直觉模糊集的减法和除法运算的新定义

A New Definition of Subtraction and Division Operations for Intuitionistic Fuzzy Sets

摘要: 模糊集合的概念自提出以来，其扩展形式不断得到丰富，其中最为经典的是直觉模糊集和区间直觉模糊集两个概念，后续直觉模糊集和区间直觉模糊集理论被广泛应用于处理决策信息不确定的问题。我们对直觉模糊集的关系和基本运算进行了梳理，发现现有的研究成果中没有定义直觉模糊集的减法和除法运算。因此，我们通过基于直觉模糊集已有的基本运算规则定义了减法和除法两种新运算，证明了基于直觉模糊数的减法和除法是封闭的，并进一步将直觉模糊集上的基本运算规则推广到区间直觉模糊集和区间直觉模糊数上，证明了基于区间直觉模糊数的和、差、积、商运算是封闭的。本文的最重要的贡献是使得直觉模糊集和区间直觉模糊集有了完整的四则运算，从而丰富了其理论基础。

Abstract: Since the concept of fuzzy sets was proposed, its extended form has been continuously enriched, among which the most classical ones are the concepts of intuitionistic fuzzy sets and interval intuitionistic fuzzy sets, and the subsequent theories of intuitionistic fuzzy sets and interval intuitionistic fuzzy sets have been widely used to deal with the problems of uncertainty of decision-making information. We have sorted out the relations and basic operations of intuitionistic fuzzy sets and found that the subtraction and division operations of intuitionistic fuzzy sets are not defined in the existing research results. Therefore, we prove that subtraction and division based on intuitionistic fuzzy numbers are closed by defining two new operations, subtraction and divi-sion, based on the existing basic operation rules of intuitionistic fuzzy sets, and further extend the basic operation rules on intuitionistic fuzzy sets to interval intuitionistic fuzzy sets and in-terval intuitionistic fuzzy numbers, and prove that the sum, difference, product, and quotient operations based on interval intuitionistic fuzzy numbers are closed. The most important contribution of this paper is to make intuitionistic fuzzy sets and interval intuitionistic fuzzy sets have complete four-rule operations, which enriches their theoretical foundation.

1. 引言

模糊数学是针对模糊性现象的一门数学分支 [1] ，基于模糊集合论。它提供了一种新方法来处理不确定性和不精确性问题，有效描述人脑处理模糊信息的思维过程。这一学科适用于“硬”和“软”科学，并已应用于模糊控制 [2] 、模糊决策 [3] [4] [5] 、模糊识别 [6] 、模糊聚类分析 [7] 等领域。传统的模糊集理论在面对社会发展中更多呈现模糊性和不确定性的事物特征时表现不足，直觉模糊集理论 [8] 因此产生，强调事物的隶属度、非隶属度和犹豫度，更有效地处理不确定信息和模糊信息。Atanassov和Gargov [9] 扩展了直觉模糊集的概念，提出了区间直觉模糊集，其中隶属度和非隶属度以区间数形式表示，其相关理论已在环境治理 [10] 、水利工程 [11] [12] 、供应商选择 [13] [14] 和风险评估 [15] [16] [17] 等领域得到广泛应用，因其实用性而备受关注。

文章主要通过阐述直觉模糊集与区间直觉模糊集的基本概念、基本运算规则等内容，发现直觉模糊数与区间直觉模糊数缺少减法和除法运算规则。针对发现的问题，文章给出了一种直觉模糊数和区间直觉模糊数的减法和除法运算定义，并证明了基于直觉模糊数和区间直觉模糊数的减法和除法是封闭的。最后，通过两个具体计算例子对直觉模糊数和区间直觉模糊数的基本运算规则进行了演示。

2. 直觉模糊集的理论及运算规则

2.1. 直觉模糊集的理论

定义1 [8] 设 $X=\left\{x_1,x_2,\cdots ,x_n\right\}$ 是一个非空集合，则下列 $\tilde{A}$ 为定义在X上的直觉模糊集：

$\tilde{A}\text{=}\left\{\left(x_i,{\mu }_{\tilde{A}}\left(x_i\right),{\nu }_{\tilde{A}}\left(x_i\right)\right)|x_i\in X\right\}$ (1)

其中 ${\mu }_{\tilde{A}}\left(x_i\right)$ 、 ${\nu }_{\tilde{A}}\left(x_i\right)$ 分别表示元素 $x_i$ 属于X的隶属度和非隶属度，且 ${\mu }_{\tilde{A}}\left(x_i\right)\in \left[0,1\right]$ ， ${\nu }_{\tilde{A}}\left(x_i\right)\in \left[0,1\right]$ ，满足 $0\le {\mu }_{\tilde{A}}\left(x_i\right)+{\nu }_{\tilde{A}}\left(x_i\right)\le 1$ ， $\forall x_i\in X$ ，元素 $x$ 属于X的犹豫度可表示为 ${\pi }_{\tilde{A}}\left(x_i\right)$ ，且满足 ${\pi }_{\tilde{A}}\left(x_i\right)=1-{\mu }_{\tilde{A}}\left(x_i\right)-{\nu }_{\tilde{A}}\left(x_i\right)$ ， $\forall x_i\in X$ 。当 ${\pi }_{\tilde{A}}\left(x_i\right)=0$ 时，则 $\tilde{A}$ 退化为Zadeh的模糊集。在文献 [8] 中，称下列为直觉模糊数：

$\tilde{\alpha }=\left(\mu ,\nu \right)$ (2)

其中 $\mu ,\nu \in \left[0,1\right]$ 且 $0\le \mu +\nu \le 1$ ， $\pi =1-\mu -\nu$ 。

2.2. 直觉模糊集的运算规则

在模糊集运算规则的基础上，Atanassov [8] 定义了以下运算规则：

定义2 设 $\tilde{A}=\left\{\left(x_i,{\mu }_{\tilde{A}}\left(x_i\right),{\nu }_{\tilde{A}}\left(x_i\right)\right)|x_i\in X\right\}$ ， $\tilde{B}=\left\{\left(x_i,{\mu }_{\tilde{B}}\left(x_i\right),{\nu }_{\tilde{B}}\left(x_i\right)\right)|x_i\in X\right\}$ 为任意两个直觉模糊集，定义以下运算规则：

1) $\overline{\stackrel{˜}{A}}=\left\{\left(x_i,{\nu }_{\tilde{A}}\left(x_i\right),{\mu }_{\tilde{A}}\left(x_i\right)\right)|x_i\in X\right\}$ ；

2) $\tilde{A}+\tilde{B}=\left\{\left(x_i,{\mu }_{\tilde{A}}\left(x_i\right)+{\mu }_{\tilde{B}}\left(x_i\right)-{\mu }_{\tilde{A}}\left(x_i\right){\mu }_{\tilde{B}}\left(x_i\right),{\nu }_{\tilde{A}}\left(x_i\right){\nu }_{\tilde{B}}\left(x_i\right)\right)|x_i\in X\right\}$ ；

3) $\tilde{A}\cdot \tilde{B}=\left\{\left(x_i,{\mu }_{\tilde{A}}\left(x_i\right){\mu }_{\tilde{B}}\left(x_i\right),{\nu }_{\tilde{A}}\left(x_i\right)+{\nu }_{\tilde{B}}\left(x_i\right)-{\nu }_{\tilde{A}}\left(x_i\right){\nu }_{\tilde{B}}\left(x_i\right)\right)|x_i\in X\right\}$ ；

