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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.
Method
The paper defines d_TR for n-tuples of TFNs by integrating Linear Rescaling directly into the distance calculation before comparison.
Results
d_TR is formally shown to be a metric, bounded, scale-invariant, and origin-invariant.
Takeaways & Limitations
The distance can support heterogeneous fuzzy-data applications including synthetic indicators, decision methods, clustering, and classification, with weighting for prioritizing dimensions.
Takeaways & Limitations
The current formulation is specific to TFNs, and interpreting distance values requires careful consideration of the problem domain.
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.
- Motivation: Distance measures support applications including fuzzy clustering, pattern recognition, and decision-making.
- 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.
- 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.
- 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.
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.
- Fuzzy numbers: A fuzzy number is a fuzzy set on ℝ whose membership function satisfies normality, convexity, upper semi-continuity, and bounded support.
- 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.
- Distance and metrics: A metric is a function satisfying non-negativity, identity of indiscernibles, symmetry, and the triangle inequality.
- 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.
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].
- 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.
- 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.
- 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.
- Invariance: 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.
- 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.
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.
- 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.
- Results: 55.90 is the reported d_TR value for λ=2 in the example.Repeating the calculation with λ=1 gives d_TR=50.00.
- Results: 50.00 is the reported d_TR value when the same example is evaluated with λ=1.This value is explicitly contrasted with the λ=2 result.
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.
- Application setup: Equal weights of 0.2 are assigned to all five strategic indicators in the aggregation.
- 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.
- Aggregated results: 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.
- Aggregated results: 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.
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.
- Conclusion: d_TR is formally shown to be a metric, bounded, scale-invariant, and origin-invariant.Its computational complexity is linear, O(n).
- 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.
- 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.
- Limitations and future work: Future work includes empirical validation in diverse real-world scenarios, comparison with existing distances, parameter tuning, and extension to other fuzzy-number types.