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
DOI: 10.12677/ORF.2023.136761, PDF,   
作者: 马 虹：广东金融学院金融数学与统计学院，广东 广州；谢小军：广州工商学院通识教育学院，广东 广州
关键词: 模糊数学；直觉模糊集；区间直觉模糊集；运算规则；Fuzzy Mathematics； Intuitionistic Fuzzy Sets； Interval Intuitionistic Fuzzy Sets； Arithmetic Rules
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.
文章引用：马虹, 谢小军. 直觉模糊集的减法和除法运算的新定义[J]. 运筹与模糊学, 2023, 13(6): 7789-7796. https://doi.org/10.12677/ORF.2023.136761

参考文献

[1] 莫智文, 舒兰, 许彪. 模糊数学理论及其应用评述[J]. 四川师范大学学报(自然科学版), 1998, 21(3): 330-335.
[2] 黄军辉, 傅沈文. 模糊控制理论的发展及应用[J]. 中国科技信息, 2006(12): 303-304, 306.
[3] 张利亭, 冯涛, 李欢. 不完备信息系统的直觉模糊决策粗糙集[J]. 郑州大学学报(理学版), 2021, 53(2): 57-65.
[4] 柯强, 冯康宏, 曹心德, 等. 基于模糊数学的场地重金属污染风险综合评价[J]. 中国环境科学, 2023, 43(1): 415-423.
[5] 陈静. 模糊数学方法在旅游吸引力评价中的应用[J]. 浙江大学学报(理学版), 2021, 48(1): 118-123.
[6] 张幼振, 张宁, 邵俊杰, 等. 基于钻进参数聚类的含煤地层岩性模糊识别[J]. 煤炭学报, 2019, 44(8): 2328-2335.
[7] 刘芹, 孔维丽, 崔筱, 等. 基于模糊数学和聚类分析的香菇品种综合品质评价[J]. 中国瓜菜, 2021, 34(11): 37-46.
[8] Atanassov, K. (1986) Intuitionistic Fuzzy Sets. Fuzzy Sets and Systems, 20, 87-96. [Google Scholar] [CrossRef] 
[9] Atanassov, K. and Gargov, G. (1989) Interval Valued Intuitionistic Fuzzy Set. Fuzzy Sets System, 31, 343-349. [Google Scholar] [CrossRef] 
[10] 李慧敏, 吕乐琳, 汪伦焰, 等. 基于区间直觉模糊集的水环境治理PPP项目可持续性评价[J]. 节水灌溉, 2019(3): 59-66, 70.
[11] 陈璇, 孙开畅, 马文俊, 等. 基于区间灰色直觉模糊语言集的水利工程应急能力评价[J]. 水力发电, 2019, 45(5): 103-107.
[12] 马文俊, 孙开畅, 颜鑫, 等. 水利工程应急方案直觉模糊多属性决策研究[J]. 水力发电, 2019, 45(2): 79-82.
[13] 周清华, 王琦. 基于直觉模糊集和VIKOR的低碳供应商选择方法研究[J]. 科技管理研究, 2017, 37(10): 233-237.
[14] 商克俭, 冯东梅, 李乃文. 基于直觉模糊集的公私合营项目方案选择[J]. 统计与决策, 2019, 35(3): 178-181.
[15] 田永超, 于丽英. 基于区间值直觉模糊集的技术创新风险评价方法[J]. 上海大学学报(自然科学版), 2020, 26(2): 290-300.
[16] 李沃源, 乔剑敏. 基于区间直觉模糊集的双边匹配决策模型及应用[J]. 模糊系统与数学, 2021, 35(3): 71-77.
[17] 杨敏, 张仕斌, 张航. 基于新模糊熵和得分函数的区间直觉模糊多属性决策[J]. 运筹与管理, 2021, 30(6): 63-68.

为你推荐