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Triangular Fuzzy Rescaling Distance
Eddy Soria, Aida Valls, Ana Beatriz Hernández-Lara
TL;DR
Existing fuzzy distances may require separate normalization when comparing TFNs across heterogeneous scales and units. The paper introduces d_TR, which integrates Linear Rescaling into the distance calculation, and proves metric, boundedness, scale-invariance, and origin-invariance properties. The resulting construction is positioned for heterogeneous fuzzy-data applications, while its current TFN-specific formulation and need for domain-context interpretation remain limitations.
Problem
Many fuzzy distance measures do not intrinsically handle differing scales or origins, complicating comparisons across heterogeneous attributes with different units or ranges.
Source: §1 · p. 2Method
The paper defines d_TR for n-tuples of TFNs by integrating Linear Rescaling directly into the distance calculation before comparison.
Source: §1 · p. 2Results
d_TR is formally shown to be a metric, bounded, scale-invariant, and origin-invariant.
Source: §3.1 · pp. 4, 6, 7, §6 · p. 11Takeaways & Limitations
The distance can support heterogeneous fuzzy-data applications including synthetic indicators, decision methods, clustering, and classification, with weighting for prioritizing dimensions.
Source: §6 · p. 11The current formulation is specific to TFNs, and interpreting distance values requires careful consideration of the problem domain.
Source: §6 · p. 11
Abstract
from arXiv · showhide
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.
1 Introduction
The introduction frames fuzzy distances as important for uncertain decision-making but limited by mismatched scales and origins. It presents d_TR as a metric that embeds linear rescaling to compare heterogeneous TFN data.
Motivation
- TFNs provide a framework for representing and manipulating imprecise information in complex-system decision-making and analysis.Examples include expert market opinions and imprecise environmental measurements.Source: §1 · p. 1
- Distance measures support applications including fuzzy clustering, pattern recognition, and decision-making.Source: §1 · p. 1
Research gap
- Many existing fuzzy distances do not intrinsically address differences in scales or origins, risking distorted comparisons across heterogeneous attributes.The issue is especially relevant when attributes use different units or ranges.Source: §1 · p. 2
Contribution
- d_TR integrates Linear Rescaling directly into distance calculation, normalizing n-tuples of TFNs before comparison.The approach is designed to mitigate biases introduced by differing scales and units.Source: §1 · p. 2
Paper organization
- The paper formally develops d_TR and illustrates it through mathematical preliminaries, metric properties, a numerical example, and a case study.The stated application scope includes heterogeneous fuzzy data and potential synthetic-indicator construction.Source: §1 · p. 2
2 Preliminaries
The preliminaries define fuzzy sets, fuzzy numbers, TFNs, metrics, and linear rescaling. These concepts establish the representation and normalization foundations used by the proposed distance.
Fuzzy sets
- A fuzzy set assigns each element x in a universe X a membership degree μΨ(x) between 0 and 1.Membership degree 0 denotes no membership, while 1 denotes full membership.Source: §2.1 · p. 2
Fuzzy numbers
- A fuzzy number is a fuzzy set on ℝ whose membership function satisfies normality, convexity, upper semi-continuity, and bounded support.Source: §2.1 · p. 2
Triangular fuzzy numbers
- A TFN is represented by an ordered triplet (f(1), f(2), f(3)) with f(1) ≤ f(2) ≤ f(3), using a linearly increasing then decreasing membership function.Source: §2.1 · p. 3
Distance and metrics
- A metric is a function satisfying non-negativity, identity of indiscernibles, symmetry, and the triangle inequality.Source: §2.2 · p. 3
Linear rescaling
- Linear Rescaling maps values from an original range [a,b] into a dimensionless range [m,M] using the attribute minimum and maximum.The transformation produces normalized values from the original data range.Source: §2.3 · p. 3
3 Triangular Fuzzy Rescaling Distance
The paper defines d_TR for n-tuples of TFNs by incorporating linear rescaling into distance calculation. It establishes metric properties and proves boundedness, scale invariance, origin invariance, and normalization.