4) $\tilde{A}\cup \tilde{B}=\left\{\left(x_i,\max \left\{{\mu }_{\tilde{A}}\left(x_i\right),{\mu }_{\tilde{B}}\left(x_i\right)\right\},\min \left\{{\nu }_{\tilde{A}}\left(x_i\right),{\nu }_{\tilde{B}}\left(x_i\right)\right\}\right)|x_i\in X\right\}$ ；

5) $\tilde{A}\cap \tilde{B}=\left\{\left(x_i,\min \left\{{\mu }_{\tilde{A}}\left(x_i\right),{\mu }_{\tilde{B}}\left(x_i\right)\right\},\max \left\{{\nu }_{\tilde{A}}\left(x_i\right),{\nu }_{\tilde{B}}\left(x_i\right)\right\}\right)|x_i\in X\right\}$ ；

6) $\lambda \tilde{A}=\left\{\left(x_i,1-{\left(1-{\mu }_{\tilde{A}}\left(x_i\right)\right)}^{\lambda },{\left({\nu }_{\tilde{A}}\left(x_i\right)\right)}^{\lambda }\right)|x_i\in X\right\},\lambda >0$ ；

7) ${\tilde{A}}^{\lambda }=\left\{\left(x_i,{\left({\mu }_{\tilde{A}}\left(x_i\right)\right)}^{\lambda },1-{\left(1-{\nu }_{\tilde{A}}\left(x_i\right)\right)}^{\lambda }\right)|x_i\in X\right\},\lambda >0$ 。

类似上述运算规则，可以定义以下直觉模糊数的运算规则：

定义3 [8] 设 $\tilde{\alpha }=\left({\mu }_{\tilde{\alpha }},{\nu }_{\tilde{\alpha }}\right)$ ， $\tilde{\beta }=\left({\mu }_{\tilde{\beta }},{\nu }_{\tilde{\beta }}\right)$ 为任意两个直觉模糊数，定义以下运算规则：

1) $\overline{\stackrel{˜}{\alpha }}=\left({\nu }_{\tilde{\alpha }},{\mu }_{\tilde{\alpha }}\right)$ ；

2) $\tilde{\alpha }\oplus \tilde{\beta }=\left({\mu }_{\tilde{\alpha }}+{\mu }_{\tilde{\beta }}-{\mu }_{\tilde{\alpha }}{\mu }_{\tilde{\beta }},{\nu }_{\tilde{\alpha }}{\nu }_{\tilde{\beta }}\right)$ ；

3) $\tilde{\alpha }\otimes \tilde{\beta }=\left({\mu }_{\tilde{\alpha }}{\mu }_{\tilde{\beta }},{\nu }_{\tilde{\alpha }}+{\nu }_{\tilde{\beta }}-{\nu }_{\tilde{\alpha }}{\nu }_{\tilde{\beta }}\right)$ ；

4) $\tilde{\alpha }\cup \tilde{\beta }=\left(\max \left\{{\mu }_{\tilde{\alpha }},{\mu }_{\tilde{\beta }}\right\},\min \left\{{\nu }_{\tilde{\alpha }},{\nu }_{\tilde{\beta }}\right\}\right)$ ；

5) $\tilde{\alpha }\cap \tilde{\beta }=\left(\min \left\{{\mu }_{\tilde{\alpha }},{\mu }_{\tilde{\beta }}\right\},\max \left\{{\nu }_{\tilde{\alpha }},{\nu }_{\tilde{\beta }}\right\}\right)$ ；

6) $\lambda \tilde{\alpha }=\left(1-{\left(1-{\mu }_{\tilde{\alpha }}\right)}^{\lambda },{\left({\nu }_{\tilde{\alpha }}\right)}^{\lambda }\right),\lambda >0$ ；

7) ${\tilde{\alpha }}^{\lambda }=\left({\left({\mu }_{\tilde{\alpha }}\right)}^{\lambda },1-{\left(1-{\nu }_{\tilde{\alpha }}\right)}^{\lambda }\right),\lambda >0$ 。

上述运算规则为直觉模糊数方面的应用奠定了良好的理论基础。且式(1)至(7)的运算结果仍为直觉模糊数。

2.3. 直觉模糊集的减法和除法运算规则定义

我们尝试通过类比于实数集合中的减法和除法的运算规则给出直觉模糊集减法和除法的定义。

定义4 设 $\tilde{A}=\left\{\left(x_i,{\mu }_{\tilde{A}}\left(x_i\right),{\nu }_{\tilde{A}}\left(x_i\right)\right)|x_i\in X\right\}$ ， $\tilde{B}=\left\{\left(x_i,{\mu }_{\tilde{B}}\left(x_i\right),{\nu }_{\tilde{B}}\left(x_i\right)\right)|x_i\in X\right\}$ 为任意两个直觉模糊集，定义以下减法和除法运算规则：

1) $\tilde{A}-\tilde{B}=\tilde{A}\text{+}\overline{\stackrel{˜}{B}}=\left\{\left(x_i,{\mu }_{\tilde{A}}\left(x_i\right)+{\nu }_{\tilde{B}}\left(x_i\right)-{\mu }_{\tilde{A}}\left(x_i\right){\nu }_{\tilde{B}}\left(x_i\right),{\nu }_{\tilde{A}}\left(x_i\right){\mu }_{\tilde{B}}\left(x_i\right)\right)|x_i\in X\right\}$ ；

2) $\tilde{A}÷\tilde{B}=\tilde{A}\cdot \overline{\stackrel{˜}{B}}=\left\{\left(x_i,{\mu }_{\tilde{A}}\left(x_i\right){\nu }_{\tilde{B}}\left(x_i\right),{\nu }_{\tilde{A}}\left(x_i\right)\text{+}{\mu }_{\tilde{B}}\left(x_i\right)-{\nu }_{\tilde{A}}\left(x_i\right){\mu }_{\tilde{B}}\left(x_i\right)\right)|x_i\in X\right\}$ 。

类似上述运算规则，定义直觉模糊数相反数、减法和除法运算规则：

定义5 设 $\tilde{\alpha }=\left({\mu }_{\tilde{\alpha }},{\nu }_{\tilde{\alpha }}\right)$ ， $\tilde{\beta }=\left({\mu }_{\tilde{\beta }},{\nu }_{\tilde{\beta }}\right)$ 为任意两个直觉模糊数，定义以下运算规则：

1) 定义 $-\tilde{\alpha }$ 称为 $\tilde{\alpha }$ 的“相反数”， $-\tilde{\alpha }=\overline{\stackrel{˜}{\alpha }}=\left({\nu }_{\tilde{\alpha }},{\mu }_{\tilde{\alpha }}\right)$ ；

2) $\tilde{\alpha }-\tilde{\beta }=\tilde{\alpha }+\left(-\tilde{\beta }\right)=\tilde{\alpha }+\overline{\stackrel{˜}{\beta }}=\left({\mu }_{\tilde{\alpha }}+{\nu }_{\tilde{\beta }}-{\mu }_{\tilde{\alpha }}{\nu }_{\tilde{\beta }},{\nu }_{\tilde{\alpha }}{\mu }_{\tilde{\beta }}\right)$ ；

