[IEEE TFS] Decoding the Logic of Words: Optimizing the Numerical Scale in 2-Tuple Fuzzy Linguistic Models

TL;DR

In the realm of Computing with Words (CW), translating human qualitative labels (like "Very High" or "Small Chance") into precise math is notoriously difficult. This paper revolutionizes the 2-tuple fuzzy linguistic representation model by introducing a dynamic Numerical Scale. Instead of forcing words into rigid, predefined fuzzy shapes (like the Herrera-Martínez or Wang-Hao models), this method uses an optimization model to "learn" the best numerical values from a decision maker's own logic—specifically through a "Transitive Calibration Matrix."

The Evolution of Computing with Words

For years, decision science has relied on linguistic variables. However, early models suffered from two fatal flaws:

1. Information Loss: Approximating results back into "words" often led to imprecise outcomes.
2. Rigid Distributions: Most models assumed words were perfectly symmetrical and evenly spaced—a rarity in human psychology.

While the classic 2-tuple model solved the approximation problem by adding a "symbolic translation" (a tiny numerical offset to a word), it still lacked the flexibility to adapt to how different people perceive the "distance" between terms like "Fair" and "Excellent."

The Core Insight: Transitive Calibration

The authors argue that the "heart" of CW isn't the dictionary definition of a word, but the Transitive Consistency of the person using it.

If a decision maker says:

• The gap between A and B is "Low" ($s_2$)
• The gap between B and C is "Medium" ($s_3$)
• Then the gap between A and C should logically be "High" ($s_5$).

By collecting these logical triplets into a Transitive Calibration Matrix ($E$), the authors create a benchmark for human consistency.

Methodology: Mapping Words to Numbers

The paper extends the 2-tuple model by defining a function $NS(s_i)$—a Numerical Scale. This scale isn't arbitrary; it is the solution to a bi-objective optimization problem:

• Objective 1: Maximize consistency within the Transitive Calibration Matrix.
• Objective 2: Minimize the distance from the "initial intuition" (the starting fuzzy values).

The optimization model (5) balances human logical consistency with prior semantic knowledge.

This is converted into a Linear Programming problem, ensuring it can be solved efficiently even for large linguistic sets.

Key Properties

• Generalization: If you set the scale to integers ($0,1,\mathrm{2...}$), you get the Herrera-Martínez model. If you use Canonical Characteristic Values (CCVs), you get the Wang-Hao model.
• Order Preservation: The model guarantees that if a word is "greater" than another linguistically, its optimized numerical value will also be higher.
• Personalization: Two different experts can use the same words but have different underlying numerical scales based on their unique Transitive Calibration Matrices.

Experimental Proof: Better Consistency

The authors tested their model against the Wang-Hao model, which was the previous state-of-the-art for non-uniform labels.

Table II shows the shift from standard CCV values to the optimized NS values.

In Example 1, the consistency index improved by nearly 50% (reducing from 0.1177 to 0.0614). This proves that the "logic" of words is better captured when the math adapts to the user, rather than the user being forced into a fixed fuzzy membership function.

Conclusion & Perspective

This paper provides the "missing link" for 2-tuple linguistic models. By decoupling the fuzzy representation from a fixed distribution and using optimization to find the logical "scale," it allows for highly accurate, personalized decision support systems.

Future Outlook: The next frontier is applying this to Unbalanced Linguistic Term Sets—where a scale might have more "positive" words than "negative" ones (e.g., Poor, Fair, Good, Very Good, Excellent). This numerical scale approach provides a clean mathematical path to handle such asymmetries without complex hierarchical re-mapping.