毕达哥拉斯模糊环境下海明距离测度的证明及推广  

The proof and generalization of Hamming distance measure in the Pythagorean fuzzy environment

作　　者：李丹 王贵君[2] LI Dan;WANG Guijun(School of Science and Technology,University of Sanya,Sanya 572000,Hainan Province,China;School of Mathematical Science,Tianjin Normal University,Tianjin 300387,China)

机构地区：[1]三亚学院理工学院,海南三亚572000 [2]天津师范大学数学科学学院,天津300387

出　　处：《浙江大学学报（理学版）》2023年第4期402-408,共7页Journal of Zhejiang University（Science Edition）

基　　金：国家自然科学基金资助项目(61374009)。

摘　　要：距离测度是毕达哥拉斯(Pythagorean)模糊决策的一个基本概念,其在用权重向量进行有序集成运算、有序加权距离和相似度构造时具有重要作用。然而,早期发表于International Journal of Intelligent Systems中的文章Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets,在首次引入海明距离测度时,对距离公理化条件的证明存在不妥。为此,通过反例和分析指出了其在证明公理化条件的有界性和三点不等式时存在的错误,并统筹考虑犹豫度的绝对值项和其他绝对值项,通过分情况讨论给出了海明距离测度的严格证明,进一步,证明了其推广形式(欧几里得距离测度)的公理化条件。Distance measure is a basic concept in Pythagorean fuzzy decision making,it plays an important role in the ordered aggregation operation,ordered weighted distance evaluation and construction of similarity through a weight vector.However,in the early article Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets published in the famous journal International Journal of Intelligent Systems,which is the first paper to introduce the Hamming distance measure,and many authors cited the concept of distance measure later,it is found that the method of proving the axiomatization conditions of distance is inappropriate.We demonstrate counterexamples to show that there are two serious errors in proving the boundness and three-point inequality in the axiomatization conditions.Alternatively,we consider the absolute value term of hesitation degree and other absolute value terms as a whole,and an improved strict proof of the Hamming distance measure is given by analysis and discussion.In addition,it is proved that the extended form of Hamming distance(Euclid distance measure)satisfies the axiomatic conditions.

关 键 词：毕达哥拉斯模糊数 海明距离测度 欧几里得距离测度 公理化条件 

分 类 号：O159[理学—数学]