3) $\tilde{\alpha }÷\tilde{\beta }=\tilde{\alpha }\otimes \overline{\stackrel{˜}{\beta }}=\left({\mu }_{\tilde{\alpha }}{\nu }_{\tilde{\beta }},{\nu }_{\tilde{\alpha }}+{\mu }_{\tilde{\beta }}-{\nu }_{\tilde{\alpha }}{\mu }_{\tilde{\beta }}\right)$ 。

定理1 设 $\tilde{\alpha }=\left({\mu }_{\tilde{\alpha }},{\nu }_{\tilde{\alpha }}\right)$ ， $\tilde{\beta }=\left({\mu }_{\tilde{\beta }},{\nu }_{\tilde{\beta }}\right)$ 为任意两个直觉模糊数，设 ${\tilde{\alpha }}_1=-\tilde{\alpha }$ ， ${\tilde{\alpha }}_2=\tilde{\alpha }-\tilde{\beta }$ ， ${\tilde{\alpha }}_3=\tilde{\alpha }÷\tilde{\beta }$ ，则 ${\tilde{\alpha }}_i\left(i=1,2,3\right)$ 均为直觉模糊数。

证明：因为 $\tilde{\alpha }=\left({\mu }_{\tilde{\alpha }},{\nu }_{\tilde{\alpha }}\right)$ , $\tilde{\beta }=\left({\mu }_{\tilde{\beta }},{\nu }_{\tilde{\beta }}\right)$ 为直觉模糊数，则有 ${\mu }_{\tilde{\alpha }},{\nu }_{\tilde{\alpha }},{\mu }_{\tilde{\beta }},{\nu }_{\tilde{\beta }}\in \left[0,1\right]$ ，且 ${\mu }_{\tilde{\alpha }}+{\nu }_{\tilde{\alpha }}\le 1$ ， ${\mu }_{\tilde{\beta }}+{\nu }_{\tilde{\beta }}\le 1$ 。

根据上述定义5中运算规则(1) ${\tilde{\alpha }}_1\text{=}-\tilde{\alpha }=\left({\nu }_{\tilde{\alpha }},{\mu }_{\tilde{\alpha }}\right)$ ，显然 ${\tilde{\alpha }}_1$ 仍为直觉模糊数。

根据上述定义5中运算规则(2)可知 ${\mu }_{{\tilde{\alpha }}_2}={\mu }_{\tilde{\alpha }}+{\nu }_{\tilde{\beta }}-{\mu }_{\tilde{\alpha }}{\nu }_{\tilde{\beta }}={\mu }_{\tilde{\alpha }}\left(1-{\nu }_{\tilde{\beta }}\right)+{\nu }_{\tilde{\beta }}\ge 0$ ， ${\nu }_{{\tilde{\alpha }}_2}={\nu }_{\tilde{\alpha }}{\mu }_{\tilde{\beta }}\ge 0$ ， ${\mu }_{{\tilde{\alpha }}_2}+{\nu }_{{\tilde{\alpha }}_2}={\mu }_{\tilde{\alpha }}+{\nu }_{\tilde{\beta }}-{\mu }_{\tilde{\alpha }}{\nu }_{\tilde{\beta }}+{\nu }_{\tilde{\alpha }}{\mu }_{\tilde{\beta }}\le {\mu }_{\tilde{\alpha }}+{\nu }_{\tilde{\beta }}-{\mu }_{\tilde{\alpha }}{\nu }_{\tilde{\beta }}+\left(1-{\mu }_{\tilde{\alpha }}\right)\left(1-{\nu }_{\tilde{\beta }}\right)=1$ ，所以 ${\tilde{\alpha }}_2$ 为直觉模糊数。

根据上述定义5中运算规则(3)可知 ${\mu }_{{\tilde{\alpha }}_3}={\mu }_{\tilde{\alpha }}{\nu }_{\tilde{\beta }}\ge 0$ ， ${\nu }_{{\tilde{\alpha }}_3}={\nu }_{\tilde{\alpha }}+\left(1-{\nu }_{\tilde{\alpha }}\right){\mu }_{\tilde{\beta }}\ge 0$ ， ${\mu }_{{\tilde{\alpha }}_3}\text{+}{\nu }_{{\tilde{\alpha }}_3}={\mu }_{\tilde{\alpha }}{\nu }_{\tilde{\beta }}+{\nu }_{\tilde{\alpha }}+{\mu }_{\tilde{\beta }}-{\nu }_{\tilde{\alpha }}{\mu }_{\tilde{\beta }}\le {\nu }_{\tilde{\alpha }}+{\mu }_{\tilde{\beta }}-{\nu }_{\tilde{\alpha }}{\mu }_{\tilde{\beta }}\text{+}\left(\text{1}-{\nu }_{\tilde{\alpha }}\right)\left(\text{1}-{\mu }_{\tilde{\beta }}\right)\text{=}1$ ，所以 ${\tilde{\alpha }}_3$ 为直觉模糊数。

3. 区间直觉模糊集的理论及运算规则

3.1. 直觉模糊集的理论

Atanassov和Gargov [9] 将直觉模糊集进一步拓展到区间直觉模糊集，区间直觉模糊集的隶属度和非隶属度其值是一个区间而不是一个精确的数，具体定义如下：

定义6 [9] 设 $X$ 是一个非空集合，则称下列 $\tilde{A}$ 为区间直觉模糊集：

$\tilde{A}\text{=}\left\{\left(x,{\mu }_{\tilde{A}}\left(x\right),{\nu }_{\tilde{A}}\left(x\right)\right)|x\in X\right\}$ (3)

其中 ${\mu }_{\tilde{A}}\left(x\right)、{\nu }_{\tilde{A}}\left(x\right)$ 分别表示元素 $x$ 属于 $X$ 的隶属度和非隶属度。用区间数表示为： ${\mu }_{\tilde{A}}\left(x\right)=\left[{\mu }_{\tilde{A}}^l\left(x\right),{\mu }_{\tilde{A}}^u\left(x\right)\right]\subseteq \left[0,1\right]$ ， ${\nu }_{\tilde{A}}\left(x\right)=\left[{\nu }_{\tilde{A}}^l\left(x\right),{\nu }_{\tilde{A}}^u\left(x\right)\right]\subseteq \left[0,1\right]$ ，其中 ${\mu }_{\tilde{A}}^l\left(x\right)=\inf {\mu }_{\tilde{A}}\left(x\right)$ ， ${\mu }_{\tilde{A}}^u\left(x\right)=\sup {\mu }_{\tilde{A}}\left(x\right)$ ， ${\nu }_{\tilde{A}}^l\left(x\right)=\inf {\nu }_{\tilde{A}}\left(x\right)$ ， ${\nu }_{\tilde{A}}^u\left(x\right)=\sup {\nu }_{\tilde{A}}\left(x\right)$ ，且满足 ${\mu }_{\tilde{A}}^u\left(x\right)+{\nu }_{\tilde{A}}^u\left(x\right)\le 1$ ，元素 $x$ 属于 $X$ 的犹豫度可表示为： ${\pi }_{\tilde{A}}\left(x\right)=\left[{\pi }_{\tilde{A}}^l\left(x\right),{\pi }_{\tilde{A}}^u\left(x\right)\right]$ 且满足 ${\pi }_{\tilde{A}}^l\left(x\right)=1-{\mu }_{\tilde{A}}^u\left(x\right)-{\nu }_{\tilde{A}}^u\left(x\right)$ ， ${\pi }_{\tilde{A}}^u\left(x\right)=1-{\mu }_{\tilde{A}}^l\left(x\right)-{\nu }_{\tilde{A}}^l\left(x\right)$ 。当 ${\mu }_{\tilde{A}}\left(x\right)={\mu }_{\tilde{A}}^l\left(x\right)={\mu }_{\tilde{A}}^u\left(x\right)$ ， ${\nu }_{\tilde{A}}\left(x\right)={\nu }_{\tilde{A}}^l\left(x\right)={\nu }_{\tilde{A}}^u\left(x\right)$ 时，即退化为直觉模糊集。在文献 [9] 中，称下列为区间直觉模糊数：

