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arxiv: 2503.01413 · v2 · submitted 2025-03-03 · 💻 cs.AI · math.OC

Building Interval Type-2 Fuzzy Membership Function: A Deck of Cards based Co-constructive Approach

Bapi Dutta , Diego Garc\'ia-Zamora , Jos\'e Rui Figueira , Luis Mart\'inez This is my paper

Pith reviewed 2026-05-23 01:52 UTC · model grok-4.3

classification 💻 cs.AI math.OC
keywords interval type-2 fuzzy setsdeck of cards methodmembership function constructionmulticriteria decision makingco-constructive approachlinguistic termsuncertainty modelingpreference elicitation
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The pith

Decision makers actively construct interval type-2 fuzzy sets using a modified deck-of-cards process to capture ambiguity in their judgments.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper introduces a two-phase socio-technical method in which decision makers work interactively with an analyst to build interval type-2 fuzzy sets. A modified deck-of-cards technique first places linguistic terms on a ratio scale to form type-1 membership functions, then extends the placement to express hesitation as intervals. A sympathetic reader would care because conventional construction methods leave the decision maker passive, while this approach aims to make the resulting models reflect the decision maker's own semantic understanding for use in multicriteria problems.

Core claim

The authors present an interactive modified Deck-of-Cards process that first elicits type-1 fuzzy membership functions on a ratio scale and then incorporates ambiguity to produce interval type-2 fuzzy set models of linguistic terms; these models are then equipped with mathematical representations, aggregation rules, and an ordering principle so they can be applied directly in multicriteria decision making while preserving the decision maker's personalized semantics.

What carries the argument

The modified Deck-of-Cards method, an interactive ratio-scale elicitation process extended to model ambiguity in membership degrees.

If this is right

  • Decision makers gain direct control over the semantic placement of linguistic terms as interval type-2 fuzzy sets.
  • The resulting models explicitly represent hesitation through interval membership degrees rather than single values.
  • Aggregation and ordering operations are supplied so the constructed sets can be used immediately in multicriteria decision procedures.
  • Fuzzy decision models become more aligned with the individual decision maker's subjective judgments of uncertainty.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method could be compared against existing construction techniques in applied multicriteria cases to check whether the added decision-maker involvement produces different final rankings.
  • It might be adapted to elicit other forms of uncertainty representations beyond interval type-2 sets.
  • The ratio-scale deck-of-cards step could be examined for consistency across repeated elicitations with the same decision maker.

Load-bearing premise

That an interactive modified deck-of-cards process on a ratio scale can be extended to incorporate ambiguity while preserving interpretability and producing usable interval type-2 fuzzy set models for multicriteria decision making.

What would settle it

An experiment in which the same decision makers construct models both with the proposed method and with a standard passive method, and the interval type-2 sets from the new method show no greater alignment with the decision makers' stated levels of hesitation.

Figures

Figures reproduced from arXiv: 2503.01413 by Bapi Dutta, Diego Garc\'ia-Zamora, Jos\'e Rui Figueira, Luis Mart\'inez.

Figure 1
Figure 1. Figure 1: Support and Core of A – All the alternatives are on the right of the core of the fuzzy set A. We will show how to compute for the alternatives on the right, the procedure for the left is similar. 2. We want to elicit the subjective ratios A(xp) A(xp−1) , A(xp) A(xp−2) , . . . , A(x2) A(x1) . The comparison of A(x1) with the rest is insignificant in the sense that A(x1) belongingness to the fuzzy set is alm… view at source ↗
Figure 2
Figure 2. Figure 2: Blank Cards x1, and 4 cards between x3 and x2. The example tells that the difference in the membership values when moving from x3 to x1 is greater than the one from x2 to x1 and this can be modelled by inserting 4 cards and 1 card, in the intervals between the first two pairs of alternative and the second pair. x3 x2 x1 x3 x2 x1 4 1 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: An example of blank cards insertion 4. Compute the non-normalized values. From this information provided by the decision-maker, the non-normalized values of memberships of the alternatives in A, A¯ : {x1, ..., xp} → [0, ∞) can be determined from the following relations: A¯(xr) = A¯(x1) +Xr−1 h=1 (eh + 1), for r = 2, . . . p, where A¯(x1), the non-normalized value of the right endpoint of the support is zer… view at source ↗
Figure 4
Figure 4. Figure 4: Ratio Table where a s r = A¯(xs)/A¯(xr) for all s, r = p, . . . 2 and s < r. Afterward the analyst tries to know the decision-maker’s preference on the ratio table in the following way: − the analyst asks the decision-maker whether she/he feels satisfied with the ratios between membership values of the different alternatives. If the answer is YES. The membership values of the alternatives are computed from… view at source ↗
Figure 5
Figure 5. Figure 5: Right-hand membership function in ratio scale [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Value function for label of Eg from [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Core and support of the fuzzy concept lg,r 3.4.3. Identify left-right upper and lower membership function Now, we attempt to construct the left-right upper and lower membership functions part of IT2MF Alg,r = (Alg,r , A¯ lg,r ) by coupling the idea of constructing T1MF proposed in Section 3.1 with the interpersonal uncertainty appears in the human subjective judgments. The construction phase con￾sists of t… view at source ↗
Figure 8
Figure 8. Figure 8: Right-hand membership function construction from uncertainty into the number of blank cards [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: DoC-IT2MF from DoC-T1MFs 1. A, A are fuzzy numbers. 2. A(x) ⩽ A(x) ∀ x ∈ R Note that A1 ⊆ A1. Definition 7 (DoC-IT2MF). A DoC-IT2MF is a mapping A = (A, A) : R → I such that: 1. A, A are DoC-MFs 2. A(x) ⩽ A(x) ∀ x ∈ R Note that any DoC-IT2MF is a Type 2 fuzzy set. D2 is the set of all the DoC-IT2MFs on [0, 1]. Definition 8. Let A = (A, A). Then, given α ∈]0, 1], we can define its α-cut by Aα = (Aα, Aα). Of… view at source ↗
read the original abstract

