\sum }}{\omega }_i}=1$，则

${\tilde{P}}^L\le {\text{IVPHFHWM}}^{p,q}\left({\tilde{P}}_1,{\tilde{P}}_2,{\tilde{P}}_3,\cdots ,{\tilde{P}}_n\right)\le {\tilde{P}}^U$

其中 ${\tilde{P}}^L=\underset{i}{\min }\left({\omega }_i{\tilde{P}}_i\right)$，${\tilde{P}}^U=\underset{i}{\max }\left({\omega }_i{\tilde{P}}_i\right)$。

证明 $\begin{array}{c}{\text{IVPHFHWM}}^{p,q}\left({\tilde{P}}_1,{\tilde{P}}_2,{\tilde{P}}_3,\cdots ,{\tilde{P}}_n\right)={\left(\frac{2}{n\left(n+1\right)}\underset{i=1,j=i}{\overset{n}{\oplus }}\left({\left({\omega }_i{\tilde{P}}_i\right)}^p\otimes {\left({\omega }_j{\tilde{P}}_j\right)}^q\right)\right)}^{\frac{1}{p+q}}\\ \le {\left(\frac{2}{n\left(n+1\right)}\underset{i=1,j=i}{\overset{n}{\oplus }}\left({\left({\tilde{P}}^U\right)}^p\otimes {\left({\tilde{P}}^U\right)}^q\right)\right)}^{\frac{1}{p+q}}\\ ={\left(\frac{2}{n\left(n+1\right)}\frac{n\left(n+1\right)}{2}{\left({\tilde{P}}^U\right)}^{p+}{}^q\right)}^{\frac{1}{p+q}}\\ ={\tilde{P}}^U\end{array}$

同理可证得 ${\tilde{P}}^L\le {\text{IVPHFHWM}}^{p,q}\left({\tilde{P}}_1,{\tilde{P}}_2,{\tilde{P}}_3,\cdots ,{\tilde{P}}_n\right)$。

即 ${\tilde{P}}^L\le {\text{IVPHFHWM}}^{p,q}\left({\tilde{P}}_1,{\tilde{P}}_2,{\tilde{P}}_3,\cdots ,{\tilde{P}}_n\right)\le {\tilde{P}}^U$

定理4 (单调性)设 ${\tilde{P}}_i={\displaystyle \underset{\begin{array}{l}\left[{\mu }_i^L,{\mu }_i^U\right]\in {\tilde{\Gamma }}_i\\ \left[{\nu }_i^L,{\nu }_i^U\right]\in {\tilde{\Psi }}_i\end{array}}{\cup }\left\{\langle {\tilde{\Gamma }}_i,{\tilde{\Psi }}_i\rangle \right\}}$ 和 $P_i^{\ast }={\displaystyle \underset{\begin{array}{l}\left[{\mu }_i^L,{\mu }_i^U\right]\in {\tilde{\Gamma }}_i\\ \left[{\nu }_i^L,{\nu }_i^U\right]\in {\tilde{\Psi }}_i\end{array}}{\cup }\left\{\langle {\tilde{\Gamma }}_i,{\tilde{\Psi }}_i\rangle \right\}}$ 为两个区间毕达哥拉斯犹豫模糊数，如果对任意的 $i\left(i=1,2,\cdots ,n\right)$ 都有 ${\tilde{\Gamma }}_i\ge {\Gamma }_i^{*}$，${\tilde{\Psi }}_i\le {\Psi }_i^{*}$，则

${\text{IVPHFHWM}}^{p,q}\left({\tilde{P}}_1,{\tilde{P}}_2,{\tilde{P}}_3,\cdots ,{\tilde{P}}_n\right)\ge {\text{IVPHFHWM}}^{p,q}\left(P_1^{*},P_2^{*},P_3^{*},\cdots ,P_n^{*}\right)$

证明：因为 $\left[{\tilde{\mu }}^L,{\tilde{\mu }}^U\right]={\tilde{\Gamma }}_i\ge {\Gamma }_i^{*}=\left[{}^{*}\mu {}^L,{}^{*}\mu {}^U\right]$

