基于区间值毕达哥拉斯Sugeno-Weber Softmax算子的WASPAS多属性决策方法

WASPAS Multi-Attribute Decision-Making Method Based on Interval Valued Pythagorean Sugeno-Weber Softmax Operator

摘要: 针对属性值为区间值毕达哥拉斯模糊数且权重信息完全未知的决策问题，本文提出基于Sugeno-Weber Softmax算子的区间值毕达哥拉斯模糊多属性决策方法。首先，定义了基于Sugeno-Weber三角模的区间值毕达哥拉斯模糊数运算法则。其次，为考虑属性间的优先关系，提出四种区间值毕达哥拉斯模糊Sugeno-Weber Softmax平均和几何集成算子并讨论所提算子的幂等性、有界性和单调性等性质。为确定属性的权重信息，提出基于区间值毕达哥拉斯模糊PSI (Preference Selection Index)方法确定属性的客观权重。进一步提出基于Sugeno-Weber Softmax算子的区间值毕达哥拉斯模糊WASPAS (Weighted Aggregated Sum Product Assessment)方法确定备选方案的排序。通过实际案例验证所提方法的有效性及合理性，并由比较分析和参数分析讨论所提方法的鲁棒性和优越性。

Abstract: This paper addresses decision-making problems characterized by attribute values expressed as interval-valued Pythagorean fuzzy numbers (IVPFNs) and completely unknown weight information. We propose a novel interval-valued Pythagorean fuzzy multi-attribute decision-making (MADM) methodology based on the Sugeno-Weber Softmax operator. Firstly, arithmetic operations for IVPFNs are defined utilizing the Sugeno-Weber t-norm and t-conorm. Secondly, to account for priority relationships among attributes, four Pythagorean fuzzy Sugeno-Weber Softmax averaging and geometric aggregation operators are introduced. Fundamental properties of these operators, including idempotency, boundedness, and monotonicity, are rigorously investigated. To resolve the unknown weight issue, an objective weight determination method for attributes is developed using the Pythagorean fuzzy Preference Selection Index approach. Furthermore, an enhanced Pythagorean fuzzy WASPAS (Weighted Aggregated Sum Product Assessment) method, integrated with the proposed Sugeno-Weber Softmax operators, is presented to determine the ranking of alternatives. The effectiveness and rationality of the proposed methodology are validated through an empirical case study. Comparative analysis and parameter sensitivity analysis further demonstrate its superior robustness and performance over existing methods.

1. 引言

现实决策问题的复杂性、模糊性及决策者认知能力的不确定性使得决策者难以用精确的数值表达其偏好信息。为此，诸多学者从不确定性角度出发，通过模糊数、模糊语言和区间数等模型表征实际问题中存在的模糊不确定性。1965年，Zadeh [1]首次提出模糊集理论，通过隶属度函数刻画元素属于给定目标的模糊性。此后，Atanassov [2]从隶属度、非隶属度和犹豫度的角度提出直觉模糊集理论以更加精确的表征模糊不确定现象。但直觉模糊集在应用时具有极高的限制条件，使得决策者并不能完全表达其不确定性评估信息。为增大决策者表达偏好的范围和自由度，Yager [3] [4]提出毕达哥拉斯模糊集理论，将直觉模糊集的隶属度和非隶属度和小于等于1的条件扩充至隶属度和非隶属度的平方和小于等于1。毕达哥拉斯模糊集的提出使得其在决策分析和信息融合等领域产生广泛的应用前景并取得丰硕的成果[5]-[8]。但决策者在表达隶属度和非隶属度时往往存在不确定性而难以给出准确的隶属度和非隶属度，因此Peng和Yang [9]提出区间值毕达哥拉斯模糊集的概念，通过区间值来表达决策者提供隶属度和非隶属度的深层次不确定性。此后，诸多学者针对区间值毕达哥拉斯模糊集的理论和应用展开了广泛研究[10] [11]。

集成算子是模糊信息融合过程的重要组成部分并被广泛用于模糊信息融合和决策方法构建过程中。集成算子的基础是基于不同的三角模定义不同模糊环境下的基本运算法则，然后基于优先算子、幂算子和考虑属性间相互关系的Hamy均值、Heronian均值等聚合函数提出一系列新的集成算子。Sugeno-Weber [12]作为一种新型的三角模且具有极高的灵活性，目前已被拓展到不同的模糊环境中定义集成算子并构建决策方法[13]-[16]。Softmax函数是机器学习和深度学习中多分类问题的核心函数，它可以将一个包含任意实数的向量压缩成一个概率分布[17]。考虑Softmax函数的优势，目前已被用于决策分析中构建集成算子以考虑融合数据间的优先关系[17] [18]。在多属性决策方法中，WASPAS作为一种融合加权和和加权积模型的决策方法，同时兼顾了加权和和加权积模型的优势并获得更佳合理的决策分析结果[19]。WASPAS方法自提出以来备受学者关注并将其与模糊集理论和不确定性理论结合，提出了不同特征下的扩展WASPAS方法，并被广泛应用于不同领域的实际决策中[20]-[22]。据文献所知，目前尚未有关于Sugeno-Weber Softmax集成算子，WASPAS方法在区间值毕达哥拉斯模糊环境中的研究。

基于上述讨论和分析，本文针对属性值为区间值毕达哥拉斯模糊数且权重信息完全未知的决策问题，首先定义区间值毕达哥拉斯模糊数Sugeno-Weber运算法则并联合Softmax函数提出四种区间值毕达哥拉斯模糊Sugeno-Weber Softmax集成算子，同时讨论所提算子的幂等性、有界性和单调性等性质。为确定属性的权重，基于区间值毕达哥拉斯模糊数得分函数，提出改进的PSI方法[23]确定属性的客观权重。进一步提出基于区间值毕达哥拉斯模糊Sugeno-Weber Softmax算子的WASPAS方法确定备选方案的优先级。

2. 预备知识

定义1 [9]设$Y$ 为一个非空经典集合，则区间值毕达哥拉斯模糊集$F$ 表示为：

$F=\{\langle y_i,\left(\left[{\alpha }_F^{-}\left(y_i\right),{\alpha }_F^{+}\left(y_i\right)\right],\left[{\beta }_F^{-}\left(y_i\right),{\beta }_F^{+}\left(y_i\right)\right]\right)\rangle |y_i\in Y\},$ (1)

其中${\alpha }_F^{-}\left(y_i\right),{\alpha }_F^{+}\left(y_i\right)\in \left[0,1\right]$ ，${\beta }_F^{-}\left(y_i\right),{\beta }_F^{+}\left(y_i\right)\in \left[0,1\right]$ 且$0\le {\left({\alpha }_F^{+}\left(y_i\right)\right)}^2+{\left({\beta }_F^{+}\left(y_i\right)\right)}^2\le 1$ 。 ${\alpha }_F^{}\left(y_i\right)=\left[{\alpha }_F^{-}\left(y_i\right),{\alpha }_F^{+}\left(y_i\right)\right]$ 和${\beta }_F^{}\left(y_i\right)=\left[{\beta }_F^{-}\left(y_i\right),{\beta }_F^{+}\left(y_i\right)\right]$ 分别表示元素$y_i\in Y$ 的区间隶属度和非隶属度。${\pi }_F^{}\left(y_i\right)=\left[{\pi }_F^{-}\left(y_i\right),{\pi }_F^{+}\left(y_i\right)\right]$ 表示元素$y_i\in Y$ 的区间犹豫度，其中${\pi }_F^{-}\left(y_i\right)=\sqrt{1-{\left({\alpha }_F^{+}\left(y_i\right)\right)}^2-{\left({\beta }_F^{+}\left(y_i\right)\right)}^2}$ ，${\pi }_F^{+}\left(y_i\right)=\sqrt{1-{\left({\alpha }_F^{-}\left(y_i\right)\right)}^2-{\left({\beta }_F^{-}\left(y_i\right)\right)}^2}$ 。为方便起见，称$\kappa =\left(\left[{\alpha }_{\kappa }^{-},{\alpha }_{\kappa }^{+}\right],\left[{\beta }_{\kappa }^{-},{\beta }_{\kappa }^{+}\right]\right)$ 为一个区间值毕达哥拉斯模糊数(IVPFN)。

定义2 [9]设${\kappa }_1=\left(\left[{\alpha }_{{\kappa }_1}^{-},{\alpha }_{{\kappa }_1}^{+}\right],\left[{\beta }_{{\kappa }_1}^{-},{\beta }_{{\kappa }_1}^{+}\right]\right)$ 为一个IVPFN。则其得分函数和精确函数分别定义为：