$\tilde{\alpha }=\left(\left[{\mu }^l,{\mu }^u\right],\left[{\nu }^l,{\nu }^u\right]\right)$ (4)

其中 $\left[{\mu }^l,{\mu }^u\right],\left[{\nu }^l,{\nu }^u\right]\subseteq \left[0,1\right]$ 且 $0\le {\mu }^u\text{+}{\nu }^u\le 1$ ， ${\pi }^l=1-{\mu }^u-{\nu }^u$ ， ${\pi }^u=1-{\mu }^l-{\nu }^l$ 。

3.2. 区间直觉模糊集的运算规则

在直觉模糊集运算规则的基础上，进一步定义了以下关于区间直觉模糊集的运算规则 [9] ：

定义7 设 $\tilde{A}=\left\{\left(x_i,\left[{\mu }_{\tilde{A}}^l\left(x\right),{\mu }_{\tilde{A}}^u\left(x\right)\right],\left[{\nu }_{\tilde{A}}^l\left(x\right),{\nu }_{\tilde{A}}^u\left(x\right)\right]\right)|x_i\in X\right\}$ ， $\tilde{B}=\left\{\left(x_i,\left[{\mu }_{\tilde{B}}^l\left(x\right),{\mu }_{\tilde{B}}^u\left(x\right)\right],\left[{\nu }_{\tilde{B}}^l\left(x\right),{\nu }_{\tilde{B}}^u\left(x\right)\right]\right)|x_i\in X\right\}$ 为任意两个区间直觉模糊集，定义以下运算规则：

1) $\overline{\stackrel{˜}{A}}=\left\{\left(x_i,\left[{\nu }_{\tilde{A}}^l\left(x\right),{\nu }_{\tilde{A}}^u\left(x\right)\right],\left[{\mu }_{\tilde{A}}^l\left(x\right),{\mu }_{\tilde{A}}^u\left(x\right)\right]\right)|x_i\in X\right\}$ ；

2) $\tilde{A}+\tilde{B}=\left\{\left(\begin{array}{l}{x}_i,\left[{\mu }_{\tilde{A}}^l\left(x\right)+{\mu }_{\tilde{B}}^l\left(x\right)-{\mu }_{\tilde{A}}^l\left(x\right){\mu }_{\tilde{B}}^l\left(x\right),{\mu }_{\tilde{A}}^u\left(x\right)+{\mu }_{\tilde{B}}^u\left(x\right)-{\mu }_{\tilde{A}}^u\left(x\right){\mu }_{\tilde{B}}^u\left(x\right)\right],\\ \left[{\nu }_{\tilde{A}}^l\left(x\right){\nu }_{\tilde{B}}^l\left(x\right),{\nu }_{\tilde{A}}^u\left(x\right){\nu }_{\tilde{B}}^u\left(x\right)\right]\end{array}\right)|x_i\in X\right\}$ ；

3) $\tilde{A}\cdot \tilde{B}=\left\{\left(\begin{array}{l}{x}_i,\left[{\mu }_{\tilde{A}}^l\left(x\right){\mu }_{\tilde{B}}^l\left(x\right),{\mu }_{\tilde{A}}^u\left(x\right){\mu }_{\tilde{B}}^u\left(x\right)\right],\\ \left[{\nu }_{\tilde{A}}^l\left(x\right)+{\nu }_{\tilde{B}}^l\left(x\right)-{\nu }_{\tilde{A}}^l\left(x\right){\nu }_{\tilde{B}}^l\left(x\right),{\nu }_{\tilde{A}}^u\left(x\right)+{\nu }_{\tilde{B}}^u\left(x\right)-{\nu }_{\tilde{A}}^u\left(x\right){\nu }_{\tilde{B}}^u\left(x\right)\right]\end{array}\right)|x_i\in X\right\}$ ；

4) $\tilde{A}\cup \tilde{B}=\left\{\left(\begin{array}{l}{x}_i,\left[\max \left\{{\mu }_{\tilde{A}}^l\left(x\right),{\mu }_{\tilde{B}}^l\left(x\right)\right\},\max \left\{{\mu }_{\tilde{A}}^u\left(x\right),{\mu }_{\tilde{B}}^u\left(x\right)\right\}\right],\\ \left[\min \left\{{\nu }_{\tilde{A}}^l\left(x\right),{\nu }_{\tilde{B}}^l\left(x\right)\right\},\min \left\{{\nu }_{\tilde{A}}^u\left(x\right),{\nu }_{\tilde{B}}^u\left(x\right)\right\}\right]\end{array}\right)|x_i\in X\right\}$ ；

5) $\tilde{A}\cap \tilde{B}=\left\{\left(\begin{array}{l}{x}_i,\left[\min \left\{{\mu }_{\tilde{A}}^l\left(x\right),{\mu }_{\tilde{B}}^l\left(x\right)\right\},\min \left\{{\mu }_{\tilde{A}}^u\left(x\right),{\mu }_{\tilde{B}}^u\left(x\right)\right\}\right],\\ \left[\max \left\{{\nu }_{\tilde{A}}^l\left(x\right),{\nu }_{\tilde{B}}^l\left(x\right)\right\},\max \left\{{\nu }_{\tilde{A}}^u\left(x\right),{\nu }_{\tilde{B}}^u\left(x\right)\right\}\right]\end{array}\right)|x_i\in X\right\}$ ；

6) $\lambda \tilde{A}=\left\{\left(x_i,\left[1-{\left(1-{\mu }_{\tilde{A}}^l\left(x\right)\right)}^{\lambda },1-{\left(1-{\mu }_{\tilde{A}}^u\left(x\right)\right)}^{\lambda }\right],\left[{\left({\nu }_{\tilde{A}}^l\left(x\right)\right)}^{\lambda },{\left({\nu }_{\tilde{A}}^u\left(x\right)\right)}^{\lambda }\right]\right)|x_i\in X\right\},\lambda >0$ ；

7) ${\tilde{A}}^{\lambda }=\left\{\left(x_i,\left[{\left({\mu }_{\tilde{A}}^l\left(x\right)\right)}^{\lambda },{\left({\mu }_{\tilde{A}}^u\left(x\right)\right)}^{\lambda }\right],\left[1-{\left(1-{\nu }_{\tilde{A}}^l\left(x\right)\right)}^{\lambda },1-{\left(1-{\nu }_{\tilde{A}}^u\left(x\right)\right)}^{\lambda }\right]\right)|x_i\in X\right\},\lambda >0$ 。

