Building Interval Type-2 Fuzzy Membership Function: A Deck of Cards based Co-constructive Approach
Bapi Dutta, Diego García-Zamora, José Rui Figueira, Luis Martínez
TL;DR
This work addresses modelling uncertainty in linguistic assessments within multicriteria decision-making by using Interval Type-2 Fuzzy Sets (IT2FS). It introduces a socio-technical, co-constructive framework that couples a modified Deck-of-Cards method with ratio-scale elicitation to capture decision-maker hesitation, yielding IT2MF representations for linguistic terms. The methodology unfolds in two phases: (i) interactive DoC-based construction of Type-1 membership functions on a ratio scale, incorporating hesitation to produce IT2MFs, and (ii) formalization of IT2MFs for MCDM, including arithmetic operations and an admissible ordering principle. Key contributions include the DoC-T2MF construction, a consistent arithmetic and ordering framework for IT2FS in MCDM, and a pathway toward more interpretable, personalized linguistic scales that better reflect subjective judgments. This framework enhances the reliability and interpretability of fuzzy decision-support systems when expert opinions are uncertain or hesitant, enabling more robust decision outcomes in complex, real-world settings.
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.