$SF\left({\kappa }_1\right)=\frac{1}{2}\left(\frac{1}{2}\left({\left({\alpha }_{{\kappa }_1}^{-}\right)}^2+{\left({\alpha }_{{\kappa }_1}^{+}\right)}^2-{\left({\beta }_{{\kappa }_1}^{-}\right)}^2-{\left({\beta }_{{\kappa }_1}^{+}\right)}^2\right)+1\right),$ (2)

$AF\left({\kappa }_1\right)=\frac{1}{2}\left({\left({\alpha }_{{\kappa }_1}^{-}\right)}^2+{\left({\alpha }_{{\kappa }_1}^{+}\right)}^2-{\left({\beta }_{{\kappa }_1}^{-}\right)}^2-{\left({\beta }_{{\kappa }_1}^{+}\right)}^2\right).$ (3)

定义3 [9]设${\kappa }_1=\left(\left[{\alpha }_{{\kappa }_1}^{-},{\alpha }_{{\kappa }_1}^{+}\right],\left[{\beta }_{{\kappa }_1}^{-},{\beta }_{{\kappa }_1}^{+}\right]\right)$ 和${\kappa }_2=\left(\left[{\alpha }_{{\kappa }_2}^{-},{\alpha }_{{\kappa }_2}^{+}\right],\left[{\beta }_{{\kappa }_2}^{-},{\beta }_{{\kappa }_2}^{+}\right]\right)$ 为两个IVPFNs。若 $SF\left({\kappa }_1\right)>SF\left({\kappa }_2\right)$ ，则${\kappa }_1\succ {\kappa }_2$ ；若$SF\left({\kappa }_1\right)=SF\left({\kappa }_2\right)$ 且$AF\left({\kappa }_1\right)>AF\left({\kappa }_2\right)$ ，则${\kappa }_1\succ {\kappa }_2$ ；若$SF\left({\kappa }_1\right)=SF\left({\kappa }_2\right)$ 且$AF\left({\kappa }_1\right)<AF\left({\kappa }_2\right)$ ，则${\kappa }_1\prec {\kappa }_2$ ；若$SF\left({\kappa }_1\right)=SF\left({\kappa }_2\right)$ 且$AF\left({\kappa }_1\right)=AF\left({\kappa }_2\right)$ ，则${\kappa }_1~{\kappa }_2$ 。

定义4 [17]逻辑函数Softmax定义如下：

${\psi }^{\xi }\left(j,T_1,T_2,\cdots ,T_n\right)={\psi }_j^{\xi }=\exp \left(T_j/\xi \right)/\left({\displaystyle \sum _{j=1}^n\exp \left(T_j/\xi \right)}\right),\xi >0$ (4)

其中$\xi$ 为参数，对任意区间值毕达哥拉斯模糊数，$T_j={\displaystyle {\prod }_{l=1}^{j-1}SF}\left({\kappa }_j\right),\left(j=2,3,\cdots ,n\right)$ ，$T_1=1$ 且$SF\left({\kappa }_j\right)$ 是${\kappa }_j$ 的得分函数。

定义5 [12]. Sugeno-Weber T-范数${\left(T_{sw}^{\aleph }\right)}_{\aleph \in \left[0,\infty \right)}$ 和S-范数${\left(S_{sw}^{\aleph }\right)}_{\aleph \in \left[0,\infty \right)}$ 定义如下：

$T_{sw}^{\aleph }\left(a,b\right)=\{\begin{array}{l}{T}_D\left(a,b\right),\text{if}\aleph =-1\\ \max \left(0,\frac{a+b-1+\aleph ab}{1+\aleph }\right),\aleph \in \left(-1,+\infty \right)\\ T_P\left(a,b\right),\text{if}\aleph =+\infty \end{array},$ $S_{sw}^{\aleph }\left(a,b\right)=\{\begin{array}{l}{S}_D\left(a,b\right),\text{if}\aleph =-1\\ \min \left(1,a+b-\frac{\aleph }{1+\aleph }ab\right),\aleph \in \left(-\text{1,}+\infty \right)\\ S_P\left(a,b\right),\text{if}\aleph =+\infty \end{array}$

其中$T_D^{}\left(a,b\right)$ 和$S_D^{}\left(a,b\right)$ 表示Drastic T-范数和S-范数，$T_P^{}\left(a,b\right)$ 和$S_P^{}\left(a,b\right)$ 分别表示和Product T-范数和S-范数。

3. 区间值毕达哥拉斯模糊环境下的Sugeno-Weber算子

本节定义区间值毕达哥拉斯模糊Sugeno-Weber运算法则。基于Softmax函数提出几种区间值毕达哥拉斯模糊Sugeno-Weber Softmax集成算子并讨论所提算子的相关性质。$\Omega$ 表示区间值毕达哥拉斯模糊数

集合，${\varpi }_j\left(j=1\left(1\right)n\right)$ 是区间值毕达哥拉斯模糊数${\kappa }_j$ 的权重且满足${\displaystyle {\sum }_{j=1}^n{\varpi }_j=1},{\varpi }_j\in \left[0,1\right]$ 。

3.1. 基于Sugeno-Weber模的区间值毕达哥拉斯模糊数运算法则

基于Sugeno-Weber模的区间值毕达哥拉斯模糊运算法则定义如下。

定义6 设${\kappa }_1=\left(\left[{\alpha }_{{\kappa }_1}^{-},{\alpha }_{{\kappa }_1}^{+}\right],\left[{\beta }_{{\kappa }_1}^{-},{\beta }_{{\kappa }_1}^{+}\right]\right)$ 和${\kappa }_2=\left(\left[{\alpha }_{{\kappa }_2}^{-},{\alpha }_{{\kappa }_2}^{+}\right],\left[{\beta }_{{\kappa }_2}^{-},{\beta }_{{\kappa }_2}^{+}\right]\right)$ 为两个IVPFNs。则：

1) ${\kappa }_1{\oplus }_{sw}{\kappa }_2=\left(\left[\begin{array}{l}\sqrt[]{{\left({\alpha }_{{\kappa }_1}^{-}\right)}^2+{\left({\alpha }_{{\kappa }_{12}}^{-}\right)}^2-\frac{\aleph }{1+\aleph }{\left({\alpha }_{{\kappa }_1}^{-}\right)}^2{\left({\alpha }_{{\kappa }_2}^{-}\right)}^2},\\ \sqrt[]{{\left({\alpha }_{{\kappa }_1}^{+}\right)}^2+{\left({\alpha }_{{\kappa }_{12}}^{+}\right)}^2-\frac{\aleph }{1+\aleph }{\left({\alpha }_{{\kappa }_1}^{+}\right)}^2{\left({\alpha }_{{\kappa }_2}^{+}\right)}^2}\end{array}\right],\left[\begin{array}{l}\sqrt[]{\frac{{\left({\beta }_{{\kappa }_1}^{-}\right)}^2+{\left({\beta }_{{\kappa }_2}^{-}\right)}^2-1+\aleph {\left({\beta }_{{\kappa }_1}^{-}\right)}^2{\left({\beta }_{{\kappa }_2}^{-}\right)}^2}{1+\aleph }},\\ \sqrt[]{\frac{{\left({\beta }_{{\kappa }_1}^{+}\right)}^2+{\left({\beta }_{{\kappa }_2}^{+}\right)}^2-1+\aleph {\left({\beta }_{{\kappa }_1}^{+}\right)}^2{\left({\beta }_{{\kappa }_2}^{+}\right)}^2}{1+\aleph }}\end{array}\right]\right)$ ；

2) ${\kappa }_1{\otimes }_{sw}{\kappa }_2=\left(,\left[\begin{array}{l}\sqrt[]{\frac{{\left({\alpha }_{{\kappa }_1}^{-}\right)}^2+{\left({\alpha }_{{\kappa }_2}^{-}\right)}^2-1+\aleph {\left({\alpha }_{{\kappa }_1}^{-}\right)}^2{\left({\alpha }_{{\kappa }_2}^{-}\right)}^2}{1+\aleph }},\\ \sqrt[]{\frac{{\left({\alpha }_{{\kappa }_1}^{+}\right)}^2+{\left({\alpha }_{{\kappa }_2}^{+}\right)}^2-1+\aleph {\left({\alpha }_{{\kappa }_1}^{+}\right)}^2{\left({\alpha }_{{\kappa }_2}^{+}\right)}^2}{1+\aleph }}\end{array}\right]\left[\begin{array}{l}\sqrt[]{{\left({\beta }_{{\kappa }_1}^{-}\right)}^2+{\left({\beta }_{{\kappa }_{12}}^{-}\right)}^2-\frac{\aleph }{1+\aleph }{\left({\beta }_{{\kappa }_1}^{-}\right)}^2{\left({\beta }_{{\kappa }_2}^{-}\right)}^2},\\ \sqrt[]{{\left({\beta }_{{\kappa }_1}^{+}\right)}^2+{\left({\beta }_{{\kappa }_{12}}^{+}\right)}^2-\frac{\aleph }{1+\aleph }{\left({\beta }_{{\kappa }_1}^{+}\right)}^2{\left({\beta }_{{\kappa }_2}^{+}\right)}^2}\end{array}\right]\right)$ ；

