Triangular Fuzzy Rescaling Distance

Abstract: 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.

摘要： 复杂系统中的决策往往涉及处理不精确或不确定的信息，这些信息通常使用模糊集（特别是三角模糊数，TFNs）来表示。许多模糊方法的一个关键方面是量化 TFNs 之间的距离。

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.

许多距离度量假设所有数值都在同一尺度上，当应用于具有不同尺度或单位的异构属性时，需要进行初步的归一化阶段。本文提出了三角模糊重缩放距离（$d_{TR}$），这是一种旨在解决这一挑战的度量标准。

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.

$d_{TR}$ 独特地将线性重缩放（LRE）直接集成到距离计算中，确保了在比较模糊数时的归一化。我们正式证明了 $d_{TR}$ 满足度量空间的属性，包括非负性、同一性、对称性和三角不等式。

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.

此外，我们证明了 $d_{TR}$ 是有界的、尺度不变的和原点不变的。这些属性结合用于优先考虑维度的权重向量，使得 $d_{TR}$ 适用于涉及异构模糊数据的应用，例如合成指标的构建、基于距离的机器学习算法或多准则决策辅助。

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Paper Details:

• Authors: Eddy Soria, Aida Valls, Ana Beatriz Hernández-Lara
• Submitted: 4 Aug 2026
• Subject: Machine Learning (cs.LG)
• DOI: 10.48550/arXiv.2608.19234

论文详情：

• 作者： Eddy Soria, Aida Valls, Ana Beatriz Hernández-Lara
• 提交日期： 2026年8月4日
• 学科： 机器学习 (cs.LG)
• DOI： 10.48550/arXiv.2608.19234