摘要：Today, I will bring you the 1.3 An overview of fuzzy numbers of ‘Research on fuzzy multi-objective decision-making theory, method
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“平学（63）：精读博士论文《模糊多目标决策理论、方法及其应用研究 》1.3 模糊数综述”
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Today, the editor brings you an article
"Pingxue(63):Intensive reading of the doctoral dissertation ‘Research on fuzzy multi-objective decision-making theory, method and its application’ 1.3 An overview of fuzzy numbers"
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一、内容摘要（Summary of content）
今天小编将从思维导图、精读内容、知识补充三个板块为大家带来博士论文《模糊多目标决策理论、方法及其应用研究》中的1.3 模糊数综述。
Today, I will bring you the 1.3 An overview of fuzzy numbers of ‘Research on fuzzy multi-objective decision-making theory, method and its application’ from the three sections of Thinking Maps, Intensive Reading Content, and Knowledge Supplement.
二、思维导图（Mind mapping）
三、精读内容（Intensive reading content）
1. 隶属函数的基本概念（Basic Concepts of Affiliation functions）
隶属函数是用于表征模糊集合的数学工具。它通过在区间中取值来描述元素属于某个模糊集合的“真实程度”。隶属函数的值越接近1，表示元素属于该模糊集合的程度越高。值越接近0，表示元素属于该集合的程度越低。具体如下图所示：
An affiliation function is a mathematical tool used to characterize a fuzzy set. It describes the “true degree” to which an element belongs to a fuzzy set by taking values in an interval. The closer the value of the affiliation function is to 1, the higher the degree to which the element belongs to the fuzzy set. The closer the value is to 0, the lower the degree to which the element belongs to the set. This is shown in the following figure:
2. 特殊模糊数的代数运算规则（Algebraic rules for special fuzzy numbers）
这篇文章给出了模糊数的一些运算规则，并且用一张表来进行了汇总，具体如下图所示：
This article gives some arithmetic rules for fuzzy numbers and summarizes them in a table as shown below:
3. 模糊距离（Blurring distance）
为度量两个模糊数之间的差异，可将泛函分析中的距离概念拓展到模糊集合上模糊数之间的距离概念。距离的计算可以使用汉明距离、欧式距离、切比雪夫距离和闵可夫斯基距离公式。这些距离的计算公式和使用情况如下图所示：
To measure the difference between two fuzzy numbers, the concept of distance in generalized function analysis can be extended to the concept of distance between fuzzy numbers on a fuzzy set. Distances can be calculated using the Hamming distance, Euclidean distance, Chebyshev distance and Minkowski distance formulas. The formulas and use of these distances are shown below:
4. 决策问题中应用模糊数的意义（The significance of applying fuzzy numbers in decision making problems）
模糊性普遍存在，尤其是在定性目标的描述上。模糊数作为一种实数域上的模糊子集，能够有效描述这种模糊性。模糊数的运算具有自身特点，依赖于特定的模糊算子，反映了决策者的偏好和风险意识。将模糊数去模糊化虽然简化了决策过程，但可能会忽略决策的模糊性。模糊数的比较和距离测度与精确数有很大差异，需要根据实际情况选择合适的方法。合理的方法是具体问题具体分析，避免在非模糊情况下强行使用模糊决策。
Ambiguity is prevalent, especially in the description of qualitative goals. Fuzzy numbers, as a fuzzy subset on the domain of real numbers, can effectively describe this ambiguity. The operation of fuzzy numbers has its own characteristics and relies on specific fuzzy operators that reflect the decision maker's preferences and risk awareness. Defuzzification of fuzzy numbers, while simplifying the decision-making process, may ignore the ambiguity of the decision. Comparison and distance measures of fuzzy numbers are very different from exact numbers, and need to choose the appropriate method according to the actual situation. A reasonable method is to analyze specific problems and avoid forcing fuzzy decisions in non-fuzzy situations.
四、知识补充（Knowledge supplementation）
我们之前听说过欧式距离、切比雪夫距离，那么汉明距离是什么呢？
We've heard of the Euclidean distance and the Chebyshev distance before, so what is the Hamming distance?
汉明距离是一种衡量两个等长字符串之间差异的度量方式。它表示两个相同长度的字符串在相同位置上不同字符的数量。具体来说，汉明距离是通过将两个字符串进行，并统计结果为1的个数来计算的。这个数就是汉明距离。
Hamming distance is a metric that measures the difference between two equal length strings. It represents the number of different characters in the same position of two strings of the same length. Specifically, the Hamming distance is calculated by taking two strings and counting the number of times the result is 1. This number is the Hamming distance.
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参考资料：Google翻译，百度，通义千问
参考文献：黄宪成. 模糊多目标决策理论、方法及其应用研究 [D]. 大连：大连理工大学, 2003.
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来源：LearningYard学苑