N-AROMAN - Neutrosophic extension of AROMAN

1. 1. Shifted SVN score s'_ij = (3 + T − 2I − F)/4 in [0, 1]; direction is consumed in Step 5 as in crisp AROMAN, so no complement is applied.

   $$s_{ij}^{\prime }=\frac{3+T_{ij}-2I_{ij}-F_{ij}}{4}$$

2. 2. Direction-aware normalisations of the score matrix: min-max T1 (cost columns inverted: (max − s)/(max − min)) and vector T2 (cost: 1 − s/‖s‖). Every normalised value is then "higher is better".

   $$T{1}_{ij}=\{\begin{array}{ll}\frac{s_{ij}^{\prime }-{\min }_i{s}_{ij}^{\prime }}{{\max }_i{s}_{ij}^{\prime }-{\min }_i{s}_{ij}^{\prime }}& j\in {\Omega }_b\\ \frac{{\max }_i{s}_{ij}^{\prime }-s_{ij}^{\prime }}{{\max }_i{s}_{ij}^{\prime }-{\min }_i{s}_{ij}^{\prime }}& j\in {\Omega }_c\end{array},T{2}_{ij}=\{\begin{array}{ll}\frac{s_{ij}^{\prime }}{\sqrt{{\sum }_i{s}_{ij}^{\prime 2}}}& j\in {\Omega }_b\\ 1-\frac{s_{ij}^{\prime }}{\sqrt{{\sum }_i{s}_{ij}^{\prime 2}}}& j\in {\Omega }_c\end{array}$$

3. 3. Aggregated normalisation Z = (α·T1 + (1−α)·T2)/2, α = 0.5 by default.

   $$Z_{ij}=\frac{\alpha T{1}_{ij}+(1-\alpha )T{2}_{ij}}{2}$$

4. 4. Weighted matrix V = Z · w.

   $$V_{ij}=w_j{Z}_{ij}$$

5. 5. Benefit sum σ⁺_i and cost sum σ⁻_i; AROMAN score L_i = (σ⁻_i)^λ + (σ⁺_i)^(1−λ), λ = 0.5 by default; rank descending (Bošković et al. 2023, crisp AROMAN chain).

   $${\sigma }_i^{+}=\sum _{j\in {\Omega }_b}{V}_{ij},{\sigma }_i^{-}=\sum _{j\in {\Omega }_c}{V}_{ij},L_i=({\sigma }_i^{-}{)}^{\lambda }+({\sigma }_i^{+}{)}^{1-\lambda }$$