3) $\lambda {\kappa }_1=\left(\left[\begin{array}{l}\sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\left(1-{\left({\alpha }_{{\kappa }_1}^{-}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{\lambda }\right)},\\ \sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\left(1-{\left({\alpha }_{{\kappa }_1}^{+}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{\lambda }\right)}\end{array}\right],\left[\begin{array}{l}\sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\left(\frac{\aleph {\left({\beta }_{{\kappa }_1}^{-}\right)}^2+1}{1+\aleph }\right)}^{\lambda }-1\right)},\\ \sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\left(\frac{\aleph {\left({\beta }_{{\kappa }_1}^{+}\right)}^2+1}{1+\aleph }\right)}^{\lambda }-1\right)}\end{array}\right]\right),\lambda >0$ ；

4) ${\left({\kappa }_1\right)}^{\lambda }=\left(\left[\begin{array}{l}\sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\left(\frac{\aleph {\left({\alpha }_{{\kappa }_1}^{-}\right)}^2+1}{1+\aleph }\right)}^{\lambda }-1\right)},\\ \sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\left(\frac{\aleph {\left({\alpha }_{{\kappa }_1}^{+}\right)}^2+1}{1+\aleph }\right)}^{\lambda }-1\right)}\end{array}\right],\left[\begin{array}{l}\sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\left(1-{\left({\beta }_{{\kappa }_1}^{-}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{\lambda }\right)},\\ \sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\left(1-{\left({\beta }_{{\kappa }_1}^{+}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{\lambda }\right)}\end{array}\right]\right),\lambda >0$ ；

基于区间值毕达哥拉斯模糊Sugeno-Weber运算法则，下面提出区间值毕达哥拉斯模糊环境下的Sugeno-Weber Softmax集成算子。

3.2. 区间值毕达哥拉斯模糊环境下的Sugeno-Weber Softmax算子

定义7 设${\kappa }_j=\left(\left[{\alpha }_{{\kappa }_j}^{-},{\alpha }_{{\kappa }_j}^{+}\right],\left[{\beta }_{{\kappa }_j}^{-},{\beta }_{{\kappa }_j}^{+}\right]\right)\left(j=1,2,\cdots ,n\right)$ 为一组IVPFNs。则区间值毕达哥拉斯模糊Sugeno-Weber Softmax (IVPFSWSF)算子$\text{IVPFSWSF}:{\Omega }^n\to \Omega$ 定义如下：

$\text{IVPFSWSF}\left({\kappa }_1,{\kappa }_2,\cdots ,{\kappa }_n\right)=\underset{j=1}{\overset{n}{{\oplus }_{sw}}}{\psi }_j^{\xi }{\kappa }_j$ (5)

其中${\psi }_j^{\xi }=\exp \left(T_j/\xi \right)/{\displaystyle {\sum }_{j=1}^n\exp \left(T_j/\xi \right)}$ 满足${\displaystyle {\sum }_{j=1}^n{\psi }_j^{\xi }=1},{\psi }_j^{\xi }\in \left[0,1\right]$ ，$\xi >0$ 。$T_1=1$ ， $T_j={\displaystyle {\prod }_{l=1}^{j-1}SF}\left({\kappa }_{\text{j}}\right),\left(j=2,3,\cdots ,n\right)$ ，$SF\left({\kappa }_j\right)$ 是 ${\kappa }_j$ 的得分函数。

定理1 设${\kappa }_j=\left(\left[{\alpha }_{{\kappa }_j}^{-},{\alpha }_{{\kappa }_j}^{+}\right],\left[{\beta }_{{\kappa }_j}^{-},{\beta }_{{\kappa }_j}^{+}\right]\right)\left(j=1,2,\cdots ,n\right)$ 为一组IVPFNs。则利用IVPFSWSF算子集成这组区间值毕达哥拉斯模糊数后的结果仍是区间值毕达哥拉斯模糊数且

$\begin{array}{l}\text{IVPFSWSF}\left({\kappa }_1,{\kappa }_2,\cdots ,{\kappa }_n\right)=\underset{j=1}{\overset{n}{{\oplus }_{sw}}}{\psi }_j^{\xi }{\kappa }_j\\ =\left(\left[\begin{array}{l}\sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\alpha }_{{\kappa }_j}^{-}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{{\psi }_j^{\xi }}}\right)},\\ \sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\alpha }_{{\kappa }_j}^{+}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{{\psi }_j^{\xi }}}\right)}\end{array}\right],\left[\begin{array}{l}\sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\displaystyle \prod _{j=1}^n{\left(\frac{\aleph {\left({\beta }_{{\kappa }_j}^{-}\right)}^2+1}{1+\aleph }\right)}^{{\psi }_j^{\xi }}}-1\right)},\\ \sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\displaystyle \prod _{j=1}^n{\left(\frac{\aleph {\left({\beta }_{{\kappa }_j}^{+}\right)}^2+1}{1+\aleph }\right)}^{{\psi }_j^{\xi }}}-1\right)}\end{array}\right]\right)\end{array}$ (6)

证明：下面通过数学归纳法证明公式(6)成立。当$n=2$ 时，根据定义7可得：

$\begin{array}{l}\text{IVPFSWSFWA}\left({\kappa }_1,{\kappa }_2,\cdots ,{\kappa }_n\right)={\psi }_1^{\xi }{\kappa }_1{\oplus }_{sw}{\psi }_2^{\xi }{\kappa }_2\\ =\left(\left[\begin{array}{l}\sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\left(1-{\left({\alpha }_{{\kappa }_1}^{-}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{{\psi }_1^{\xi }}\right)},\\ \sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\left(1-{\left({\alpha }_{{\kappa }_1}^{+}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{{\psi }_1^{\xi }}\right)}\end{array}\right],\left[\begin{array}{l}\sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\left(\frac{\aleph {\left({\beta }_{{\kappa }_1}^{-}\right)}^2+1}{1+\aleph }\right)}^{{\psi }_1^{\xi }}-1\right)},\\ \sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\left(\frac{\aleph {\left({\beta }_{{\kappa }_1}^{+}\right)}^2+1}{1+\aleph }\right)}^{{\psi }_1^{\xi }}-1\right)}\end{array}\right]\right)\\ {\oplus }_{sw}\left(\left[\begin{array}{l}\sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\left(1-{\left({\alpha }_{{\kappa }_2}^{-}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{{\psi }_2^{\xi }}\right)},\\ \sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\left(1-{\left({\alpha }_{{\kappa }_2}^{+}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{{\psi }_2^{\xi }}\right)}\end{array}\right],\left[\begin{array}{l}\sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\left(\frac{\aleph {\left({\beta }_{{\kappa }_2}^{-}\right)}^2+1}{1+\aleph }\right)}^{{\psi }_2^{\xi }}-1\right)},\\ \sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\left(\frac{\aleph {\left({\beta }_{{\kappa }_2}^{+}\right)}^2+1}{1+\aleph }\right)}^{{\psi }_2^{\xi }}-1\right)}\end{array}\right]\right)\end{array}$

$=\left(\left[\begin{array}{l}\sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\displaystyle \prod _{j=1}^2{\left(1-{\left({\alpha }_{{\kappa }_j}^{-}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{{\psi }_j^{\xi }}}\right)},\\ \sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\displaystyle \prod _{j=1}^2{\left(1-{\left({\alpha }_{{\kappa }_j}^{+}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{{\psi }_j^{\xi }}}\right)}\end{array}\right],\left[\begin{array}{l}\sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\displaystyle \prod _{j=1}^2{\left(\frac{\aleph {\left({\beta }_{{\kappa }_j}^{-}\right)}^2+1}{1+\aleph }\right)}^{{\psi }_j^{\xi }}}-1\right)},\\ \sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\displaystyle \prod _{j=1}^2{\left(\frac{\aleph {\left({\beta }_{{\kappa }_j}^{+}\right)}^2+1}{1+\aleph }\right)}^{{\psi }_j^{\xi }}}-1\right)}\end{array}\right]\right)$

