Fuzzy set theory has been widely employed to handle uncertainty and imprecision in various fields. However, conventional fuzzy models and their extensions often operate as abstract mathematical constructs, overlooking the physical constraints that govern real-world processes in domains like engineering, medicine, and energy systems. To bridge this gap, this study introduces the Thermodynamic Fuzzy Set (TDFS), a novel extension that integrates principles from thermodynamics—such as entropy, equilibrium, and irreversibility—directly into the fuzzy membership structure. By defining membership functions as <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ \mu = f(T, S, U, H, G) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>S</mi> <mo>,</mo> <mi>U</mi> <mo>,</mo> <mi>H</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, the TDFS framework provides a physically grounded representation of uncertainty, capable of modeling both mathematical vagueness and process-dependent constraints. The paper establishes the formal foundations of TDFS, including a set of axioms for membership function design to ensure boundedness, continuity, and physical consistency. Fundamental operations such as addition, subtraction, union, intersection, complement, and generalized <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$t$</EquationSource> <EquationSource Format="MATHML"><math> <mi>t</mi> </math></EquationSource> </InlineEquation>-norms and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$t$</EquationSource> <EquationSource Format="MATHML"><math> <mi>t</mi> </math></EquationSource> </InlineEquation>-conorms are defined. Key properties and theorems, including idempotency, monotonicity, boundedness, reflexivity, and De Morgan duality, are formally proved. A group decision-making algorithm (TFGDM) based on TDFS is also proposed, designed to handle complex multi-criteria scenarios where evaluations are derived from physical parameters. The framework’s efficacy is validated through a real-life case study in personalized hyperthermia therapy planning for cancer treatment. In this application, expert evaluations are not abstract scores but are derived from thermodynamic membership functions of temperature (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$T$</EquationSource> <EquationSource Format="MATHML"><math> <mi>T</mi> </math></EquationSource> </InlineEquation>) and specific absorption rate (SAR). The results demonstrate TDFS’s superior capability to generate decisions that are not only mathematically sound but also physically interpretable and consistent with biophysical laws. Comparisons with classical, intuitionistic, and hesitant fuzzy sets reveal that TDFS enhances both the expressiveness of uncertainty representation and the interpretability of outcomes in contexts influenced by physical phenomena. The study concludes by discussing advantages, limitations, and promising future research directions, establishing TDFS as a principled tool for advanced uncertainty modeling in thermodynamics-driven domains.

Thermodynamic Fuzzy Set: A Novel Framework for Uncertainty Modeling and Decision-Making with Application of Personalized Hyperthermia Therapy Planning

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发表日期：

2026年3月3日

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Muhammad Bilal,
Li Chaoqian,
Ioan Lucian-Popa

摘要

Fuzzy set theory has been widely employed to handle uncertainty and imprecision in various fields. However, conventional fuzzy models and their extensions often operate as abstract mathematical constructs, overlooking the physical constraints that govern real-world processes in domains like engineering, medicine, and energy systems. To bridge this gap, this study introduces the Thermodynamic Fuzzy Set (TDFS), a novel extension that integrates principles from thermodynamics—such as entropy, equilibrium, and irreversibility—directly into the fuzzy membership structure. By defining membership functions as $ \mu = f(T, S, U, H, G) $ $μ = f (T, S, U, H, G)$ , the TDFS framework provides a physically grounded representation of uncertainty, capable of modeling both mathematical vagueness and process-dependent constraints. The paper establishes the formal foundations of TDFS, including a set of axioms for membership function design to ensure boundedness, continuity, and physical consistency. Fundamental operations such as addition, subtraction, union, intersection, complement, and generalized $t$ $t$ -norms and $t$ $t$ -conorms are defined. Key properties and theorems, including idempotency, monotonicity, boundedness, reflexivity, and De Morgan duality, are formally proved. A group decision-making algorithm (TFGDM) based on TDFS is also proposed, designed to handle complex multi-criteria scenarios where evaluations are derived from physical parameters. The framework’s efficacy is validated through a real-life case study in personalized hyperthermia therapy planning for cancer treatment. In this application, expert evaluations are not abstract scores but are derived from thermodynamic membership functions of temperature ( $T$ $T$ ) and specific absorption rate (SAR). The results demonstrate TDFS’s superior capability to generate decisions that are not only mathematically sound but also physically interpretable and consistent with biophysical laws. Comparisons with classical, intuitionistic, and hesitant fuzzy sets reveal that TDFS enhances both the expressiveness of uncertainty representation and the interpretability of outcomes in contexts influenced by physical phenomena. The study concludes by discussing advantages, limitations, and promising future research directions, establishing TDFS as a principled tool for advanced uncertainty modeling in thermodynamics-driven domains.