Rough-EDAS - EDAS yönteminin Rough uzantısı

Formül adımları

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

Adım 1 — Tam IRN alanında ortalama çözüm. IRN(AV_j)'nin her bileşeni, tüm m alternatifin karşılık gelen bileşenlerinin aritmetik ortalamasıdır. Defuzzifikasyon yok.

$I R N (A V_{j}) = [[\frac{1}{m} \sum_{i} x_{i j}^{l L}, \frac{1}{m} \sum_{i} x_{i j}^{l U}], [\frac{1}{m} \sum_{i} x_{i j}^{u L}, \frac{1}{m} \sum_{i} x_{i j}^{u U}]]$

LaTeX
IRN(AV_j)=\left[\!\left[\frac{1}{m}\sum_i x_{ij}^{lL},\;\frac{1}{m}\sum_i x_{ij}^{lU}\right]\!,\;\left[\frac{1}{m}\sum_i x_{ij}^{uL},\;\frac{1}{m}\sum_i x_{ij}^{uU}\right]\!\right]
Adım 2 — Tam IRN aritmetiğinde pozitif ortalamadan uzaklık (PDA). Fayda kriterleri: B_ij çapraz-eşleştirilmiş çıkarma (x-av). Maliyet kriterleri: C_ij aynı-seviye çıkarma (av-x).

$b_{i j}^{l L} = max (0, x^{l L} - a v^{u U}), b^{l U} = max (0, x^{l U} - a v^{u L}), b^{u L} = max (0, x^{u L} - a v^{l U}), b^{u U} = max (0, x^{u U} - a v^{l L}) [4pt] c^{l L} = max (0, a v^{u U} - x^{u U}), c^{l U} = max (0, a v^{u L} - x^{u L}), c^{u L} = max (0, a v^{l U} - x^{l U}), c^{u U} = max (0, a v^{l L} - x^{l L}) [4pt] I R N (P D A_{i j}) = [[\frac{b^{l L}}{a v^{u U}}, \frac{b^{l U}}{a v^{u L}}], [\frac{b^{u L}}{a v^{l U}}, \frac{b^{u U}}{a v^{l L}}]] (j \in Ω_{+}); replace b \leftrightarrow c for j \in Ω_{-}$

LaTeX
b_{ij}^{lL}=\max(0,x^{lL}\!-\!av^{uU}),\;b^{lU}=\max(0,x^{lU}\!-\!av^{uL}),\;b^{uL}=\max(0,x^{uL}\!-\!av^{lU}),\;b^{uU}=\max(0,x^{uU}\!-\!av^{lL})\\[4pt]c^{lL}=\max(0,av^{uU}\!-\!x^{uU}),\;c^{lU}=\max(0,av^{uL}\!-\!x^{uL}),\;c^{uL}=\max(0,av^{lU}\!-\!x^{lU}),\;c^{uU}=\max(0,av^{lL}\!-\!x^{lL})\\[4pt]IRN(PDA_{ij})=\left[\left[\frac{b^{lL}}{av^{uU}},\frac{b^{lU}}{av^{uL}}\right],\left[\frac{b^{uL}}{av^{lU}},\frac{b^{uU}}{av^{lL}}\right]\right]\;(j\in\Omega_+);\quad\text{replace }b\leftrightarrow c\text{ for }j\in\Omega_-
Adım 3 — Tam IRN aritmetiğinde negatif ortalamadan uzaklık (NDA). F2'nin aynası: fayda kriterleri C_ij/AV_j, maliyet kriterleri B_ij/AV_j kullanır.