类似上述运算规则，可以定义以下区间直觉模糊数的运算规则：

定义8 [9] 设 ${\tilde{\alpha }}_{\text{1}}=\left(\left[{\mu }_{\text{1}}^l,{\mu }_1^u\right],\left[{\nu }_{\text{1}}^l,{\nu }_1^u\right]\right)$ ， ${\tilde{\alpha }}_{\text{2}}=\left(\left[{\mu }_{\text{2}}^l,{\mu }_{\text{2}}^u\right],\left[{\nu }_{\text{2}}^l,{\nu }_{\text{2}}^u\right]\right)$ 为任意两个区间直觉模糊数，定义以下运算规则：

1) ${\overline{\stackrel{˜}{\alpha }}}_{\text{1}}=\left(\left[{\nu }_{\text{1}}^l,{\nu }_1^u\right],\left[{\mu }_{\text{1}}^l,{\mu }_1^u\right]\right)$ ；

2) ${\tilde{\alpha }}_{\text{1}}\oplus {\tilde{\alpha }}_{\text{2}}=\left(\left[{\mu }_{\text{1}}^l+{\mu }_2^l-{\mu }_{\text{1}}^l{\mu }_2^l,{\mu }_1^u\text{+}{\mu }_{\text{2}}^u-{\mu }_1^u{\mu }_2^u\right],\left[{\nu }_{\text{1}}^l{\nu }_2^l,{\nu }_1^u{\nu }_2^u\right]\right)$ ；

3) ${\tilde{\alpha }}_{\text{1}}\otimes {\tilde{\alpha }}_{\text{2}}=\left(\left[{\mu }_{\text{1}}^l{\mu }_2^l,{\mu }_1^u{\mu }_2^u\right]，\left[{\nu }_{\text{1}}^l+{\nu }_2^l-{\nu }_{\text{1}}^l{\nu }_2^l,{\nu }_1^u\text{+}{\nu }_{\text{2}}^u-{\nu }_1^u{\nu }_2^u\right]\right)$ ；

4) ${\tilde{\alpha }}_{\text{1}}\cup {\tilde{\alpha }}_{\text{2}}=\left(\left[\max \left\{{\mu }_{\text{1}}^l,{\mu }_2^l\right\},\max \left\{{\mu }_1^u,{\mu }_2^u\right\}\right]，\left[\min \left\{{\nu }_{\text{1}}^l,{\nu }_2^l\right\},\min \left\{{\nu }_1^u,{\nu }_{\text{2}}^u\right\}\right]\right)$ ；

5) ${\tilde{\alpha }}_{\text{1}}\cap {\tilde{\alpha }}_{\text{2}}=\left(\left[\min \left\{{\mu }_{\text{1}}^l,{\mu }_2^l\right\},\min \left\{{\mu }_1^u,{\mu }_2^u\right\}\right]，\left[\max \left\{{\nu }_{\text{1}}^l,{\nu }_2^l\right\},\max \left\{{\nu }_1^u,{\nu }_{\text{2}}^u\right\}\right]\right)$ ；

6) $\lambda {\tilde{\alpha }}_1=\left(\left[1-{\left(1-{\mu }_{\text{1}}^l\right)}^{\lambda },1-{\left(1-{\mu }_{\text{1}}^u\right)}^{\lambda }\right],\left[{\left({\nu }_{\text{1}}^l\right)}^{\lambda },{\left({\nu }_{\text{1}}^u\right)}^{\lambda }\right]\right),\lambda >0$ ；

7) ${\left({\tilde{\alpha }}_1\right)}^{\lambda }=\left(\left[{\left({\mu }_{\text{1}}^l\right)}^{\lambda },{\left({\mu }_1^u\right)}^{\lambda }\right],\left[1-{\left(1-{\nu }_{\text{1}}^l\right)}^{\lambda },1-{\left(1-{\nu }_{\text{1}}^u\right)}^{\lambda }\right]\right),\lambda >0$ 。

定理2 设 ${\tilde{\alpha }}_{\text{1}}=\left(\left[{\mu }_{\text{1}}^l,{\mu }_1^u\right],\left[{\nu }_{\text{1}}^l,{\nu }_1^u\right]\right)$ ， ${\tilde{\alpha }}_{\text{2}}=\left(\left[{\mu }_{\text{2}}^l,{\mu }_{\text{2}}^u\right],\left[{\nu }_{\text{2}}^l,{\nu }_{\text{2}}^u\right]\right)$ 为任意两个区间直觉模糊数，设 ${\tilde{\beta }}_1={\overline{\stackrel{˜}{\alpha }}}_1$ ， ${\tilde{\beta }}_2={\tilde{\alpha }}_1\oplus {\tilde{\alpha }}_2$ ， ${\tilde{\beta }}_3={\tilde{\alpha }}_1\otimes {\tilde{\alpha }}_2$ ， ${\tilde{\beta }}_4=\lambda {\tilde{\alpha }}_1$ ， ${\tilde{\beta }}_5={\left({\tilde{\alpha }}_1\right)}^{\lambda }$ ，则 ${\tilde{\alpha }}_i\left(i=1,2,3,4,5\right)$ 均为区间直觉模糊数。

证明：因为 ${\tilde{\alpha }}_{\text{1}}=\left(\left[{\mu }_{\text{1}}^l,{\mu }_1^u\right],\left[{\nu }_{\text{1}}^l,{\nu }_1^u\right]\right)$ ， ${\tilde{\alpha }}_{\text{2}}=\left(\left[{\mu }_{\text{2}}^l,{\mu }_{\text{2}}^u\right],\left[{\nu }_{\text{2}}^l,{\nu }_{\text{2}}^u\right]\right)$ 为区间直觉模糊数，则有 $\left[{\mu }_{\text{1}}^l,{\mu }_1^u\right],\left[{\nu }_{\text{1}}^l,{\nu }_1^u\right],\left[{\mu }_{\text{2}}^l,{\mu }_{\text{2}}^u\right],\left[{\nu }_{\text{2}}^l,{\nu }_{\text{2}}^u\right]\subseteq \left[0,1\right]$ ，且 ${\mu }_1^u+{\nu }_1^u\le 1,{\mu }_2^u+{\nu }_2^u\le 1$ 。

根据上述定义8中运算规则(1) ${\tilde{\beta }}_1\text{=}{\overline{\stackrel{˜}{\alpha }}}_{\text{1}}=\left(\left[{\nu }_{\text{1}}^l,{\nu }_1^u\right],\left[{\mu }_{\text{1}}^l,{\mu }_1^u\right]\right)$ ，显然 ${\tilde{\beta }}_1$ 为区间直觉模糊数。

根据上述定义5中运算规则(2)可知：

${\mu }_{{\tilde{\beta }}_2}^l={\mu }_{\text{1}}^l+\left(1-{\mu }_{\text{1}}^l\right){\mu }_2^l\ge 0$ , $0\le {\nu }_{{\tilde{\beta }}_2}^l={\nu }_1^l{\nu }_2^l\le {\nu }_{{\tilde{\beta }}_2}^u={\nu }_1^u{\nu }_2^u\le 1$ ,