因此，则当$n=2$ 时，公式(6)成立。假设当$n=\ell$ ，公式(6)成立。

$\begin{array}{l}\text{IVPFSWSFWA}\left({\kappa }_1,{\kappa }_2,\cdots ,{\kappa }_{\ell }\right)=\underset{j=1}{\overset{\ell }{{\oplus }_{sw}}}{\psi }_j^{\xi }{\kappa }_j\\ =\left(\left[\begin{array}{l}\sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\displaystyle \prod _{j=1}^{\ell }{\left(1-{\left({\alpha }_{{\kappa }_j}^{-}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{{\psi }_j^{\xi }}}\right)},\\ \sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\displaystyle \prod _{j=1}^{\ell }{\left(1-{\left({\alpha }_{{\kappa }_j}^{+}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{{\psi }_j^{\xi }}}\right)}\end{array}\right],\left[\begin{array}{l}\sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\displaystyle \prod _{j=1}^{\ell }{\left(\frac{\aleph {\left({\beta }_{{\kappa }_j}^{-}\right)}^2+1}{1+\aleph }\right)}^{{\psi }_j^{\xi }}}-1\right)},\\ \sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\displaystyle \prod _{j=1}^{\ell }{\left(\frac{\aleph {\left({\beta }_{{\kappa }_j}^{+}\right)}^2+1}{1+\aleph }\right)}^{{\psi }_j^{\xi }}}-1\right)}\end{array}\right]\right)\end{array}$

则当$n=\ell +1$ 时，

$\begin{array}{l}\text{IVPFSWSF}\left({\kappa }_1,{\kappa }_2,\cdots ,{\kappa }_{\ell }\right)=\text{IVPFSWSF}\left({\kappa }_1,{\kappa }_2,\cdots ,{\kappa }_{\ell }\right){\oplus }_{sw}{\psi }_{\ell +1}^{\xi }{\kappa }_{\ell +1}\\ =\left(\left[\begin{array}{l}\sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\displaystyle \prod _{j=1}^{\ell }{\left(1-{\left({\alpha }_{{\kappa }_j}^{-}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{{\psi }_j^{\xi }}}\right)},\\ \sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\displaystyle \prod _{j=1}^{\ell }{\left(1-{\left({\alpha }_{{\kappa }_j}^{+}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{{\psi }_j^{\xi }}}\right)}\end{array}\right],\left[\begin{array}{l}\sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\displaystyle \prod _{j=1}^{\ell }{\left(\frac{\aleph {\left({\beta }_{{\kappa }_j}^{-}\right)}^2+1}{1+\aleph }\right)}^{{\psi }_j^{\xi }}}-1\right)},\\ \sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\displaystyle \prod _{j=1}^{\ell }{\left(\frac{\aleph {\left({\beta }_{{\kappa }_j}^{+}\right)}^2+1}{1+\aleph }\right)}^{{\psi }_j^{\xi }}}-1\right)}\end{array}\right]\right)\\ {\oplus }_{sw}\left(\left[\begin{array}{l}\sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\left(1-{\left({\alpha }_{{\kappa }_{\ell +1}}^{-}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{{\psi }_{\ell +1}^{\xi }}\right)},\\ \sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\left(1-{\left({\alpha }_{{\kappa }_{\ell +1}}^{+}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{{\psi }_{\ell +1}^{\xi }}\right)}\end{array}\right],\left[\begin{array}{l}\sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\left(\frac{\aleph {\left({\beta }_{{\kappa }_{\ell +1}}^{-}\right)}^2+1}{1+\aleph }\right)}^{{\psi }_{\ell +1}^{\xi }}-1\right)},\\ \sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\left(\frac{\aleph {\left({\beta }_{{\kappa }_{\ell +1}}^{+}\right)}^2+1}{1+\aleph }\right)}^{{\psi }_{\ell +1}^{\xi }}-1\right)}\end{array}\right]\right)\end{array}$

$=\left(\left[\begin{array}{l}\sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\displaystyle \prod _{j=1}^{\ell +1}{\left(1-{\left({\alpha }_{{\kappa }_j}^{-}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{{\psi }_j^{\xi }}}\right)},\\ \sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\displaystyle \prod _{j=1}^{\ell +1}{\left(1-{\left({\alpha }_{{\kappa }_j}^{+}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{{\psi }_j^{\xi }}}\right)}\end{array}\right],\left[\begin{array}{l}\sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\displaystyle \prod _{j=1}^{\ell +1}{\left(\frac{\aleph {\left({\beta }_{{\kappa }_j}^{-}\right)}^2+1}{1+\aleph }\right)}^{{\psi }_j^{\xi }}}-1\right)},\\ \sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\displaystyle \prod _{j=1}^{\ell +1}{\left(\frac{\aleph {\left({\beta }_{{\kappa }_j}^{+}\right)}^2+1}{1+\aleph }\right)}^{{\psi }_j^{\xi }}}-1\right)}\end{array}\right]\right)$

因此，当$n=\ell +1$ 时，公式(6)成立。因此，公式(6)对所有$n$ 成立。

接下来，我们探讨IVPFSWSF算子的性质。

性质1 (幂等性)设${\kappa }_j=\left(\left[{\alpha }_{{\kappa }_j}^{-},{\alpha }_{{\kappa }_j}^{+}\right],\left[{\beta }_{{\kappa }_j}^{-},{\beta }_{{\kappa }_j}^{+}\right]\right)\left(j=1,2,\cdots ,n\right)$ 为一组区间值毕达哥拉斯模糊数。若${\kappa }_j={\kappa }_0=\left(\left[{\alpha }_{{\kappa }_0}^{-},{\alpha }_{{\kappa }_0}^{+}\right],\left[{\beta }_{{\kappa }_0}^{-},{\beta }_{{\kappa }_0}^{+}\right]\right)$ ，则$\text{IVPFSWSF}\left({\kappa }_1,{\kappa }_2,\cdots ,{\kappa }_n\right)={\kappa }_0$ 。

证明：因为${\kappa }_j={\kappa }_0=\left(\left[{\alpha }_{{\kappa }_0}^{-},{\alpha }_{{\kappa }_0}^{+}\right],\left[{\beta }_{{\kappa }_0}^{-},{\beta }_{{\kappa }_0}^{+}\right]\right)$ ，$\text{IVPFSWSF}\left({\kappa }_1,{\kappa }_2,\cdots ,{\kappa }_n\right)=\underset{j=1}{\overset{n}{{\oplus }_{sw}}}{\psi }_j^{\xi }{\kappa }_j={\kappa }_0{\displaystyle \sum _{j=1}^n{\psi }_j^{\xi }}={\kappa }_0$ 。

性质2 (单调性)设${\kappa }_j=\left(\left[{\alpha }_{{\kappa }_j}^{-},{\alpha }_{{\kappa }_j}^{+}\right],\left[{\beta }_{{\kappa }_j}^{-},{\beta }_{{\kappa }_j}^{+}\right]\right)$ 和${\tilde{\kappa }}_j=\left(\left[{\tilde{\alpha }}_{{\tilde{\kappa }}_j}^{-},{\tilde{\alpha }}_{{\tilde{\kappa }}_j}^{+}\right],\left[{\tilde{\beta }}_{{\tilde{\kappa }}_j}^{-},{\tilde{\beta }}_{{\tilde{\kappa }}_j}^{+}\right]\right)\left(j=1\left(1\right)n\right)$ 是IVPFNs。若${\alpha }_{{\kappa }_j}^{-}\ge {\tilde{\alpha }}_{{\tilde{\kappa }}_j}^{-}$ ，${\alpha }_{{\kappa }_j}^{+}\ge {\tilde{\alpha }}_{{\tilde{\kappa }}_j}^{+}$ ，${\beta }_{{\kappa }_j}^{-}\le {\tilde{\beta }}_{{\tilde{\kappa }}_j}^{-}$ ，${\beta }_{{\kappa }_j}^{+}\le {\tilde{\beta }}_{{\tilde{\kappa }}_j}^{+}$ ，则$\text{IVPFSWSF}\left({\kappa }_1,{\kappa }_2,\cdots ,{\kappa }_n\right)\ge \text{IVPFSWSF}\left({\tilde{\kappa }}_1,{\tilde{\kappa }}_2,\cdots ,{\tilde{\kappa }}_n\right)$.

