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Triangular Fuzzy Rescaling Distance

This paper proposes the Triangular Fuzzy Rescaling Distance ( $d_{TR}$ ), a novel metric that integrates linear rescaling directly into the distance calculation to enable scale-invariant comparisons of Triangular Fuzzy Numbers in heterogeneous decision-making contexts while formally satisfying all metric properties.

Original authors: Eddy Soria, Aida Valls, Ana Beatriz Hernández-Lara

Published 2026-08-21

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CC BY 4.0

Original authors: Eddy Soria, Aida Valls, Ana Beatriz Hernández-Lara

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the messy reality of the world, data is rarely perfect. Measurements come with margins of error, expert opinions vary in confidence, and information is often incomplete. To make sense of this uncertainty, scientists and decision-makers often turn to a mathematical tool called a triangular fuzzy number. Instead of pinning a value to a single, rigid point on a number line, this approach represents a value as a small, triangular shape that captures a range of possibilities, from a most likely outcome to the furthest edges of what is plausible. This method is invaluable for modeling everything from future market trends to environmental monitoring, where exactness is impossible. However, a significant problem arises when researchers try to compare these fuzzy shapes against one another, especially when the data comes from different sources with different units. Comparing a measurement of temperature in degrees to a measurement of income in dollars is like trying to add apples to oranges; the different scales and starting points distort the comparison, often leading to misleading conclusions about which option is truly better or closer to a goal.

To solve this persistent hurdle, researchers Eddy Soria, Aida Valls, and Ana Beatriz Hernández have developed a new way to measure the distance between these fuzzy shapes. They call it the Triangular Fuzzy Rescaling Distance. The core innovation of their work is that they do not treat the comparison as a two-step process where data is first cleaned and then measured. Instead, they have built the act of cleaning the data directly into the measurement itself. As the distance is calculated, the method automatically adjusts the values to a common, neutral scale, ensuring that a large number in one category does not unfairly dominate a small number in another. This allows for a fair comparison of complex, mixed data sets without the need for separate, error-prone preprocessing steps.

The researchers proved that this new method is mathematically sound, satisfying all the strict rules required for a reliable distance measure. It behaves predictably: the distance is never negative, it is zero only when the two things being compared are identical, and it follows the logical rule that the path from point A to point C cannot be shorter than going through point B. Crucially, the method is immune to the arbitrary choices of scale or starting point. Whether the data is measured in thousands or millions, or whether the scale starts at zero or one hundred, the resulting distance remains the same. The method also guarantees that the final result will always fall within a specific, understandable range, making it easy to interpret the degree of difference between two sets of information.

To demonstrate the power of this approach, the team applied it to a hypothetical scenario involving the sustainability performance of five different cities. They evaluated each city based on five distinct indicators, such as carbon dioxide emissions, access to green space, and public transportation usage. These indicators had vastly different units and ranges, from percentages to tonnes per year. By using their new distance measure, the researchers could calculate how far each city was from an ideal, perfect state of sustainability. The results showed that the method could handle this diverse data seamlessly, producing a clear ranking of the cities. The study also revealed that the final ranking could shift depending on how much weight was given to the largest discrepancies, offering decision-makers a way to see not just the average performance, but also to identify critical weaknesses in specific areas.

This work provides a robust tool for anyone trying to make decisions with uncertain or mixed data. By integrating the normalization of data directly into the distance calculation, the Triangular Fuzzy Rescaling Distance removes the bias introduced by different units and scales. It allows for the construction of synthetic indicators that accurately reflect complex realities, such as quality of life or economic health, without distorting the picture. While the current method is specifically designed for triangular fuzzy numbers, its success suggests a promising path forward for handling imprecise information in fields ranging from machine learning to policy-making, offering a clearer view of the distance between where we are and where we want to be.

Technical Summary: Triangular Fuzzy Rescaling Distance

Problem Statement
Decision-making in complex systems frequently relies on imprecise or uncertain information, often modeled using Triangular Fuzzy Numbers (TFNs). A critical challenge in applying fuzzy methods is quantifying the distance between TFNs, particularly when dealing with heterogeneous attributes that possess different scales, units, or ranges. Traditional distance measures often assume uniform scales, necessitating a preliminary normalization stage. This separation can introduce biases or distortions, especially in contexts like synthetic indicator construction or multi-criteria decision aiding where data heterogeneity is inherent.

