提出新型模糊数运算方法，解决传统模糊运算复杂且结果失真问题。
Fuzzy numbers revisited: operations on extensional fuzzy numbers
- 采用扩展性模糊数定义新运算规则，提升计算效率
- 运算结果保持原有模糊特征，避免模糊度递增
- 适用于需要稳定模糊推理的工程与决策场景
模糊数常以模糊集形式表示，用于更好刻画不确定数据。但其运算不如精确数直接：通常使用扎德扩展法则，导致计算复杂度高，且某些操作（如两个三角模糊集相乘）结果不再是同类型模糊集。此外，连续运算会使结果模糊度不断增大，严重限制应用范围。本文重新审视该问题，提出基于扩展性模糊数的新运算方法，定义了其上的等价、大于等关系运算。通过多个应用实例验证有效性，相关C++代码已开源至公共GitHub仓库。
原文摘要 · Abstract (English)
Fuzzy numbers are commonly represented with fuzzy sets. Their objective is to better represent imprecise data. However, operations on fuzzy numbers are not as straightforward as maths on crisp numbers. Commonly, the Zadeh's extension rule is applied to elaborate a result. This can produce two problems: (1) high computational complexity and (2) for some fuzzy sets and some operations the results is not a fuzzy set with the same features (eg. multiplication of two triangular fuzzy sets does not produce a triangular fuzzy set). One more problem is the fuzzy spread -- fuzziness of the result increases with the number of operations. These facts can severely limit the application field of fuzzy numbers. In this paper we would like to revisite this problem with a different kind of fuzzy numbers -- extensional fuzzy numbers. The paper defines operations on extensional fuzzy numbers and relational operators (=, >, >=, <, <=) for them. The proposed approach is illustrated with several applicational examples. The C++ implementation is available from a public GitHub repository.
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