| Summary: | The Neutrosophic Set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>s</mi><mi>e</mi><mi>t</mi></mrow></msub><mo>)</mo></mrow></semantics></math></inline-formula> represents the uncertainty in data with fuzzy attributes beyond true and false values independently. The problem arises when the summation of true <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="bold-script">T</mi><mi mathvariant="bold-italic">r</mi><mo>)</mo></mrow></semantics></math></inline-formula>, false <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="bold-script">F</mi><mi mathvariant="bold-italic">a</mi><mo>)</mo></mrow></semantics></math></inline-formula>, and indeterminacy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced separators="|"><mrow><mi mathvariant="bold-script">I</mi><mi mathvariant="bold-italic">n</mi></mrow></mfenced></mrow></semantics></math></inline-formula> values crosses the membership value of one, that is, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold-script">T</mi><mi mathvariant="bold-italic">r</mi><mo>+</mo><mi mathvariant="bold-script">I</mi><mi mathvariant="bold-italic">n</mi><mo>+</mo><mi mathvariant="bold-script">F</mi><mi mathvariant="bold-italic">a</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>. It becomes more crucial during decision-making processes like medical diagnoses or any data sets where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold-script">T</mi><mi mathvariant="bold-italic">r</mi><mo>+</mo><mi mathvariant="bold-script">I</mi><mi mathvariant="bold-italic">n</mi><mo>+</mo><mi mathvariant="bold-script">F</mi><mi mathvariant="bold-italic">a</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>. To achieve this goal, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>s</mi><mi>e</mi><mi>t</mi></mrow></msub></mrow></semantics></math></inline-formula> is recently introduced. This study employs the Interval-Valued Fermatean Neutrosophic Set (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mi>V</mi><mi>F</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>s</mi><mi>e</mi><mi>t</mi></mrow></msub></mrow></semantics></math></inline-formula>) as its chosen framework to address instances of partial ignorance within the domains of truth, falsehood, or uncertainty. This selection stands out due to its unique approach to managing such complexities within multi-decision processes when compared to alternative methodologies. Furthermore, the proposed method reduces the propensity for information loss often encountered in other techniques. IVFNS excels at preserving intricate relationships between variables even when dealing with incomplete or vague information. In the present work, we introduce the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mi>V</mi><mi>F</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>s</mi><mi>e</mi><mi>t</mi></mrow></msub></mrow></semantics></math></inline-formula>, which deals with partial ignorance in true, false, or uncertain regions independently for multi-decision processes. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mi>V</mi><mi>F</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>s</mi><mi>e</mi><mi>t</mi></mrow></msub></mrow></semantics></math></inline-formula> contains the interval-valued <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi mathvariant="bold-script">T</mi><mi mathvariant="bold-italic">r</mi></mrow><mrow><mi mathvariant="bold-italic">m</mi><mi mathvariant="bold-italic">e</mi><mi mathvariant="bold-italic">m</mi><mi mathvariant="bold-italic">b</mi><mi mathvariant="bold-italic">e</mi><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">s</mi><mi mathvariant="bold-italic">h</mi><mi mathvariant="bold-italic">i</mi><mi mathvariant="bold-italic">p</mi></mrow></msub></mrow></semantics></math></inline-formula> value, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi mathvariant="bold-script">I</mi><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">m</mi><mi mathvariant="bold-italic">e</mi><mi mathvariant="bold-italic">m</mi><mi mathvariant="bold-italic">b</mi><mi mathvariant="bold-italic">e</mi><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">s</mi><mi mathvariant="bold-italic">h</mi><mi mathvariant="bold-italic">i</mi><mi mathvariant="bold-italic">p</mi></mrow></msub></mrow></semantics></math></inline-formula> value, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi mathvariant="bold-script">F</mi><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">m</mi><mi mathvariant="bold-italic">e</mi><mi mathvariant="bold-italic">m</mi><mi mathvariant="bold-italic">b</mi><mi mathvariant="bold-italic">e</mi><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">s</mi><mi mathvariant="bold-italic">h</mi><mi mathvariant="bold-italic">i</mi><mi mathvariant="bold-italic">p</mi></mrow></msub></mrow></semantics></math></inline-formula> for knowledge representation. The algebraic properties and set theory between the interval-valued <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>s</mi><mi>e</mi><mi>t</mi></mrow></msub></mrow></semantics></math></inline-formula> have also been presented with an illustrative example.
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