Triangular Fuzzy Rescaling Distance

arXiv cs.LG Papers
fuzzy-sets distance-measures normalization machine-learning decision-making

Summary

This paper proposes the Triangular Fuzzy Rescaling Distance (d_TR), a metric that integrates normalization directly into distance calculations for comparing triangular fuzzy numbers, suitable for heterogeneous data applications.

arXiv:2608.19234v1 Announce Type: new 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. 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.
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# Triangular Fuzzy Rescaling Distance
Source: [https://arxiv.org/abs/2608.19234](https://arxiv.org/abs/2608.19234)
[View PDF](https://arxiv.org/pdf/2608.19234)

> 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\. 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\.

## Submission history

From: Eddy Soria \[[view email](https://arxiv.org/show-email/f832a6e5/2608.19234)\] **\[v1\]**Tue, 4 Aug 2026 20:20:13 UTC \(369 KB\)

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