AI 中文总结
本文针对异构属性下三角模糊数距离计算需预先归一化的问题，提出三角模糊重标距离$d_{TR}$，其整合线性重标于距离计算，满足度量特性且适用于异构模糊数据相关应用。
AI 中文摘要
复杂系统中的决策常涉及处理不精确或不确定信息，这类信息通常用模糊集表示，尤其是三角模糊数（Triangular Fuzzy Numbers, TFNs）。许多模糊方法的关键环节是对TFNs之间的距离进行量化。现有多数距离度量假设所有值处于同一尺度，当应用于具有不同尺度或单位的异构属性时，需要预先进行归一化阶段。本文提出三角模糊重标距离（$d_{TR}$），这一指标旨在解决该挑战。$d_{TR}$独特地将线性重标（Linear Rescaling, LRE）直接整合到距离计算中，确保在模糊数比较过程中完成归一化。我们正式证明$d_{TR}$满足度量的特性，包括非负性、同一性、对称性和三角不等式。此外，我们证明$d_{TR}$具有有界性、尺度不变性和原点不变性。这些特性结合用于优先考虑维度的权重向量，使$d_{TR}$适用于涉及异构模糊数据的应用，例如综合指标构建、基于距离的机器学习算法或多准则决策辅助。
英文摘要
Decision-making in complex systems often involves dealing with imprecise or uncertain information, frequently represented using fuzzy sets, particularly Triangular Fuzzy Numbers (TFNs). A crucial aspect of many fuzzy methods is the quantification of distance between TFNs. Many distance measures assume that all values are in the same scale, requiring a preliminary normalization stage when applied to heterogeneous attributes with different scales or units. This paper proposes the Triangular Fuzzy Rescaling Distance (d_{TR}), a metric designed to address this challenge. The d_{TR} uniquely integrates Linear Rescaling (LRE) directly into the distance calculation, ensuring normalization during the comparison of fuzzy numbers. We formally prove that d_{TR} satisfies the properties of a metric, including non-negativity, identity, symmetry, and the triangle inequality. Furthermore, we demonstrate that d_{TR} is bounded, scale-invariant, and origin-invariant. These properties, combined with a weighting vector for prioritizing dimensions, make d_{TR} suitable for applications involving heterogeneous fuzzy data, such as the construction of synthetic indicators, distance-based machine learning algorithms or multicriteria-decision aiding.
Comments12 pages, 2 tables
DOI：10.1007/978-3-032-00891-6_10
