IVN-MULTIMOORA

IVN-MULTIMOORA - Aralık Değerli Nötrosofik Sayı Ortamında MULTIMOORA

Nötrosofik MULTIMOORA - Aralık Değerli Nötrosofik Sayı (IVNN: <[tl,tu],[il,iu],[fl,fu]>)

Formül adımları

Analiz motorunun yöntem bildirimindeki (manifest F.steps) adımlar; raporlardaki formüllerle aynı kaynaktır.

Transform K individual SVNN decision matrices into a group IVNN matrix using the boundary-median approach (Eq.44): for each (i,j), lower bound = mean of ratings ≤ median, upper bound = mean of ratings ≥ median, applied separately for t, i, f components.

$x ̃ i j = < [t l_{i} j, t u_{i} j], [i l_{i} j, i u_{i} j], [f l_{i} j, f u_{i} j] > w h e r e t l_{i} j = a v g (t k_{i} j | t k_{i} j \leq m e d (t i j)), t u_{i} j = a v g (t k_{i} j | t k_{i} j \geq m e d (t i j)); a n a l o g o u s l y f o r i, f$

LaTeX
x̃ij = <[tl_ij, tu_ij],[il_ij, iu_ij],[fl_ij, fu_ij]> where tl_ij = avg(tk_ij | tk_ij ≤ med(tij)), tu_ij = avg(tk_ij | tk_ij ≥ med(tij)); analogously for i,f
Apply the IVNWA operator (Ch.18 Eq.18) to the benefit columns (Omega_max) and, separately, to the cost columns (Omega_min) of row i. The two aggregates are kept apart; F3 subtracts one from the other.

$Y_{m} a x_{i} = I V N W A (x_{i} j : j i n O m e g a_{m} a x, w n o r m a l i s e d w i t h i n t h e s i d e); Y_{m} i n_{i} = I V N W A (x_{i} j : j i n O m e g a_{m} i n, w n o r m a l i s e d w i t h i n t h e s i d e); W_{m} a x = s u m (w j : j i n O m e g a_{m} a x), W_{m} i n = s u m (w j : j i n O m e g a_{m} i n)$

LaTeX
Y_max_i = IVNWA(x_ij : j in Omega_max, w normalised within the side); Y_min_i = IVNWA(x_ij : j in Omega_min, w normalised within the side); W_max = sum(wj : j in Omega_max), W_min = sum(wj : j in Omega_min)
Convert Y_max_i and Y_min_i to crisp scores with the score function (Ch.18 Eq.16), weight each by its side's mass, and take the difference: yi = W_max*s(Y_max_i) - W_min*s(Y_min_i). Rank alternatives in descending order of yi.

$y i = W_{m} a x * s (Y_{m} a x_{i}) - W_{m} i n * s (Y_{m} i n_{i})$

LaTeX
yi = W_max*s(Y_max_i) - W_min*s(Y_min_i)
For each criterion j the reference point r*j is the ideal cell of that column, taken in the direction the criterion points.

$r * j = < [m a x_{i} t l_{i} j, m a x_{i} t u_{i} j], [m i n_{i} i l_{i} j, m i n_{i} i u_{i} j], [m i n_{i} f l_{i} j, m i n_{i} f u_{i} j] >$

LaTeX
r*j = <[max_i tl_ij, max_i tu_ij], [min_i il_ij, min_i iu_ij], [min_i fl_ij, min_i fu_ij]>
Compute weighted IVNN distance from each alternative to the reference point for each criterion, then take the maximum.

$d_{i} j = w j \cdot d (x ̃ i j, r * j) w h e r e d (x 1, x 2) = (1 / 3) (| t l 1 - t l 2 | + | t u 1 - t u 2 | + | i l 1 - i l 2 | + | i u 1 - i u 2 | + | f l 1 - f l 2 | + | f u 1 - f u 2 |); d^{m} a x_{i} = m a x_{j} (d_{i} j)$

LaTeX
d_ij = wj · d(x̃ij, r*j) where d(x1,x2) = (1/3)(|tl1−tl2|+|tu1−tu2|+|il1−il2|+|iu1−iu2|+|fl1−fl2|+|fu1−fu2|); d^max_i = max_j(d_ij)
Rank alternatives in ascending order of d^max_i; lowest distance = best rank.

$r_{i}^{R P} = {r a n k}_{↑} (d_{i}^{max}); best = \arg {min}_{i} d_{i}^{max}$

LaTeX
r^{RP}_i = \mathrm{rank}_{\uparrow}\!\left(d^{\max}_i\right);\quad \text{best} = \arg\min_i d^{\max}_i
Apply the IVNWG operator (Ch.18 Eq.19) to the benefit columns (Omega_max) and, separately, to the cost columns (Omega_min) of row i, each with its weights renormalised within the side. F8 divides one by the other.

$U_{m} a x_{i} = I V N W G (x_{i} j : j i n O m e g a_{m} a x); U_{m} i n_{i} = I V N W G (x_{i} j : j i n O m e g a_{m} i n)$

LaTeX
U_max_i = IVNWG(x_ij : j in Omega_max); U_min_i = IVNWG(x_ij : j in Omega_min)
Convert U_max_i and U_min_i to crisp scores with the score function (Ch.18 Eq.16), then take ui = s(U_max_i) / s(U_min_i) and rank alternatives in descending order of ui.

$u i = s (U_{m} a x_{i}) / s (U_{m} i n_{i})$

LaTeX
ui = s(U_max_i) / s(U_min_i)
Combine three sub-rankings (RS, RP, FMF) using dominance theory: the alternative that ranks first (or at highest positions) across all three sub-approaches is selected as best. Ties broken by majority.

$Dominance theory (Brauers-Zavadskas): a^{*} = \arg ⋂_{k \in {R S, R P, F M F}} t o p (r_{i}^{(k)}); ties broken by majority across the three sub-rankings$

LaTeX
\text{Dominance theory (Brauers-Zavadskas):}\;\; a^* = \arg\!\!\!\bigcap_{\,k \in \{RS, RP, FMF\}} \mathrm{top}\!\left(r^{(k)}_i\right);\;\; \text{ties broken by majority across the three sub-rankings}

Yöntem ayrıntıları kaynak kütüphanedeki özgün (İngilizce) metindir.

Sezgi

Sonucu okuma: IVN-MULTIMOORA ranks alternatives based on performance scores. Higher score = better rank.