Since its inception, Fuzzy Set has been widely used to handle uncertainty and imprecision in decision-making. However, conventional fuzzy sets, often referred to as type-1 fuzzy sets (T1FSs) have limitations in capturing higher levels of uncertainty, particularly when decision-makers (DMs) express hesitation or ambiguity in membership degree. To address this, Interval Type-2 Fuzzy Sets (IT2FSs) have been introduced by incorporating uncertainty in membership degree allocation, which enhanced flexibility in modelling subjective judgments. Despite their advantages, existing IT2FS construction methods often lack active involvement from DMs and that limits the interpretability and effectiveness of decision models. This study proposes a socio-technical co-constructive approach for developing IT2FS models of linguistic terms by facilitating the active involvement of DMs in preference elicitation and its application in multicriteria decision-making (MCDM) problems. Our methodology is structured in two phases. The first phase involves an interactive process between the DM and the decision analyst, in which a modified version of Deck-of-Cards (DoC) method is proposed to construct T1FS membership functions on a ratio scale. We then extend this method to incorporate ambiguity in subjective judgment and that resulted in an IT2FS model that better captures uncertainty in DM's linguistic assessments. The second phase formalizes the constructed IT2FS model for application in MCDM by defining an appropriate mathematical representation of such information, aggregation rules, and an admissible ordering principle. The proposed framework enhances the reliability and effectiveness of fuzzy decision-making not only by accurately representing DM's personalized semantics of linguistic information.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript proposes a two-phase socio-technical co-constructive method for constructing Interval Type-2 Fuzzy Sets (IT2FS) membership functions of linguistic terms. Phase 1 uses an interactive modified Deck-of-Cards (DoC) process to first build Type-1 Fuzzy Sets (T1FS) on a ratio scale, then extends the process to incorporate ambiguity and produce IT2FS models. Phase 2 defines mathematical representation, aggregation rules, and an ordering principle to apply the resulting IT2FS in multicriteria decision-making (MCDM). The central claim is that active DM involvement yields IT2FS models that more accurately capture personalized semantics and thereby improve reliability and effectiveness of fuzzy decision-making.

Significance. If the extension step from T1FS to IT2FS can be shown to preserve interpretability and ratio-scale properties while adding usable interval uncertainty, the approach would address a documented limitation of existing IT2FS construction techniques that minimize DM participation. The co-constructive framing and explicit ratio-scale elicitation are potentially valuable contributions to the MCDM literature if supported by concrete illustrations.

major comments (1)
  1. [Abstract] Abstract (phase-one description): the claim that the modified DoC extension 'resulted in an IT2FS model that better captures uncertainty' is presented without any derivation, numerical example, or validation data showing how ambiguity is incorporated on the ratio scale while preserving interpretability and producing usable models for MCDM. This absence makes the central claim that the framework enhances reliability unverifiable from the supplied text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed review and constructive feedback on our manuscript. We address the single major comment below and are willing to revise the abstract accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract (phase-one description): the claim that the modified DoC extension 'resulted in an IT2FS model that better captures uncertainty' is presented without any derivation, numerical example, or validation data showing how ambiguity is incorporated on the ratio scale while preserving interpretability and producing usable models for MCDM. This absence makes the central claim that the framework enhances reliability unverifiable from the supplied text.

    Authors: We agree that the abstract, as a concise summary, states the outcome of the extension without including the supporting derivation, example, or validation steps. The full manuscript describes the modified Deck-of-Cards process, the incorporation of ambiguity while retaining the ratio scale, and the resulting IT2FS representation in the methodology and application sections. To make the abstract's claim verifiable at the summary level, we will revise the abstract wording to more accurately reflect the process described in the body of the paper and to avoid overstating the immediate evidence provided in the abstract itself. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper describes a two-phase socio-technical methodology for constructing IT2FS membership functions via a modified Deck-of-Cards elicitation process followed by formalization for MCDM. No equations, fitted parameters, predictions, or self-citations appear in the provided text that reduce any central claim to its inputs by construction. The derivation is presented as directly resulting from interactive DM input and subsequent mathematical representation, with no self-definitional loops, renamed empirical patterns, or load-bearing self-citations. This is the most common honest finding for a purely methodological proposal without quantitative reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, invented entities, or non-standard axioms are stated.

axioms (1)
  • domain assumption Decision makers can meaningfully express relative preferences via a deck-of-cards procedure on a ratio scale.
    Invoked as the basis for constructing the T1FS membership functions in phase one.

pith-pipeline@v0.9.0 · 5841 in / 1190 out tokens · 53177 ms · 2026-05-23T01:52:51.968350+00:00 · methodology

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Reference graph

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