${\mu }_{{\tilde{\beta }}_2}^u={\mu }_1^u\text{+}{\mu }_{\text{2}}^u\left(1-{\mu }_1^u\right)\ge {\mu }_1^u\text{+}{\mu }_{\text{2}}^l\left(1-{\mu }_1^u\right)={\mu }_1^u\left(1-{\mu }_{\text{2}}^l\right)\text{+}{\mu }_{\text{2}}^l\ge {\mu }_1^l\left(1-{\mu }_{\text{2}}^l\right)\text{+}{\mu }_{\text{2}}^l\text{=}{\mu }_{{\tilde{\beta }}_2}^l\ge 0$ ,

$\begin{array}{c}{\mu }_{{\tilde{\beta }}_2}^u+{\nu }_{{\tilde{\beta }}_2}^u={\mu }_1^u+{\mu }_2^u-{\mu }_1^u{\mu }_2^u+{\nu }_1^u{\nu }_2^u={\mu }_1^u+{\mu }_2^u\left(1-{\mu }_1^u\right)+{\nu }_1^u{\nu }_2^u\le {\mu }_1^u+\left(1-{\nu }_2^u\right)\left(1-{\mu }_1^u\right)+{\nu }_1^u{\nu }_2^u\\ =1-{\nu }_2^u+{\mu }_1^u{\nu }_2^u+{\nu }_1^u{\nu }_2^u=1-{\nu }_2^u+\left({\mu }_1^u+{\nu }_1^u\right){\nu }_2^u\le 1\end{array}$

所以 ${\tilde{\beta }}_2$ 为区间直觉模糊数。

根据上述定义5中运算规则(3)可知：

${\mu }_{{\tilde{\beta }}_3}^l={\mu }_{\text{1}}^l{\mu }_2^l\ge 0$ , ${\mu }_{\text{1}}^l{\mu }_2^l\le {\mu }_{{\tilde{\beta }}_3}^u={\mu }_1^u{\mu }_2^u\le 1$ ,

${\nu }_{{\tilde{\beta }}_3}^u={\nu }_1^u\text{+}\left(1-{\nu }_1^u\right){\nu }_2^u\ge {\nu }_1^u\text{+}\left(1-{\nu }_1^u\right){\nu }_2^l={\nu }_1^u\left(1-{\nu }_2^l\right)\text{+}{\nu }_2^l\text{+}{\mu }_{\text{2}}^l\ge {\nu }_1^l\left(1-{\nu }_2^l\right)\text{+}{\nu }_2^l={\nu }_{{\tilde{\beta }}_3}^l\ge 0$

$\begin{array}{c}{\mu }_{{\tilde{\beta }}_3}^u+{\nu }_{{\tilde{\beta }}_3}^u={\mu }_1^u{\mu }_2^u+{\nu }_1^u+{\nu }_2^u-{\nu }_1^u{\nu }_2^u={\mu }_1^u{\mu }_2^u+{\nu }_1^u+\left(1-{\nu }_1^u\right){\nu }_2^u\le {\mu }_1^u{\mu }_2^u+{\nu }_1^u+\left(1-{\nu }_1^u\right)\left(1-{\mu }_2^u\right)\\ =1-{\mu }_2^u+{\mu }_1^u{\mu }_2^u+{\nu }_1^u{\mu }_2^u=1-{\mu }_2^u+\left({\mu }_1^u+{\nu }_1^u\right){\mu }_2^u\le 1\end{array}$

所以 ${\tilde{\beta }}_3$ 为区间直觉模糊数。

根据上述定义5中运算规则(6)可知：

$0\le {\mu }_{{\tilde{\beta }}_4}^l=1-{\left(1-{\mu }^l\right)}^{\lambda }\le {\mu }_{{\tilde{\beta }}_4}^u=1-{\left(1-{\mu }^u\right)}^{\lambda }\le 1$ , $0\le {\nu }_{{\tilde{\beta }}_4}^l={\left({\nu }^l\right)}^{\lambda }\le {\nu }_{{\tilde{\beta }}_4}^u={\left({\nu }^u\right)}^{\lambda }\le 1$

${\mu }_{{\tilde{\beta }}_4}^u+{\nu }_{{\tilde{\beta }}_4}^u=1-{\left(1-{\mu }^u\right)}^{\lambda }+{\left({\nu }^u\right)}^{\lambda }\le 1-{\left(1-{\mu }^u\right)}^{\lambda }+{\left(1-{\mu }^u\right)}^{\lambda }=1$

所以 ${\tilde{\beta }}_4$ 为区间直觉模糊数。

根据上述定义5中运算规则(6)可知：

$0\le {\mu }_{{\tilde{\beta }}_5}^l={\left({\mu }_{\text{1}}^l\right)}^{\lambda }\le {\mu }_{{\tilde{\beta }}_5}^u={\left({\mu }_1^u\right)}^{\lambda }\le 1$ , $0\le {\nu }_{{\tilde{\beta }}_5}^l=1-{\left(1-{\nu }_1^l\right)}^{\lambda }\le {\nu }_{{\tilde{\beta }}_5}^u=1-{\left(1-{\nu }_1^u\right)}^{\lambda }\le 1$

${\mu }_{{\tilde{\beta }}_5}^u+{\nu }_{{\tilde{\beta }}_5}^u={\left({\mu }_1^u\right)}^{\lambda }+1-{\left(1-{\nu }_1^u\right)}^{\lambda }\le {\left(1-{\nu }_1^u\right)}^{\lambda }+1-{\left(1-{\nu }_1^u\right)}^{\lambda }=1$

${\tilde{\beta }}_5$ 所以为区间直觉模糊数。

3.3. 区间直觉模糊集的减法和除法运算规则定义

类比于上述定义4中直觉模糊数的减法和除法的定义给出区间直觉模糊集的减法和除法的定义。

定义9 设 $\tilde{A}=\left\{\left(x_i,\left[{\mu }_{\tilde{A}}^l\left(x\right),{\mu }_{\tilde{A}}^u\left(x\right)\right],\left[{\nu }_{\tilde{A}}^l\left(x\right),{\nu }_{\tilde{A}}^u\left(x\right)\right]\right)|x_i\in X\right\}$ ， $\tilde{B}=\left\{\left(x_i,\left[{\mu }_{\tilde{B}}^l\left(x\right),{\mu }_{\tilde{B}}^u\left(x\right)\right],\left[{\nu }_{\tilde{B}}^l\left(x\right),{\nu }_{\tilde{B}}^u\left(x\right)\right]\right)|x_i\in X\right\}$ 为任意两个区间直觉模糊集，定义以下减法和除法运算规则：

1) $\tilde{A}-\tilde{B}=\tilde{A}+\overline{\stackrel{˜}{B}}=\left\{\left(\begin{array}{l}{x}_i,\left[{\mu }_{\tilde{A}}^l\left(x\right)+{\nu }_{\tilde{B}}^l\left(x\right)-{\mu }_{\tilde{A}}^l\left(x\right){\nu }_{\tilde{B}}^l\left(x\right),{\mu }_{\tilde{A}}^u\left(x\right)+{\nu }_{\tilde{B}}^l\left(x\right)-{\mu }_{\tilde{A}}^u\left(x\right){\nu }_{\tilde{B}}^l\left(x\right)\right],\\ \left[{\nu }_{\tilde{A}}^l\left(x\right){\mu }_{\tilde{B}}^l\left(x\right),{\nu }_{\tilde{A}}^u\left(x\right){\mu }_{\tilde{B}}^u\left(x\right)\right]\end{array}\right)|x_i\in X\right\}$ ；