证明：因为${\alpha }_{{\kappa }_j}^{-}\ge {\tilde{\alpha }}_{{\tilde{\kappa }}_j}^{-}$ ，则$1-\left(\aleph /\left(1+\aleph \right)\right)\cdot {\alpha }_{{\kappa }_j}^{-}\le 1-\left(\aleph /\left(1+\aleph \right)\right)\cdot {\tilde{\alpha }}_{{\tilde{\kappa }}_j}^{-}$ 。此外， $1-{\displaystyle \prod _{j=1}^n{\left(1-\left(\frac{\aleph }{1+\aleph }\right)\cdot {\alpha }_{{\kappa }_j}^{-}\right)}^{{\psi }_j^{\xi }}}\ge 1-{\displaystyle \prod _{j=1}^n{\left(1-\left(\frac{\aleph }{1+\aleph }\right)\cdot {\tilde{\alpha }}_{{\tilde{\kappa }}_j}^{-}\right)}^{{\psi }_j^{\xi }}}$ ，则

$\begin{array}{l}\frac{1+\aleph }{\aleph }\left(1-{\displaystyle \prod _{j=1}^n{\left(1-\left(\frac{\aleph }{1+\aleph }\right)\cdot {\alpha }_{{\kappa }_j}^{-}\right)}^{{\psi }_j^{\xi }}}\right)\ge \frac{1+\aleph }{\aleph }\left(1-{\displaystyle \prod _{j=1}^n{\left(1-\left(\frac{\aleph }{1+\aleph }\right)\cdot {\tilde{\alpha }}_{{\tilde{\kappa }}_j}^{-}\right)}^{{\psi }_j^{\xi }}}\right),\\ \frac{1+\aleph }{\aleph }\left(1-{\displaystyle \prod _{j=1}^n{\left(1-\left(\frac{\aleph }{1+\aleph }\right)\cdot {\alpha }_{{\kappa }_j}^{+}\right)}^{{\psi }_j^{\xi }}}\right)\ge \frac{1+\aleph }{\aleph }\left(1-{\displaystyle \prod _{j=1}^n{\left(1-\left(\frac{\aleph }{1+\aleph }\right)\cdot {\tilde{\alpha }}_{{\tilde{\kappa }}_j}^{+}\right)}^{{\psi }_j^{\xi }}}\right).\end{array}$

此外，因为${\beta }_{{\kappa }_j}^{-}\le {\tilde{\beta }}_{{\tilde{\kappa }}_j}^{-}$ ，则$\left(\aleph \cdot {\beta }_{{\kappa }_j}^{-}+1\right)/\left(1+\aleph \right)\le \left(\aleph \cdot {\tilde{\beta }}_{{\tilde{\kappa }}_j}^{-}+1\right)/\left(1+\aleph \right)$ ，则

$\begin{array}{l}\frac{1}{\aleph }\left(\left(1+\aleph \right){\displaystyle \prod _{j=1}^n{\left(\frac{\aleph \cdot {\beta }_{{\kappa }_j}^{-}+1}{1+\aleph }\right)}^{{\psi }_j^{\xi }}}-1\right)\le \frac{1}{\wp }\left(\left(1+\wp \right){\displaystyle \prod _{j=1}^n{\left(\frac{\aleph \cdot {\tilde{\beta }}_{{\tilde{\kappa }}_j}^{-}+1}{1+\aleph }\right)}^{{\psi }_j^{\xi }}}-1\right),\\ \frac{1}{\aleph }\left(\left(1+\aleph \right){\displaystyle \prod _{j=1}^n{\left(\frac{\aleph \cdot {\beta }_{{\kappa }_j}^{+}+1}{1+\aleph }\right)}^{{\psi }_j^{\xi }}}-1\right)\le \frac{1}{\wp }\left(\left(1+\wp \right){\displaystyle \prod _{j=1}^n{\left(\frac{\aleph \cdot {\tilde{\beta }}_{{\tilde{\kappa }}_j}^{+}+1}{1+\aleph }\right)}^{{\psi }_j^{\xi }}}-1\right).\end{array}$

基于定义2可得$\text{IVPFSWSF}\left({\kappa }_1,{\kappa }_2,\cdots ,{\kappa }_n\right)\ge \text{IVPFSWSF}\left({\tilde{\kappa }}_1,{\tilde{\kappa }}_2,\cdots ,{\tilde{\kappa }}_n\right)$。

性质3 (有界性)设${\kappa }_j=\left(\left[{\alpha }_{{\kappa }_j}^{-},{\alpha }_{{\kappa }_j}^{+}\right],\left[{\beta }_{{\kappa }_j}^{-},{\beta }_{{\kappa }_j}^{+}\right]\right)\left(j=1,2,\cdots ,n\right)$ 为一组区间值毕达哥拉斯模糊数。若${\kappa }^{-}=\left(\left[\underset{1\le j\le n}{\min }\left\{{\alpha }_{{\kappa }_j}^{-}\right\},\underset{1\le j\le n}{\min }\left\{{\alpha }_{{\kappa }_j}^{+}\right\}\right],\left[\underset{1\le j\le n}{\max }\left\{{\beta }_{{\kappa }_j}^{-}\right\},\underset{1\le j\le n}{\max }\left\{{\beta }_{{\kappa }_j}^{+}\right\}\right]\right)$ ， ${\kappa }^{+}=\left(\left[\underset{1\le j\le n}{\max }\left\{{\alpha }_{{\kappa }_j}^{-}\right\},\underset{1\le j\le n}{\max }\left\{{\alpha }_{{\kappa }_j}^{+}\right\}\right],\left[\underset{1\le j\le n}{\min }\left\{{\beta }_{{\kappa }_j}^{-}\right\},\underset{1\le j\le n}{\min }\left\{{\beta }_{{\kappa }_j}^{+}\right\}\right]\right)$ 。则${\kappa }^{-}\le \text{IVPFSWSF}\left({\kappa }_1,{\kappa }_2,\cdots ,{\kappa }_n\right)\le {\kappa }^{+}$ 。

定义8 设${\kappa }_j=\left(\left[{\alpha }_{{\kappa }_j}^{-},{\alpha }_{{\kappa }_j}^{+}\right],\left[{\beta }_{{\kappa }_j}^{-},{\beta }_{{\kappa }_j}^{+}\right]\right)\left(j=1,2,\cdots ,n\right)$ 为一组IVPFNs。则区间值毕达哥拉斯模糊Sugeno-Weber Softmax加权平均(IVPFSWSFWA)算子$\text{IVPFSWSFWA}:{\Omega }^n\to \Omega$ 定义如下：

$\text{IVPFSWSFWA}\left({\kappa }_1,{\kappa }_2,\cdots ,{\kappa }_n\right)=\underset{j=1}{\overset{n}{{\oplus }_{sw}}}{\Phi }_j^{\xi }{\kappa }_j$ (7)

其中${\Phi }_j^{\xi }={\varpi }_j\exp \left(T_j/\xi \right)/{\displaystyle {\sum }_{j=1}^n\left({\varpi }_j\exp \left(T_j/\xi \right)\right)}\left(\xi >0\right)$ 满足${\displaystyle {\sum }_{j=1}^n{\psi }_j^{\xi }=1},{\psi }_j^{\xi }\in \left[0,1\right]$ ，$\xi$ 为参数且$\xi >0$ 。$T_j={\displaystyle {\prod }_{l=1}^{j-1}SF}\left({\kappa }_1\right),\left(j=2,3,\cdots ,n\right)$ ，$T_1=1$ 且$SF\left({\kappa }_j\right)$ 是${\kappa }_j$ 的得分函数。

定理2 设${\kappa }_j=\left(\left[{\alpha }_{{\kappa }_j}^{-},{\alpha }_{{\kappa }_j}^{+}\right],\left[{\beta }_{{\kappa }_j}^{-},{\beta }_{{\kappa }_j}^{+}\right]\right)\left(j=1,2,\cdots ,n\right)$ 为一组IVPFNs。则利用IVPFSWSFWA算子集成这组区间值毕达哥拉斯模糊数后的结果仍是区间值毕达哥拉斯模糊数且

$\begin{array}{l}\text{IVPFSWSFWA}\left({\kappa }_1,{\kappa }_2,\cdots ,{\kappa }_n\right)=\underset{j=1}{\overset{n}{{\oplus }_{sw}}}{\Phi }_j^{\xi }{\kappa }_j\\ =\left(\left[\begin{array}{l}\sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\alpha }_{{\kappa }_j}^{-}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{{\Phi }_j^{\xi }}}\right)},\\ \sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\alpha }_{{\kappa }_j}^{+}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{{\Phi }_j^{\xi }}}\right)}\end{array}\right],\left[\begin{array}{l}\sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\displaystyle \prod _{j=1}^n{\left(\frac{\aleph {\left({\beta }_{{\kappa }_j}^{-}\right)}^2+1}{1+\aleph }\right)}^{{\Phi }_j^{\xi }}}-1\right)},\\ \sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\displaystyle \prod _{j=1}^n{\left(\frac{\aleph {\left({\beta }_{{\kappa }_j}^{+}\right)}^2+1}{1+\aleph }\right)}^{{\Phi }_j^{\xi }}}-1\right)}\end{array}\right]\right)\end{array}$ (8)

