$ \mathfrak{q} $-Rung simplified neutrosophic set: A generalization of intuitionistic, Pythagorean and Fermatean neutrosophic sets

This research explored a significant and specific case in the definition of neutrosophic sets (N sets), where truth and falsehood degrees were dependent, representing $ \mathfrak{q} $-Rung orthopair fuzzy ($ \mathfrak{q} $-ROF) values, while indeterminacy acted independently. Based on this assertion, a new hybrid model called the $ \mathfrak{q} $-Rung simplified neutrosophic set ($ \mathfrak{q} $-RSN) set was proposed. This model extended the capabilities of existing frameworks such as the simplified intuitionistic neutrosophic set (Simplified IN) set and Pythagorean neutrosophic set (PyN) set, while simultaneously harnessing the advantages of both N sets and $ \mathfrak{q} $-ROF sets. To lay the foundation, we established the formal definition of the $ \mathfrak{q} $-RSN set and outlined its algebraic operations. The properties of these operations were provided, accompanied by their respective proofs. Moving forward, we introduced a method for comparing $ \mathfrak{q} $-RSN numbers, employing the score function (SF) and accuracy function (AF) as effective tools. To enable the aggregation of $ \mathfrak{q} $-RSN numbers, we proposed two types of operators: $ \mathfrak{q} $-RSN weighted averaging operators ($ \mathfrak{q} $-RSNWAOs) and $ \mathfrak{q} $-RSN weighted geometric operators ($ \mathfrak{q} $-RSNWGOs). We established and validated the desirable properties of these operators, including idempotency, boundedness, and monotonicity. Expanding on the proposed aggregation operators (AOs), we presented an algorithmic approach for multi-criteria decision making (MCDM) and provide a practical implementation to showcase the effectiveness of the algorithm. Within the given MCDM problem context, we employed the proposed $ \mathfrak{q} $-RSNWAOs, $ \mathfrak{q} $-RSNWGOs, and SFs to ensure a comprehensive evaluation. Additionally, we conducted a sensitivity analysis of the problem parameters to examine the behavior of the optimal solution. Furthermore, we conducted a comparative examination, pitting the proposed method against other related methods. The results obtained from this comparative examination demonstrated a clear and significant superiority of the proposed method over the alternatives.