2) $\tilde{A}÷\tilde{B}=\tilde{A}\cdot \overline{\stackrel{˜}{B}}=\left\{\left(\begin{array}{l}{x}_i,\left[{\mu }_{\tilde{A}}^l\left(x\right){\nu }_{\tilde{B}}^l\left(x\right),{\mu }_{\tilde{A}}^u\left(x\right){\nu }_{\tilde{B}}^u\left(x\right)\right],\\ \left[{\nu }_{\tilde{A}}^l\left(x\right)+{\mu }_{\tilde{B}}^l\left(x\right)-{\nu }_{\tilde{A}}^l\left(x\right){\mu }_{\tilde{B}}^l\left(x\right),{\nu }_{\tilde{A}}^u\left(x\right)+{\mu }_{\tilde{B}}^u\left(x\right)-{\nu }_{\tilde{A}}^u\left(x\right){\mu }_{\tilde{B}}^u\left(x\right)\right]\end{array}\right)|x_i\in X\right\}$ 。

类似上述运算规则，定义区间直觉模糊数相反数、减法和除法运算规则：

定义10 设 ${\tilde{\alpha }}_{\text{1}}=\left(\left[{\mu }_{\text{1}}^l,{\mu }_1^u\right],\left[{\nu }_{\text{1}}^l,{\nu }_1^u\right]\right)$ ， ${\tilde{\alpha }}_{\text{2}}=\left(\left[{\mu }_{\text{2}}^l,{\mu }_{\text{2}}^u\right],\left[{\nu }_{\text{2}}^l,{\nu }_{\text{2}}^u\right]\right)$ 为任意两个区间直觉模糊数，定义以下运算规则：

1) 定义 $-{\tilde{\alpha }}_1$ 称为 ${\tilde{\alpha }}_{\text{1}}$ 的“相反数”， $-{\tilde{\alpha }}_1={\overline{\stackrel{˜}{\alpha }}}_1=\left(\left[{\nu }_{\text{1}}^l,{\nu }_1^u\right],\left[{\mu }_{\text{1}}^l,{\mu }_1^u\right]\right)$ ；

2) ${\tilde{\alpha }}_{\text{1}}-{\tilde{\alpha }}_{\text{2}}={\tilde{\alpha }}_{\text{1}}\oplus {\overline{\stackrel{˜}{\alpha }}}_{\text{2}}=\left(\left[{\mu }_{\text{1}}^l+{\nu }_2^l-{\mu }_{\text{1}}^l{\nu }_2^l,{\mu }_1^u\text{+}{\nu }_2^u-{\mu }_1^u{\nu }_2^u\right],\left[{\nu }_{\text{1}}^l{\mu }_2^l,{\nu }_1^u{\mu }_2^u\right]\right)$ ；

3) ${\tilde{\alpha }}_{\text{1}}÷{\tilde{\alpha }}_{\text{2}}={\tilde{\alpha }}_{\text{1}}\otimes {\overline{\stackrel{˜}{\alpha }}}_{\text{2}}\text{=}\left(\left[{\mu }_{\text{1}}^l{\nu }_2^l,{\mu }_1^u{\nu }_2^u\right]，\left[{\nu }_{\text{1}}^l+{\mu }_2^l-{\nu }_{\text{1}}^l{\mu }_2^l,{\nu }_1^u\text{+}{\mu }_2^u-{\nu }_1^u{\mu }_2^u\right]\right)$ 。

定理3 设 ${\tilde{\alpha }}_{\text{1}}=\left(\left[{\mu }_{\text{1}}^l,{\mu }_1^u\right],\left[{\nu }_{\text{1}}^l,{\nu }_1^u\right]\right)$ ， ${\tilde{\alpha }}_{\text{2}}=\left(\left[{\mu }_{\text{2}}^l,{\mu }_{\text{2}}^u\right],\left[{\nu }_{\text{2}}^l,{\nu }_{\text{2}}^u\right]\right)$ 为任意两个区间直觉模糊数，设 ${\tilde{\beta }}_6=-{\tilde{\alpha }}_1$ ， ${\tilde{\beta }}_7={\tilde{\alpha }}_1-{\tilde{\alpha }}_2$ ， ${\tilde{\beta }}_8={\tilde{\alpha }}_1÷{\tilde{\alpha }}_2$ ，则 ${\tilde{\alpha }}_i\left(i=6,7,8\right)$ 均为区间直觉模糊数。

证明：因为 ${\tilde{\alpha }}_{\text{1}}=\left(\left[{\mu }_{\text{1}}^l,{\mu }_1^u\right],\left[{\nu }_{\text{1}}^l,{\nu }_1^u\right]\right)$ ， ${\tilde{\alpha }}_{\text{2}}=\left(\left[{\mu }_{\text{2}}^l,{\mu }_{\text{2}}^u\right],\left[{\nu }_{\text{2}}^l,{\nu }_{\text{2}}^u\right]\right)$ 为区间直觉模糊数，则有 $\left[{\mu }_{\text{1}}^l,{\mu }_1^u\right],\left[{\nu }_{\text{1}}^l,{\nu }_1^u\right],\left[{\mu }_{\text{2}}^l,{\mu }_{\text{2}}^u\right],\left[{\nu }_{\text{2}}^l,{\nu }_{\text{2}}^u\right]\subseteq \left[0,1\right]$ ，且 ${\mu }_1^u+{\nu }_1^u\le 1,{\mu }_2^u+{\nu }_2^u\le 1$ 。

根据上述定义10中运算规则(1) ${\tilde{\beta }}_6\text{=}-{\tilde{\alpha }}_1={\overline{\stackrel{˜}{\alpha }}}_1=\left(\left[{\nu }_{\text{1}}^l,{\nu }_1^u\right],\left[{\mu }_{\text{1}}^l,{\mu }_1^u\right]\right)$ ，显然 ${\tilde{\beta }}_6$ 为区间直觉模糊数。

根据上述定义10中运算规则(2)可知 ${\tilde{\alpha }}_{\text{1}}-{\tilde{\alpha }}_{\text{2}}={\tilde{\alpha }}_{\text{1}}\oplus {\overline{\stackrel{˜}{\alpha }}}_{\text{2}}$ ，由定理1中的结论 ${\overline{\stackrel{˜}{\alpha }}}_{\text{2}}$ 仍为区间直觉模糊数，再由定理1中两个区间直觉模糊数的和 ${\tilde{\alpha }}_{\text{1}}\oplus {\overline{\stackrel{˜}{\alpha }}}_{\text{2}}$ 仍为区间直觉模糊数，所以 ${\tilde{\beta }}_7$ 为区间直觉模糊数。

根据上述定义10中运算规则(3)可知 ${\tilde{\alpha }}_{\text{1}}÷{\tilde{\alpha }}_{\text{2}}={\tilde{\alpha }}_{\text{1}}\otimes {\overline{\stackrel{˜}{\alpha }}}_{\text{2}}$ ，由定理1中的结论 ${\overline{\stackrel{˜}{\alpha }}}_{\text{2}}$ 仍为区间直觉模糊数，再由定理1中两个区间直觉模糊数的乘积 ${\tilde{\alpha }}_{\text{1}}\otimes {\overline{\stackrel{˜}{\alpha }}}_{\text{2}}$ 仍为区间直觉模糊数，所以 ${\tilde{\beta }}_8$ 为区间直觉模糊数。