Definition
- d_TR maps n-tuples of triangular fuzzy numbers to a bounded interval using a weighting vector whose weights sum to one.The domain consists of TFN tuples with components constrained to intervals [a_i, b_i].Source: §3.1 · p. 4
Metric properties
- d_TR is proven to satisfy non-negativity, identity of indiscernibles, symmetry, and the triangle inequality.The triangle inequality proof uses scalar absolute-value inequalities, convexity for λ≥1, weighted summation, and Minkowski's inequality.Source: §3.1 · pp. 4–5
Boundedness
- d_TR is bounded within [0, M] for any two tuples of TFNs in its domain.The proof derives the upper bound from interval-constrained component differences and nonnegative weighted terms.Source: §3.1 · pp. 6–7
Invariance
- d_TR remains unchanged when both compared TFNs and their intervals are multiplied by a positive constant.Scaling multiplies both numerator differences and interval lengths consistently.Source: §3.1 · p. 7
- d_TR remains unchanged when both compared TFNs and their intervals are shifted by the same constant.The shifted interval length is unchanged, while corresponding differences remain unaffected.Source: §3.1 · pp. 7–8
Normalization
- d_TR normalizes characteristic TFN values through Linear Rescaling before comparison.The normalization proposition links the transformation to the established bounds 0 ≤ d_TR ≤ M.Source: §3.1 · p. 8
4 Numerical Example
The numerical example applies d_TR to two-dimensional TFN tuples with specified intervals, equal weights, λ values, and M. The reported distance is 55.90 for λ=2 and 50.00 for λ=1.
Setup
- The example compares two-dimensional TFN tuples over intervals [1.00, 3.00] and [−10.00, 0.00].The tuples use equal weights ω1 = 0.50 and ω2 = 0.50, with λ=2 and M=100.Source: §4 · p. 9
Calculation
- The calculation applies component-wise TFN differences and then the weighting vector.These are presented as the first two calculation steps in the example.Source: §4 · p. 9
5 An Application to Fuzzy Multi-Criteria Decision Making
The application evaluates five cities against five sustainability indicators using d_TR, with heterogeneous ranges and equal indicator weights. Results remain normalized within [0, M], while λ changes sensitivity to large discrepancies and can shift city rankings.
Application setup
- Five hypothetical cities are assessed across five sustainability indicators, including emissions, green space, transportation, inequality, and recycling.The indicators use different units, ranges, and preference directions.Source: §5 · p. 10
- Equal weights of 0.2 are assigned to all five strategic indicators in the aggregation.Source: §5 · p. 10
Aggregated results
- d_TR produces distance values within [0, M] across λ=1, 2, 3, and 500, enabling direct interpretation of distances.The values are interpreted as proportional deviations from the ideal.Source: §5 · pp. 10–11
- 90.95 for C1 and 3.35 for C3 at λ=1 indicate markedly different distances from the combined ideal sustainability levels.C1 is described as 90.95% distant, whereas C3 is about 3.35% away.Source: §5 · p. 11
- Increasing λ can shift city rankings and strongly penalize the largest discrepancy, as C2’s distance rises from 22.35 to 99.68 when λ reaches 500.Lower λ values provide more balanced aggregation across indicators, while higher values emphasize critical weaknesses.Source: §5 · p. 11
6 Conclusion
The conclusion presents d_TR as a metric for heterogeneous fuzzy data that combines direct rescaling with formal metric properties and linear computational complexity. It highlights synthetic indicators and other decision-support applications while noting limitations of the TFN-specific formulation and the need for empirical validation.
Conclusion
- d_TR compares n-tuples of TFNs by integrating Linear Rescaling directly into the distance calculation.This provides inherent normalization without separate preprocessing.Source: §6 · p. 11
- d_TR is formally shown to be a metric, bounded, scale-invariant, and origin-invariant.Its computational complexity is linear, O(n).Source: §6 · p. 11
Applications
- The distance can aggregate deviations between observed and target TFN states for multidimensional synthetic indicators, with weighting vectors prioritizing dimensions.Suggested domains include sustainability and quality-of-life indicators, TOPSIS, clustering, and classification.Source: §6 · p. 11
Limitations and future work
- The current formulation is specifically defined for TFNs, limiting direct applicability when trapezoidal or Gaussian fuzzy numbers are more appropriate.Interpreting distance values also requires consideration of the problem domain.Source: §6 · p. 11
- Future work includes empirical validation in diverse real-world scenarios, comparison with existing distances, parameter tuning, and extension to other fuzzy-number types.Source: §6 · pp. 11–12