证明：与定理1相似。

3.3. 区间值毕达哥拉斯模糊环境下的Sugeno-Weber Softmax几何算子

定义9 设${\kappa }_j=\left(\left[{\alpha }_{{\kappa }_j}^{-},{\alpha }_{{\kappa }_j}^{+}\right],\left[{\beta }_{{\kappa }_j}^{-},{\beta }_{{\kappa }_j}^{+}\right]\right)\left(j=1,2,\cdots ,n\right)$ 为一组IVPFNs。则区间值毕达哥拉斯模糊Sugeno-Weber Softmax几何(IVPFSWSFG)算子$\text{IVPFSWSFG}:{\Omega }^n\to \Omega$ 定义如下：

$\text{IVPFSWSFG}\left({\kappa }_1,{\kappa }_2,\cdots ,{\kappa }_n\right)=\underset{j=1}{\overset{n}{{\otimes }_{sw}}}{\left({\kappa }_j\right)}^{{\psi }_j^{\xi }}.$ (9)

定理3 设${\kappa }_j=\left(\left[{\alpha }_{{\kappa }_j}^{-},{\alpha }_{{\kappa }_j}^{+}\right],\left[{\beta }_{{\kappa }_j}^{-},{\beta }_{{\kappa }_j}^{+}\right]\right)\left(j=1,2,\cdots ,n\right)$ 为一组IVPFNs。则利用IVPFSWSFG算子集成这组区间值毕达哥拉斯模糊数后的结果仍是区间值毕达哥拉斯模糊数且集成结果表示为

$\begin{array}{l}\text{IVPFSWSFG}\left({\kappa }_1,{\kappa }_2,\cdots ,{\kappa }_n\right)=\underset{j=1}{\overset{n}{{\otimes }_{sw}}}{\left({\kappa }_j\right)}^{{\psi }_j^{\xi }}\\ =\left(\left[\begin{array}{l}\sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\displaystyle \prod _{j=1}^n{\left(\frac{\aleph {\left({\alpha }_{{\kappa }_j}^{-}\right)}^2+1}{1+\aleph }\right)}^{{\psi }_j^{\xi }}}-1\right)},\\ \sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\displaystyle \prod _{j=1}^n{\left(\frac{\aleph {\left({\alpha }_{{\kappa }_j}^{+}\right)}^2+1}{1+\aleph }\right)}^{{\psi }_j^{\xi }}}-1\right)}\end{array}\right],\left[\begin{array}{l}\sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\beta }_{{\kappa }_j}^{-}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{{\psi }_j^{\xi }}}\right)},\\ \sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\beta }_{{\kappa }_j}^{+}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{{\psi }_j^{\xi }}}\right)}\end{array}\right]\right)\end{array}$ (10)

IVPFSWSFG算子与IVPFSWSF算子都具有幂等性、单调性和有界性性质，此处不再赘述。

定义10 设${\kappa }_j=\left(\left[{\alpha }_{{\kappa }_j}^{-},{\alpha }_{{\kappa }_j}^{+}\right],\left[{\beta }_{{\kappa }_j}^{-},{\beta }_{{\kappa }_j}^{+}\right]\right)\left(j=1,2,\cdots ,n\right)$ 为一组IVPFNs。则区间值毕达哥拉斯模糊Sugeno-Weber Softmax加权几何(IVPFSWSFWG)算子$\text{IVPFSWSFWG}:{\Omega }^n\to \Omega$ 定义如下：

$\text{IVPFSWSFWG}\left({\kappa }_1,{\kappa }_2,\cdots ,{\kappa }_n\right)=\underset{j=1}{\overset{n}{{\otimes }_{sw}}}{\left({\kappa }_j\right)}^{{\Phi }_j^{\xi }}.$ (11)

定理4 设${\kappa }_j=\left(\left[{\alpha }_{{\kappa }_j}^{-},{\alpha }_{{\kappa }_j}^{+}\right],\left[{\beta }_{{\kappa }_j}^{-},{\beta }_{{\kappa }_j}^{+}\right]\right)\left(j=1,2,\cdots ,n\right)$ 为一组IVPFNs。则利用IVPFSWSFWG算子集成这组区间值毕达哥拉斯模糊数后的结果仍是区间值毕达哥拉斯模糊数且

$\begin{array}{l}\text{IVPFSWSFWG}\left({\kappa }_1,{\kappa }_2,\cdots ,{\kappa }_n\right)=\underset{j=1}{\overset{n}{{\otimes }_{sw}}}{\left({\kappa }_j\right)}^{{\Phi }_j^{\xi }}\\ =\left(\left[\begin{array}{l}\sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\displaystyle \prod _{j=1}^n{\left(\frac{\aleph {\left({\alpha }_{{\kappa }_j}^{-}\right)}^2+1}{1+\aleph }\right)}^{{\Phi }_j^{\xi }}}-1\right)},\\ \sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\displaystyle \prod _{j=1}^n{\left(\frac{\aleph {\left({\alpha }_{{\kappa }_j}^{+}\right)}^2+1}{1+\aleph }\right)}^{{\Phi }_j^{\xi }}}-1\right)}\end{array}\right],\left[\begin{array}{l}\sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\beta }_{{\kappa }_j}^{-}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{{\Phi }_j^{\xi }}}\right)},\\ \sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\beta }_{{\kappa }_j}^{+}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{{\Phi }_j^{\xi }}}\right)}\end{array}\right]\right)\end{array}$ (12)

证明：与定理2相似。

4. 基于区间值毕达哥拉斯模糊Sugeno-Weber算子的WASPAS多属性决策方法

本章基于区间值毕达哥拉斯模糊多属性决策问题涉及的概念和符号定义如下。$G=\left\{G_i|i=1\left(1\right)m\right\}$ 为一组备选方案，$C=\left\{C_j|j=1\left(1\right)n\right\}$ 为属性集合且其权重向量为$\varpi =\left\{{\varpi }_j|j=1\left(1\right)n\right\}$ 且满足 ${\varpi }_j\in \left[0,1\right],{\displaystyle {\sum }_{j=1}^n{\varpi }_j}=1$ 。专家对备选方案$G_i$ $\left(i=1,2,\cdots ,m\right)$ 在属性$C_j$ $\left(j=1,2,\cdots ,n\right)$ 下的评价值用区间值毕达哥拉斯模糊表示并构成决策矩阵$\tilde{F}={\left({\tilde{\kappa }}_{ij}\right)}_{m\times n}$ ，${\tilde{\kappa }}_{ij}=\left(\left[{\tilde{\alpha }}_{{\tilde{\kappa }}_{ij}}^{-},{\tilde{\alpha }}_{{\tilde{\kappa }}_{ij}}^{+}\right],\left[{\tilde{\beta }}_{{\tilde{\kappa }}_{ij}}^{-},{\tilde{\beta }}_{{\tilde{\kappa }}_{ij}}^{+}\right]\right)$ $\left(i=1,2,\cdots ,m;j=1,2,\cdots ,n\right)$ 。基于上述定义和符号，所提基于区间值毕达哥拉斯模糊Sugeno-Weber Softmax算子的WASPAS多属性决策方法步骤如下：

步骤1：确定归一化决策矩阵$F={\left({\kappa }_{ij}^{}\right)}_{m\times n}$ 。

${\kappa }_{ij}=\{\begin{array}{l}{\tilde{\kappa }}_{ij}=\left(\left[{\tilde{\alpha }}_{{\tilde{\kappa }}_{ij}}^{-},{\tilde{\alpha }}_{{\tilde{\kappa }}_{ij}}^{+}\right],\left[{\tilde{\beta }}_{{\tilde{\kappa }}_{ij}}^{-},{\tilde{\beta }}_{{\tilde{\kappa }}_{ij}}^{+}\right]\right),j\in C_b\\ {\left({\tilde{\kappa }}_{ij}\right)}^c=\left(\left[{\tilde{\beta }}_{{\tilde{\kappa }}_{ij}}^{-},{\tilde{\beta }}_{{\tilde{\kappa }}_{ij}}^{+}\right],\left[{\tilde{\alpha }}_{{\tilde{\kappa }}_{ij}}^{-},{\tilde{\alpha }}_{{\tilde{\kappa }}_{ij}}^{+}\right]\right),j\in C_c\end{array}$ (13)

步骤2：确定属性权重。

步骤2.1：计算每个属性下的归一化均值：

$\begin{array}{c}{\varphi }_j=\text{IVPFSWSFWA}\left({\kappa }_{1j},{\kappa }_{2j},\cdots ,{\kappa }_{mj}\right)=\underset{i=1}{\overset{m}{{\oplus }_{sw}}}\left(1/m\right){\kappa }_{ij}\\ =\left(\left[\begin{array}{l}\sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\displaystyle \prod _{i=1}^m{\left(1-{\left({\alpha }_{{\kappa }_{ij}}^{-}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{1/m}}\right)},\\ \sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\displaystyle \prod _{i=1}^m{\left(1-{\left({\alpha }_{{\kappa }_{ij}}^{+}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{1/m}}\right)}\end{array}\right],\left[\begin{array}{l}\sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\displaystyle \prod _{i=1}^m{\left(\frac{\aleph {\left({\beta }_{{\kappa }_{ij}}^{-}\right)}^2+1}{1+\aleph }\right)}^{1/m}}-1\right)},\\ \sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\displaystyle \prod _{i=1}^m{\left(\frac{\aleph {\left({\beta }_{{\kappa }_{ij}}^{+}\right)}^2+1}{1+\aleph }\right)}^{1/m}}-1\right)}\end{array}\right]\right)\end{array}$ (14)