4. 实例

例1设 $\tilde{\alpha }=\left(0.8,0.1\right)$ ， $\tilde{\beta }=\left(0.5,0.3\right)$ 为两个区间直觉模糊数，请分别计算 $\overline{\stackrel{˜}{\alpha }}$ ， $\tilde{\alpha }\oplus \tilde{\beta }$ ， $\tilde{\alpha }\otimes \tilde{\beta }$ ， $\tilde{\alpha }\cup \tilde{\beta }$ ， $\tilde{\alpha }\cap \tilde{\beta }$ ， $2\tilde{\alpha }$ ， ${\tilde{\alpha }}^2$ ， $\tilde{\alpha }-\tilde{\beta }$ ， $\tilde{\alpha }÷\tilde{\beta }$ 。

解： $\overline{\stackrel{˜}{\alpha }}=\left(0.1,0.8\right)$

$\tilde{\alpha }\oplus \tilde{\beta }=\left(0.8+0.5-0.8\times 0.5,0.1\times 0.3\right)=\left(0.9,0.03\right)$ ，

$\tilde{\alpha }\otimes \tilde{\beta }=\left(0.8\times 0.5，0.1+0.3-0.1\times 0.3\right)=\left(0.4,0.37\right)$ ，

$\tilde{\alpha }\cup \tilde{\beta }=\left(\max \left\{0.8,0.5\right\},\min \left\{0.1,0.3\right\}\right)=\left(0.8,0.1\right)$ ，

$\tilde{\alpha }\cap \tilde{\beta }=\left(\min \left\{0.8,0.5\right\},\max \left\{0.1,0.3\right\}\right)=\left(0.5,0.3\right)$ ，

$2\tilde{\alpha }=\left(1-{\left(1-0.8\right)}^2,{\left(0.1\right)}^2\right)=\left(0.96,0.01\right)$ ，

${\tilde{\alpha }}^2=\left({\left(0.8\right)}^2,1-{\left(1-0.1\right)}^2\right)=\left(0.64,0.19\right)$ ，

$\tilde{\alpha }-\tilde{\beta }=\left(0.8+0.3-0.8\times 0.3,0.1\times 0.5\right)=\left(0.86,0.05\right)$ ，

$\tilde{\alpha }÷\tilde{\beta }=\left(0.8\times 0.3,0.1+0.5-0.1\times 0.5\right)\left(0.24,0.55\right)$ 。

例2 设 $\tilde{\alpha }=\left(\left[0.6,0.8\right],\left[0,0.1\right]\right)$ ， $\tilde{\beta }=\left(\left[0.4,0.5\right],\left[0.2,0.4\right]\right)$ 为两个区间直觉模糊数，请分别计算 $\overline{\stackrel{˜}{\alpha }}$ ， $\tilde{\alpha }\oplus \tilde{\beta }$ ， $\tilde{\alpha }\otimes \tilde{\beta }$ ， $\tilde{\alpha }\cup \tilde{\beta }$ ， $\tilde{\alpha }\cap \tilde{\beta }$ ， $2\tilde{\alpha }$ ， ${\tilde{\alpha }}^2$ ， $\tilde{\alpha }-\tilde{\beta }$ ， $\tilde{\alpha }÷\tilde{\beta }$ 。

解： $\overline{\stackrel{˜}{\alpha }}=\left(\left[0,0.1\right],\left[0.6,0.8\right]\right)$

$\tilde{\alpha }\oplus \tilde{\beta }=\left(\left[0.6+0.4-0.6\times 0.4,0.8+0.5-0.8\times 0.5\right],\left[0\times 0.2,0.1\times 0.4\right]\right)=\left(\left[0.76,0.9\right],\left[0,0.04\right]\right)$ ，

$\tilde{\alpha }\otimes \tilde{\beta }=\left(\left[0.6\times 0.4,0.8\times 0.5\right],\left[0+0.2-0\times 0.2,0.1+0.4-0.1\times 0.4\right]\right)=\left(\left[0.24,0.4\right],\left[0.2,0.46\right]\right)$ ，

$\tilde{\alpha }\cup \tilde{\beta }=\left(\left[\max \left\{0.6,0.4\right\},\max \left\{0.8,0.5\right\}\right],\left[\min \left\{0,0.2\right\},\min \left\{0.1,0.4\right\}\right]\right)\text{=}\left(\left[0.6,0.8\right],\left[0,\text{0}\text{.1}\right]\right)$ ，

$\tilde{\alpha }\cap \tilde{\beta }=\left(\left[\min \left\{0.6,0.4\right\},\min \left\{0.8,0.5\right\}\right],\left[\max \left\{0,0.2\right\},\max \left\{0.1,0.4\right\}\right]\right)=\left(\left[0.4,0.5\right],\left[0.2,0.4\right]\right)$ ，

$2\tilde{\alpha }=\left(\left[1-{\left(1-0.6\right)}^2,1-{\left(1-0.8\right)}^2\right],\left[0^2,{0.1}^2\right]\right)\text{=}\left(\left[0.84,0.94\right],\left[0,0.01\right]\right)$ ，

${\tilde{\alpha }}^2=\left(\left[{0.6}^2,{0.8}^2\right],\left[1-{\left(1-0\right)}^2,1-{\left(1-0.1\right)}^2\right]\right)\text{=}\left(\left[0.36,0.64\right],\left[0,0.19\right]\right)$ ，

$\tilde{\alpha }-\tilde{\beta }=\left(\left[0.6+0.2-0.6\times 0.2,0.8+0.4-0.8\times 0.4\right],\left[0\times 0.4,0.1\times 0.5\right]\right)=\left(\left[0.68,0.88\right],\left[0,0.05\right]\right)$ ，

$\tilde{\alpha }÷\tilde{\beta }=\left(\left[0.6\times 0.2,0.8\times 0.4\right],\left[0+0.4-0\times 0.4,0.1+0.5-0.1\times 0.5\right]\right)=\left(\left[0.12,0.32\right],\left[0.4,0.55\right]\right)$ 。

5. 总结

文章对直觉模糊集和区间直觉模糊集的基本理论和基本运算进行了研究和分析。首先，在已有研究理论的基础上，新定义了直觉模糊集和直觉模糊数的减法和除法运算规则，并证明了基于直觉模糊数的减法和除法是封闭的。其次，将直觉模糊集的基本运算规则进一步推广到区间直觉模糊集和区间直觉模糊数，并给出了区间直觉模糊集和区间直觉模糊数的减法和除法运算，且进一步证明了区间直觉模糊数基于和、差、积、商等一些运算规则的封闭性。最后，通过两个具体计算例子对直觉模糊数和区间直觉模糊数的基本运算规则进行了演示。本文的研究结果对丰富直觉模糊集和区间直觉模糊集的基本理论具有重要的意义，进一步丰富了其理论基础，为直觉模糊集和区间直觉模糊集的理论发展提供了理论支撑。

当然本文的研究也还存在许多需要完善的地方，比如所提出的运算规则在实际应用中是否合理还有待进一步验证。因此下一步的研究工作将利用本文的研究结果应用到实际问题中，以验证其合理性和准确性，并丰富其应用研究。