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A hybrid approach for building fuzzy numbers based on data and expert knowledge

Diego García-Zamora, José Rui Figueira, Miguel Couceiro

TL;DR

This work addresses the challenge of constructing fuzzy numbers that reflect both empirical data and human expert semantics while preserving interpretability. It proposes a hybrid approach that couples a data-driven convex fuzzy $k$-means (C-FKM) pipeline with the Deck-of-Cards elicitation (DoC-MF), enabling intermediate outputs to be translated into card-based representations for iterative expert validation. A key contribution is the Convex FKM formulation, which enforces convex, contiguous supports so each cluster corresponds to a valid fuzzy number, paired with a three-step DoC-based refinement (value scale, cores/supports, and left/right sides) to ensure empirical consistency and semantic clarity. The method is demonstrated on real educational data and synthetic distributions, showing adaptability to symmetric, skewed, and multimodal patterns, and offering a transparent uncertainty representation suitable for decision support in diverse domains.

Abstract

This paper presents a hybrid socio-technical methodology for constructing fuzzy numbers from numerical data while incorporating expert knowledge through an interactive Deck of Cards (DoC) process. The approach extends the existing DoC membership function construction framework by introducing a data-driven pipeline based on a convex version of fuzzy $k$-Means in which each computational step produces intermediate outputs that are translated into card-based structures for expert validation and tuning. The proposed method ensures interpretability, adaptability, and consistency between empirical evidence and expert semantics.

A hybrid approach for building fuzzy numbers based on data and expert knowledge

TL;DR

This work addresses the challenge of constructing fuzzy numbers that reflect both empirical data and human expert semantics while preserving interpretability. It proposes a hybrid approach that couples a data-driven convex fuzzy -means (C-FKM) pipeline with the Deck-of-Cards elicitation (DoC-MF), enabling intermediate outputs to be translated into card-based representations for iterative expert validation. A key contribution is the Convex FKM formulation, which enforces convex, contiguous supports so each cluster corresponds to a valid fuzzy number, paired with a three-step DoC-based refinement (value scale, cores/supports, and left/right sides) to ensure empirical consistency and semantic clarity. The method is demonstrated on real educational data and synthetic distributions, showing adaptability to symmetric, skewed, and multimodal patterns, and offering a transparent uncertainty representation suitable for decision support in diverse domains.

Abstract

This paper presents a hybrid socio-technical methodology for constructing fuzzy numbers from numerical data while incorporating expert knowledge through an interactive Deck of Cards (DoC) process. The approach extends the existing DoC membership function construction framework by introducing a data-driven pipeline based on a convex version of fuzzy -Means in which each computational step produces intermediate outputs that are translated into card-based structures for expert validation and tuning. The proposed method ensures interpretability, adaptability, and consistency between empirical evidence and expert semantics.
Paper Structure (13 sections, 5 theorems, 50 equations, 7 figures)

This paper contains 13 sections, 5 theorems, 50 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fuzzy sets and fuzzy numbers
  4. The Deck-of-Cards method
  5. Fuzzy k-means clustering
  6. Convex fuzzy $k$-means
  7. Hybrid construction of fuzzy numbers integrating data and expert knowledge
  8. Step 1: Value scale construction
  9. Step 2: Identification of Cores and Supports
  10. Step 3: Fine–Tuning the Left and Right Sides
  11. Numerical Example
  12. Influence of data distribution on fuzzy uncertainty profiles
  13. Conclusions

Key Result

Theorem 1

Let ${\bf x}=(x_0,\ldots,x_n)\in[0,1]^{\,n+1}$ with $n>1$, be an $n$-tuple satisfying and let $m\in\mathbb{N}$ be such that $\lfloor 10^m x_{i-1} \rfloor < \lfloor 10^m x_i \rfloor,~ i=1,\ldots,n.$ Define the rational approximation with $N = 10^m$ being the total number of units. Then, the DoC method can represent the ordered tuple $x$ with precision $10^{-m}$, using a sequence of $N$ cards part

Figures (7)

  • Figure 1: Visualization of the data in quiz1_marks
  • Figure 2: Membership functions obtained from C-FKM
  • Figure 3: Membership functions after validating cores and supports
  • Figure 4: Membership functions obtained at the end of the process
  • Figure 5: Histograms and KDE of the data
  • ...and 2 more figures

Theorems & Definitions (9)

  • Theorem 1: DoC-MFT2
  • Proposition 1: Center update
  • proof
  • Theorem 2: Membership update
  • proof
  • Theorem 3
  • proof
  • Theorem 4: Convergence to local minimum
  • proof