步骤2.2：计算所有方案的评估值与每个属性下的归一化均值的偏差：

${\mu }_j={\displaystyle \sum _{i=1}^m{\left(SF\left({\kappa }_{ij}\right)-SF\left({\varphi }_j\right)\right)}^2},j=1,2,\cdots ,n$ (15)

步骤2.3：计算属性的权重：

${\varpi }_j^{}=\frac{{\mu }_j}{{\displaystyle \sum _{j=1}^n{\mu }_j}},j=1,2,\cdots ,n.$ (16)

步骤3：基于IVPFSWSFWA算子确定备选方案的加权和测度。

$\begin{array}{c}{}^a\kappa {}_i=\left(\left[{}^a\alpha {}_i^{-},{}^a\alpha {}_i^{-}\right],\left[{}^a\beta {}_i^{-},{}^a\beta {}_i^{-}\right]\right)=\text{IVPFSWSFWA}\left({\kappa }_{i1},{\kappa }_{i2},\cdots ,{\kappa }_{in}\right)=\underset{j=1}{\overset{n}{{\oplus }_{sw}}}{\psi }_j^{\xi }{\kappa }_{ij}\\ =\left(\left[\begin{array}{l}\sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\alpha }_{{\kappa }_{ij}}^{-}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{{\psi }_j^{\xi }}}\right)},\\ \sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\alpha }_{{\kappa }_{ij}}^{+}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{{\psi }_j^{\xi }}}\right)}\end{array}\right],\left[\begin{array}{l}\sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\displaystyle \prod _{j=1}^n{\left(\frac{\aleph {\left({\beta }_{{\kappa }_{ij}}^{-}\right)}^2+1}{1+\aleph }\right)}^{{\psi }_j^{\xi }}}-1\right)},\\ \sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\displaystyle \prod _{j=1}^n{\left(\frac{\aleph {\left({\beta }_{{\kappa }_{ij}}^{+}\right)}^2+1}{1+\aleph }\right)}^{{\psi }_j^{\xi }}}-1\right)}\end{array}\right]\right)\end{array}$ (17)

步骤4：基于IVPFSWSFWG算子确定备选方案的加权积测度。

$\begin{array}{c}{}^b\kappa {}_i=\left(\left[{}^b\alpha {}_i^{-},{}^b\alpha {}_i^{+}\right],\left[{}^b\beta {}_i^{-},{}^b\beta {}_i^{+}\right]\right)=\text{IVPFSWSFWG}\left({\kappa }_1,{\kappa }_2,\cdots ,{\kappa }_n\right)=\underset{j=1}{\overset{n}{{\otimes }_{sw}}}{\left({\kappa }_{ij}\right)}^{{\Phi }_j^{\xi }}\\ =\left(\left[\begin{array}{l}\sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\displaystyle \prod _{j=1}^n{\left(\frac{\aleph {\left({\alpha }_{{\kappa }_{ij}}^{-}\right)}^2+1}{1+\aleph }\right)}^{{\Phi }_j^{\xi }}}-1\right)},\\ \sqrt[]{\frac{1}{\aleph }\left(\left(1+\aleph \right){\displaystyle \prod _{j=1}^n{\left(\frac{\aleph {\left({\alpha }_{{\kappa }_{ij}}^{+}\right)}^2+1}{1+\aleph }\right)}^{{\Phi }_j^{\xi }}}-1\right)}\end{array}\right],\left[\begin{array}{l}\sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\beta }_{{\kappa }_{ij}}^{-}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{{\Phi }_j^{\xi }}}\right)},\\ \sqrt[]{\frac{1+\aleph }{\aleph }\left(1-{\displaystyle \prod _{j=1}^n{\left(1-{\left({\beta }_{{\kappa }_{ij}}^{+}\right)}^2\left(\frac{\aleph }{1+\aleph }\right)\right)}^{{\Phi }_j^{\xi }}}\right)}\end{array}\right]\right)\end{array}$ (18)

步骤5：计算每个备选方案的综合测度得分$H_i$ ：

$H_i=\vartheta \cdot SF\left({}^a\kappa {}_i\right)+\left(1-\vartheta \right)\cdot SF\left({}^b\kappa {}_i\right),$ (19)

其中$SF\left({}^a\kappa {}_i\right)$ 和$SF\left({}^b\kappa {}_i\right)$ 表示加权和测度和加权积测度的得分函数，$\vartheta \in \left[0,1\right]$ 为策略系数。

步骤6：根据每个备选方案的综合测度得分$H_i$ 对备选方案进行降序排列确定其优先级及最优方案。

5. 实例分析

本章通过案例分析和讨论分析所提基于区间值毕达哥拉斯模糊Sugeno-Weber算子的WASPAS多属性决策方法的有效性和鲁棒性。

5.1. 决策实施过程

为验证所提决策方法的有效性，本节通过逆向物流供应商选择案例进行讨论。某新能源企业为了从备选的五个逆向物流供应商中选择最优的供应商，基于逆向物流成本($C_1$ )、技术能力($C_2$ )、服务质量($C_3$ )信息共享能力($C_4$ )和运输能力($C_5$ )五个属性和决策者的认知确定备选供应商的优先级，考虑决策者的认知不确定性和权重完全未知的特征，通过所提区间值毕达哥拉斯模糊Sugeno-Weber算子的WASPAS多属性决策方法确定最优逆向物流供应商。决策者基于其知识背景和认知能力，以毕达哥拉斯模糊数形式给出五个逆向物流供应商在属性下的评价信息并构成决策矩阵如表1所示。

Table 1. Pythagorean fuzzy decision matrix

表1. 毕达哥拉斯糊决策矩阵

	$C_1$	$C_2$	$C_3$	$C_4$	$C_5$
$G_1$	([0.3, 0.4], [0.7, 0.8])	([0.4, 0.5], [0.6, 0.7])	([0.9, 0.95], [0.1, 0.15])	([0.5, 0.65], [0.5, 0.60])	([0.8, 0.9], [0.2, 0.35])
$G_2$	([0.2, 0.3], [0.8, 0.85])	([0.8, 0.9], [0.2, 0.35])	([0.4, 0.5], [0.6, 0.7])	([0.5, 0.65], [0.5, 0.60])	([0.5, 0.65], [0.5, 0.60])
$G_3$	([0.4, 0.5], [0.6, 0.7])	([0.5, 0.65], [0.5, 0.60])	([0.5, 0.65], [0.5, 0.60])	([0.4, 0.5], [0.6, 0.7])	([0.8, 0.9], [0.2, 0.35])
$G_4$	([0.3, 0.4], [0.7, 0.8])	([0.65, 0.8], [0.4, 0.50])	([0.4, 0.5], [0.6, 0.7])	([0.8, 0.9], [0.2, 0.35])	([0.5, 0.65], [0.5, 0.60])
$G_5$	([0.4, 0.5], [0.6, 0.7])	([0.5, 0.65], [0.5, 0.60])	([0.8, 0.9], [0.2, 0.35])	([0.4, 0.5], [0.6, 0.7])	([0.4, 0.5], [0.6, 0.7])

步骤1：通过公式(13)确定归一化决策矩阵如表2所示。

Table 2. Normalized pythagorean fuzzy decision matrix

表2. 归一化的毕达哥拉斯糊决策矩阵

	$C_1$	$C_2$	$C_3$	$C_4$	$C_5$
$G_1$	([0.7, 0.8], [0.3, 0.4])	([0.4, 0.5], [0.6, 0.7])	([0.9, 0.95], [0.1, 0.15])	([0.5, 0.65], [0.5, 0.60])	([0.8, 0.9], [0.2, 0.35])
$G_2$	([0.8, 0.85], [0.2, 0.3])	([0.8, 0.9], [0.2, 0.35])	([0.4, 0.5], [0.6, 0.7])	([0.5, 0.65], [0.5, 0.60])	([0.5, 0.65], [0.5, 0.60])
$G_3$	([0.6, 0.7] , [0.4, 0.5])	([0.5, 0.65], [0.5, 0.60])	([0.5, 0.65], [0.5, 0.60])	([0.4, 0.5], [0.6, 0.7])	([0.8, 0.9], [0.2, 0.35])
$G_4$	([0.7, 0.8], [0.3, 0.4])	([0.65, 0.8], [0.4, 0.50])	([0.4, 0.5], [0.6, 0.7])	([0.8, 0.9], [0.2, 0.35])	([0.5, 0.65], [0.5, 0.60])
$G_5$	([0.6, 0.7], [0.4, 0.5])	([0.5, 0.65], [0.5, 0.60])	([0.8, 0.9], [0.2, 0.35])	([0.4, 0.5], [0.6, 0.7])	([0.4, 0.5], [0.6, 0.7])