${\text{IVPHFHWM}}^{p,q}\left({\tilde{P}}_1,{\tilde{P}}_2,{\tilde{P}}_3,\cdots ,{\tilde{P}}_n\right)={\displaystyle \underset{\begin{array}{l}\left[{\mu }_i^L,{\mu }_i^U\right]\in {\tilde{\Gamma }}_i,\left[{\nu }_i^L,{\nu }_i^U\right]\in {\tilde{\Psi }}_i\\ \left[{\mu }_j^L,{\mu }_j^U\right]\in {\tilde{\Gamma }}_j,\left[{\nu }_j^L,{\nu }_j^U\right]\in {\tilde{\Psi }}_j\end{array}}{\cup }\left(\left[a,b\right],\left[c,d\right]\right)}$

${\text{IVPHFHWM}}^{p,q}\left(P_1^{*},P_2^{*},P_3^{*},\cdots ,P_n^{*}\right)={\displaystyle \underset{\begin{array}{l}\left[{}^{*}\mu {}_i^L,{}^{*}\mu {}_i^U\right]\in {\tilde{\Gamma }}_i，\left[{}^{*}\nu {}_i^L,{}^{*}\nu {}_i^U\right]\in {\tilde{\Psi }}_i\\ \left[{}^{*}\mu {}_j^L,{}^{*}\mu {}_j^U\right]\in {\tilde{\Gamma }}_j，\left[{}^{*}\nu {}_j^L,{}^{*}\nu {}_j^U\right]\in {\tilde{\Psi }}_j\end{array}}{\cup }\left(\left[a^{*},b^{*}\right],\left[c^{*},d^{*}\right]\right)}$

所以

$\begin{array}{c}a={\left(\sqrt{1-{\displaystyle \underset{i=1,j=1}{\overset{n}{\prod }}{\left(1-{\left({\left(\sqrt{1-{\left(1-{\left({\mu }_i^L\right)}^2\right)}^{{\omega }_i}}\right)}^p{\left(\sqrt{1-{\left(1-{\left({\mu }_j^L\right)}^2\right)}^{{\omega }_j}}\right)}^q\right)}^2\right)}^{\frac{2}{n\left(n+1\right)}}}}\right)}^{\frac{1}{p+q}}\\ >{\left(\sqrt{1-{\displaystyle \underset{i=1,j=1}{\overset{n}{\prod }}{\left(1-{\left({\left(\sqrt{1-{\left(1-{\left({}^{*}\mu {}_i^L\right)}^2\right)}^{{\omega }_i}}\right)}^p{\left(\sqrt{1-{\left(1-{\left({}^{*}\mu {}_j^L\right)}^2\right)}^{{\omega }_j}}\right)}^q\right)}^2\right)}^{\frac{2}{n\left(n+1\right)}}}}\right)}^{\frac{1}{p+q}}=a^{*}\end{array}$

同理可得 $b>b^{*}$，$c<c^{*}$ 和 $d<d^{*}$

所以 ${\text{IVPHFHWM}}^{p,q}\left({\tilde{P}}_1,{\tilde{P}}_2,{\tilde{P}}_3,\cdots ,{\tilde{P}}_n\right)\ge {\text{IVPHFHWM}}^{p,q}\left(P_1^{*},P_2^{*},P_3^{*},\cdots ,P_n^{*}\right)$

4. 区间值毕达哥拉斯犹豫模糊几何Heronian加权(IVPHFGHWM)算子

定义8设 $p,q\ge 0$ 且不同时为0， ${\tilde{P}}_i={\displaystyle \underset{\left[{\mu }_i^L,{\mu }_i^U\right]\in {\tilde{\Gamma }}_i,\left[{\nu }_i^L,{\nu }_i^U\right]\in {\tilde{\Psi }}_i}{\cup }\left\{\langle {\tilde{\Gamma }}_i,{\tilde{\Psi }}_i\rangle \right\}}\left(i=1,2,\cdots ,n\right)$ 为一组区间毕达哥拉斯犹豫模糊数。权重向量 $\omega ={\left({\omega }_1,{\omega }_2,{\omega }_3,\cdots ,{\omega }_n\right)}^{\text{T}}$ 且 ${\omega }_i\ge 0$，${\displaystyle \underset{i=1}{\overset{n}{\sum }}{\omega }_i}=1$，则IVPHFGHWM算子为

${\text{IVPHFGHWM}}^{p,q}\left({\tilde{P}}_1,{\tilde{P}}_2,{\tilde{P}}_3,\cdots ,{\tilde{P}}_n\right)=\frac{1}{p+q}{\left(\underset{i=1,j=i}{\overset{n}{\oplus }}\left(p{\tilde{P}}_i{}^{{\omega }_i}\oplus q{\tilde{P}}_j{}^{{\omega }_j}\right)\right)}^{\frac{2}{n\left(n+1\right)}}$ (5)

