Neutrosophic Appurtenance-Inclusion Algebra: A Novel Framework for Personalized Product Design Schemes Decision Modeling

Abstract

This paper introduces a novel mathematical framework called Neutrosophic 
Appurtenance-Inclusion Algebra (N-AIA). It extends classical and fuzzy algebraic 
systems by integrating two distinct relational concepts—appurtenance and inclusion—
 within a neutrosophic logic environment. Each element in the proposed algebra is 
described by two neutrosophic triplets, capturing degrees of truth, indeterminacy, and 
falsehood for both appurtenance and inclusion independently. This dual structure enables 
more nuanced modeling of ambiguous or uncertain relations between objects and 
decision criteria. To demonstrate the practical relevance of the proposed framework, we 
apply N-AIA to the problem of selecting personalized product design schemes. In such 
settings, customer preferences often involve incomplete, conflicting, or imprecise 
information. The N-AIA model allows for structured evaluation of design options using 
neutrosophic measures across both suitability and availability dimensions. The paper 
defines formal operations, algebraic laws, and comparison rules for N-AIA, and presents 
multiple realistic examples with full mathematical computations. Results show that the 
model accurately handles multi-dimensional uncertainty and supports more informed, 
flexible decision-making. This approach can be extended to other domains where human
centered design and uncertain relationships are critical.

DOI: 10.5281/zenodo.15558333