步骤2：确定属性权重。通过公式(14)~(16)计算属性的权重为

${\varpi }_1=0.0424$ ，${\varpi }_2=0.1641$ ，${\varpi }_3=0.3729$ ，${\varpi }_4=0.1867$ ，${\varpi }_5=0.2339$ 。

步骤3：通过公式(17)确定备选方案的加权和测度($\aleph =2,\xi =2$ )：

${}^a\kappa {}_1=\left(\left[0.7514,0.8322\right],\left[0.3767,0.4197\right]\right)$ ，${}^a\kappa {}_2=\left(\left[0.5714,0.6836\right],\left[0.5141,0.5785\right]\right)$ ，

${}^a\kappa {}_3=\left(\left[0.5788,0.7038\right],\left[0.4793,0.5582\right]\right)$ ，${}^a\kappa {}_4=\left(\left[0.5841,0.7057\right],\left[0.4980,0.5661\right]\right)$ ，

${}^a\kappa {}_5=\left(\left[0.6220,0.7314\right],\left[0.4608,0.5421\right]\right)$ 。

步骤4：通过公式(18)确定备选方案的加权积测度($\aleph =2,\xi =2$ )：

${}^b\kappa {}_1=\left(\left[0.7006,0.7882\right],\left[0.3698,0.4659\right]\right)$ ，${}^b\kappa {}_2=\left(\left[0.5376,0.6491\right],\left[0.3218,0.6011\right]\right)$ ，

${}^b\kappa {}_3=\left(\left[0.5534,0.6788\right],\left[0.3674,0.5769\right]\right)$ ，${}^b\kappa {}_4=\left(\left[0.5558,0.6721\right],\left[0.3111,0.5887\right]\right)$ ，

${}^b\kappa {}_5=\left(\left[0.5825,0.6876\right],\left[0.4525,0.5722\right]\right)$ 。

步骤5：通过公式(19)计算每个备选方案的综合测度得分$H_i$ ($\vartheta =0.5$ )：

$H_1=0.7122$ ，$H_2=0.5550$ ，$H_3=0.5735$ ，$H_4=0.5735$ ，$H_5=0.5870$ 。

步骤6：根据每个备选方案的综合测度得分$H_i$ 对备选方案进行降序排列得到备选逆向物流供应商的排序为$G_1\succ G_5\succ G_3~G_4\succ G_2$ 。

5.2. 灵敏度分析

本节将对所提方法中涉及的参数进行讨论，分析参数的变化对供应商排序的影响，进而讨论所提方法的稳定性。首先对区间值毕达哥拉斯模糊Sugeno-Weber Softmax集成算子中的参数进行讨论，当参数$\aleph \in \left[2,9\right]$ 时，基于不同参数$\aleph$ 值的供应商的综合测度和排序结果如表3所示，从结果可以发现不同参数$\aleph$ 值下的供应商的综合测度随着参数值的增大而增大，但是供应商的排序未发生变化，故所提决策方法不受参数$\aleph$ 的影响，具有较强的稳定性。其次，对WASPAS方法中加权和和加权积测度的平衡系数进行讨论，当参数$\vartheta \in \left[0.1,1\right]$ 时，基于不同参数$\vartheta$ 值的决策结果如图1所示。从图1可知，当参数$\vartheta$ 不断变化时，供应商的排序有细微的变化，不同参数下的供应商排序为$G_1\succ G_5\succ G_2\succ G_3\succ G_4$ ，$G_1\succ G_5\succ G_3\succ G_4\succ G_2$ ，$G_1\succ G_5\succ G_4\succ G_3\succ G_2$ 和$G_1\succ G_5\succ G_4\succ G_2\succ G_3$ ，但是最优选择均是$G_1$ ，决策者可根据其偏好选定不同的参数以确定稳定的供应商排序。

Table 3. Comprehensive measurement and ranking results of suppliers based on different parameter $\aleph$ values

表3. 基于不同参数$\aleph$ 值的供应商的综合测度和排序结果

$\aleph$	$G_1$	$G_2$	$G_3$	$G_4$	$G_5$	排序
2	0.7122	0.5550	0.5735	0.5735	0.5870	$G_1\succ G_5\succ G_3~G_4\succ G_2$
3	0.7142	0.5555	0.5742	0.5738	0.5883	$G_1\succ G_5\succ G_3\succ G_4\succ G_2$
4	0.7160	0.5559	0.5747	0.5739	0.5895	$G_1\succ G_5\succ G_3\succ G_4\succ G_2$
5	0.7175	0.5563	0.5752	0.5741	0.5905	$G_1\succ G_5\succ G_3\succ G_4\succ G_2$
6	0.7188	0.5565	0.5755	0.5742	0.5914	$G_1\succ G_5\succ G_3\succ G_4\succ G_2$
7	0.7200	0.5567	0.5759	0.5743	0.5922	$G_1\succ G_5\succ G_3\succ G_4\succ G_2$
8	0.7210	0.5569	0.5762	0.5743	0.5928	$G_1\succ G_5\succ G_3\succ G_4\succ G_2$
9	0.7219	0.5571	0.5764	0.5744	0.5935	$G_1\succ G_5\succ G_3\succ G_4\succ G_2$

5.3. 比较分析

为验证本文所提方法的有效性，本节将所提方法与基于区间值毕达哥拉斯模糊加权平均(IVPFWA)算子的决策方法和基于区间值毕达哥拉斯模糊加权几何(IVPFWG)算子决策方法进行比较，比较结果如表4所示。通过表4可以发现，所提方法确定的供应商排序与基于IVPFWA算子[9]、IVPFWA算子[9]、IVPF-WSM方法和IVPF-WSM方法所得到的供应商排序是一致的，说明所提方法是有效的，最优选择均是$G_1$ 。但是所提决策方法在确定供应商排序时考虑了属性之间的优先级和权重未知的情形，因而具有更加广泛的适用性。

Figure 1. Comprehensive evaluation of alternatives based on different parameter $\vartheta$ values

图1. 基于不同参数$\vartheta$ 值的备选方案的综合测度

Table 4. Decision results based on different methods

表4. 基于不同方法的决策结果

	$G_1$	$G_2$	$G_3$	$G_4$	$G_5$	排序
IVPFWA算子[9]	0.8089	0.5744	0.6077	0.6014	0.6423	$G_1\succ G_5\succ G_3\succ G_4\succ G_2$
IVPFWG算子[9]	0.6811	0.5061	0.5449	0.5294	0.5416	$G_1\succ G_3\succ G_5\succ G_4\succ G_2$
IVPF-WSM方法	0.7348	0.5487	0.5723	0.5677	0.6039	$G_1\succ G_5\succ G_3\succ G_4\succ G_2$
IVPF-WPM方法	0.6896	0.5613	0.5748	0.5793	0.5700	$G_1\succ G_5\succ G_3\succ G_4\succ G_2$
所提方法	0.7122	0.5550	0.5735	0.5735	0.5870	$G_1\succ G_5\succ G_3~G_4\succ G_2$

6. 结论

针对属性值为区间值毕达哥拉斯模糊数且属性权重完全未知的复杂决策问题，本研究提出了一种基于Sugeno-Weber Softmax算子的WASPAS方法。主要结论如下：1) 基于Sugeno-Weber模定义了区间值毕达哥拉斯模糊数的运算法则，扩展了模糊集理论在不确定性建模中的适用性。2) 提出四种区间值毕达哥拉斯模糊Sugeno-Weber Softmax加权平均和几何集成算子并通过数学证明验证了算子的幂等性、有界性和单调性等性质。3) 设计基于区间值毕达哥拉斯模糊PSI方法的属性客观权重确定模型，克服了传统主观赋权的局限性。4) 融合所提算子与权重模型，建立改进的区间值毕达哥拉斯模糊WASPAS方法实现了备选方案的排序。最后，通过实际案例验证了方法的有效性与合理性，比较分析表明该方法在区分度与稳定性上优于现有方法，参数分析进一步证实了模型具有强鲁棒性。进一步研究将探索Sugeno-Weber模与其他模糊算子，如，幂算子、优先算子、Choquet算子的混合形式以进一步提升信息融合的灵活性。