qR-MARCOS - q-Rung Orthopair extension of MARCOS

1. Parse the q-rung orthopair (mu, nu) matrix, check the rung condition, and extend it with the anti-ideal (AAI) and ideal (AI) rows, each cell picked from its column by the score and the criterion direction.
$X^{*} = [X; AA I; A I], A I_{j} = ar g i max S (x_{ij}) (j \in J^{+}), ar g i min S (x_{ij}) (j \in J^{-}), μ^{q} + ν^{q} \leq 1$
2. Normalize the extended matrix: benefit cells stay as they are, cost cells take the complement (nu, mu).
$n_{ij} = x_{ij} (j \in J^{+}); n_{ij} = (ν_{ij}, μ_{ij}) (j \in J^{-})$
3. Score every normalized cell and weight the score: v_ij = w_j * S(n_ij).
$S (p) = \frac{1 + μ ^{q} - ν ^{q}}{2}, v_{ij} = w_{j} S (n_{ij})$
4. Sum each row of the weighted score matrix; the last two rows give S_AI and S_AAI.
$S_{i} = j = 1 \sum n v_{ij}$
5. Utility degrees of each alternative against the ideal and the anti-ideal row sum.
$K_{i}^{+} = \frac{S _{i}}{S _{A I}}, K_{i}^{-} = \frac{S _{i}}{S _{AA I}}$
6. Utility functions of the two utility degrees.
$f (K_{i}^{+}) = \frac{K _{i}^{-}}{K _{i}^{+} + K _{i}^{-}}, f (K_{i}^{-}) = \frac{K _{i}^{+}}{K _{i}^{+} + K _{i}^{-}}$
7. Final utility function and descending ranking.
$f (K_{i}) = \frac{K _{i}^{+} + K _{i}^{-}}{1 + \frac{1 - f ( K _{i}^{+} )}{f ( K _{i}^{+} )} + \frac{1 - f ( K _{i}^{-} )}{f ( K _{i}^{-} )}}$

1. Parse the q-rung orthopair (mu, nu) matrix, check the rung condition, and extend it with the anti-ideal (AAI) and ideal (AI) rows, each cell picked from its column by the score and the criterion direction.