定理5设 ${\tilde{P}}_i={\displaystyle \underset{\left[{\mu }_i^L,{\mu }_i^U\right]\in {\tilde{\Gamma }}_i,\left[{\nu }_i^L,{\nu }_i^U\right]\in {\tilde{\Psi }}_i}{\cup }\left\{\langle {\tilde{\Gamma }}_i,{\tilde{\Psi }}_i\rangle \right\}}\left(i=1,2,\cdots ,n\right)$ 为一组区间毕达哥拉斯犹豫模糊变量， $p,q\ge 0$ 且不同时为0，权重向量 $\omega ={\left({\omega }_1,{\omega }_2,{\omega }_3,\cdots ,{\omega }_n\right)}^{\text{T}}$ 且 ${\omega }_i\ge 0$，${\displaystyle \underset{i=1}{\overset{n}{\sum }}{\omega }_i}=1$，利用IVPHFGHWM算子集成后的结果仍是区间毕达哥拉斯犹豫模糊变量，且

${\text{IVPHFGHWM}}^{p,q}\left({\tilde{P}}_1,{\tilde{P}}_2,{\tilde{P}}_3,\cdots ,{\tilde{P}}_n\right)={\displaystyle \underset{\left[{\mu }_i^L,{\mu }_i^U\right]\in {\tilde{\Gamma }}_i,\left[{\nu }_i^L,{\nu }_i^U\right]\in {\tilde{\Psi }}_i,\left[{\mu }_j^L,{\mu }_j^U\right]\in {\tilde{\Gamma }}_j,\left[{\nu }_j^L,{\nu }_j^U\right]\in {\tilde{\Psi }}_j}{\cup }\left(\left[e,f\right],\left[g,h\right]\right)}$ (6)

其中

$e=\sqrt{1-{\left(1-{\displaystyle \underset{i=1,j=1}{\overset{n}{\prod }}{\left({\left(1-{\left({\mu }_i^L\right)}^2{}^{{\omega }_i}\right)}^p{\left(1-{\left({\mu }_j^L\right)}^2{}^{{\omega }_j}\right)}^q\right)}^{\frac{2}{n\left(n+1\right)}}}\right)}^{\frac{1}{p+q}}}$

$f=\sqrt{1-{\left(1-{\displaystyle \underset{i=1,j=1}{\overset{n}{\prod }}{\left({\left(1-{\left({\mu }_i^U\right)}^2{}^{{\omega }_i}\right)}^p{\left(1-{\left({\mu }_j^U\right)}^2{}^{{\omega }_j}\right)}^q\right)}^{\frac{2}{n\left(n+1\right)}}}\right)}^{\frac{1}{p+q}}}$

$g={\left(\sqrt{{\displaystyle \underset{i=1,j=1}{\overset{n}{\prod }}{\left({\left(1-{\left(1-{\left({\nu }_i^L\right)}^2\right)}^{{\omega }_i}\right)}^p{\left(1-{\left(1-{\left({\nu }_j^L\right)}^2\right)}^{{\omega }_j}\right)}^q\right)}^{\frac{2}{n\left(n+1\right)}}}}\right)}^{\frac{1}{p+q}}$

$h={\left(\sqrt{{\displaystyle \underset{i=1,j=1}{\overset{n}{\prod }}{\left({\left(1-{\left(1-{\left({\nu }_i^U\right)}^2\right)}^{{\omega }_i}\right)}^p{\left(1-{\left(1-{\left({\nu }_j^U\right)}^2\right)}^{{\omega }_j}\right)}^q\right)}^{\frac{2}{n\left(n+1\right)}}}}\right)}^{\frac{1}{p+q}}$

定理6 (置换不变性)设 ${\tilde{P}}_i={\displaystyle \underset{\left[{\mu }_i^L,{\mu }_i^U\right]\in {\tilde{\Gamma }}_i,\left[{\nu }_i^L,{\nu }_i^U\right]\in {\tilde{\Psi }}_i}{\cup }\left\{\langle {\tilde{\Gamma }}_i,{\tilde{\Psi }}_i\rangle \right\}}\left(i=1,2,\cdots ,n\right)$ 为一组区间毕达哥拉斯犹豫模糊数， $\left(R_1,R_2,\cdots ,R_n\right)$ 为 $\left({\tilde{P}}_1,{\tilde{P}}_2,{\tilde{P}}_3,\cdots ,{\tilde{P}}_n\right)$ 的任一置换，则