Methodology
The paper proposes the Triangular Fuzzy Rescaling Distance ( $d_{TR}$ ), a novel metric designed to compare $n$ -tuples of TFNs. Unlike conventional approaches that normalize data prior to distance calculation, $d_{TR}$ uniquely integrates Linear Rescaling (LRE) directly into the distance formula.

The metric is defined for two $n$ -tuples of TFNs, $\tilde{f}$ and $\tilde{f}^*$ , where each component $\tilde{f}_i = (f_i^{(1)}, f_i^{(2)}, f_i^{(3)})$ resides within a specific interval $[a_i, b_i]$ . The distance is calculated as:

$d_{TR}(\tilde{f}, \tilde{f}^*) = \frac{M}{4} \left( \sum_{i=1}^{n} \omega_i \left( \frac{|f_i^{*(1)} - f_i^{(1)}| + 2|f_i^{*(2)} - f_i^{(2)}| + |f_i^{*(3)} - f_i^{(3)}|}{b_i - a_i} \right)^\lambda \right)^{1/\lambda}$

Where:

$\omega_i$ is a weighting vector ( $\sum \omega_i = 1$ ).
$\lambda \geq 1$ is a parameter controlling the sensitivity to large discrepancies.
$M > 0$ is a scaling parameter defining the upper bound.
The term $(b_i - a_i)$ in the denominator performs the linear rescaling, effectively normalizing the difference between the TFN components relative to their specific domain range.

Key Contributions and Theoretical Properties
The authors formally prove that $d_{TR}$ satisfies the axioms of a metric space and possesses several distinct properties:

Metric Properties: The paper demonstrates that $d_{TR}$ satisfies non-negativity, identity of indiscernibles, symmetry, and the triangle inequality (proven via the Minkowski inequality).
Boundedness: The distance is strictly bounded within the interval $[0, M]$ , regardless of the input values, provided they fall within their defined intervals.
Scale Invariance: The metric remains unchanged if both the TFNs and their defining intervals are scaled by a positive constant.
Origin Invariance: The distance is unaffected if both TFNs and their intervals are shifted by a constant value.
Inherent Normalization: The metric inherently normalizes characteristic values through the LRE mechanism, eliminating the need for separate preprocessing steps.

Results and Application
The paper validates $d_{TR}$ through two main demonstrations:

Numerical Example: A calculation with $n=2$ demonstrates the step-by-step application of the formula, showing how the distance is derived from raw TFN data across different intervals.
Case Study (Sustainability Assessment): The metric is applied to evaluate five hypothetical cities against an ideal sustainability profile across five indicators (e.g., CO2 emissions, green space, income inequality) with vastly different units and ranges.
- The results show that $d_{TR}$ produces values within the $[0, 100]$ range (with $M=100$ ), allowing for direct interpretation as a percentage deviation from the ideal.
- The study highlights the role of the parameter $\lambda$ . Lower $\lambda$ values (e.g., $\lambda=1, 2$ ) provide a balanced aggregation of deviations, while higher $\lambda$ values (e.g., $\lambda=500$ ) increase sensitivity to the largest single discrepancies. This allows decision-makers to detect critical weaknesses even when other indicators are strong.
- The ranking of cities shifts based on $\lambda$ , illustrating the metric's flexibility in prioritizing different types of decision-making strategies (balanced vs. worst-case focus).

Significance and Claims
The paper claims that $d_{TR}$ offers a robust solution for handling heterogeneous fuzzy data without requiring external normalization. Its primary significance lies in its ability to:

Mitigate biases introduced by differing scales and units in data.
Serve as an aggregation mechanism for constructing synthetic indicators (e.g., sustainability indices, quality of life measures).
Be applicable to distance-based machine learning algorithms and multicriteria decision-aiding methods (such as TOPSIS).
Maintain linear $O(n)$ computational complexity.

The authors acknowledge limitations, noting that the current formulation is specific to TFNs and may not directly apply to other fuzzy representations (e.g., trapezoidal or Gaussian). They suggest future work should focus on empirical validation in diverse real-world scenarios, performance comparison against existing measures, and extension to other fuzzy number types.

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