$I R N (N D A_{i j}) = [[\frac{c^{l L}}{a v^{u U}}, \frac{c^{l U}}{a v^{u L}}], [\frac{c^{u L}}{a v^{l U}}, \frac{c^{u U}}{a v^{l L}}]] (j \in Ω_{+}); replace c \leftrightarrow b for j \in Ω_{-} [4pt] (b, c as defined in F2)$

LaTeX
IRN(NDA_{ij})=\left[\left[\frac{c^{lL}}{av^{uU}},\frac{c^{lU}}{av^{uL}}\right],\left[\frac{c^{uL}}{av^{lU}},\frac{c^{uU}}{av^{lL}}\right]\right]\;(j\in\Omega_+);\quad\text{replace }c\leftrightarrow b\text{ for }j\in\Omega_-\\[4pt](b,c\text{ as defined in F2})
Adım 4 — VP/VN ağırlıklı IRN matrisleri, ardından SP/SN satır toplamı. Bileşen çarpımı: vp^k = w_j^k × pda^k.

$I R N (V P_{i j}) : v p_{i j}^{k} = w_{j}^{k} \cdot p d a_{i j}^{k}, k \in {l L, l U, u L, u U} [4pt] I R N (S P_{i}) = ⨁_{j = 1}^{n} I R N (V P_{i j}), s p_{i}^{k} = \sum_{j = 1}^{n} v p_{i j}^{k} [4pt] (analogously I R N (V N_{i j}) and I R N (S N_{i}) from NDA)$

LaTeX
IRN(VP_{ij}):\;vp_{ij}^k = w_j^k \cdot pda_{ij}^k,\quad k\in\{lL,lU,uL,uU\}\\[4pt]IRN(SP_i)=\bigoplus_{j=1}^n IRN(VP_{ij}),\quad sp_i^k=\sum_{j=1}^n vp_{ij}^k\\[4pt]\text{(analogously }IRN(VN_{ij})\text{ and }IRN(SN_i)\text{ from NDA)}
Adım 5 — IRN(SP) ve IRN(SN) normalleştirme. Max bileşen bazlı. NSN = 1 − oran: skaler çıkarma 1−[[a,b],[c,d]]=[[1−d,1−c],[1−b,1−a]].

$s p_{m a x}^{k} = {max}_{i} s p_{i}^{k}, I R N (N S P_{i}) = [[\frac{s p_{i}^{l L}}{s p_{m a x}^{u U}}, \frac{s p_{i}^{l U}}{s p_{m a x}^{u L}}], [\frac{s p_{i}^{u L}}{s p_{m a x}^{l U}}, \frac{s p_{i}^{u U}}{s p_{m a x}^{l L}}]] [4pt] s n_{m a x}^{k} = {max}_{i} s n_{i}^{k}, r a t i o_{i}^{k} = \frac{s n_{i}^{k}}{(cross-denominator)}, I R N (N S N_{i}) = 1 - \frac{I R N (S N_{i})}{I R N (S N_{m a x})} [4pt] 1 - [[a, b], [c, d]] = [[1 - d, 1 - c], [1 - b, 1 - a]]$

LaTeX
sp_{max}^k=\max_i sp_i^k,\quad IRN(NSP_i)=\left[\left[\frac{sp_i^{lL}}{sp_{max}^{uU}},\frac{sp_i^{lU}}{sp_{max}^{uL}}\right],\left[\frac{sp_i^{uL}}{sp_{max}^{lU}},\frac{sp_i^{uU}}{sp_{max}^{lL}}\right]\right]\\[4pt]sn_{max}^k=\max_i sn_i^k,\quad ratio_i^k=\frac{sn_i^k}{\text{(cross-denominator)}},\quad IRN(NSN_i)=1-\frac{IRN(SN_i)}{IRN(SN_{max})}\\[4pt]1-[[a,b],[c,d]]=[[1-d,\;1-c],[1-b,\;1-a]]
Adım 6 — IRN(AS_i) = (IRN(NSP_i) + IRN(NSN_i)) / 2; sıralama midpoint değil kesişim noktası yöntemi (Denklem 38-46). Bu veri için: S3>S4>S1>S2.

$I R N (A S_{i}) = \frac{I R N (N S P_{i}) \oplus I R N (N S N_{i})}{2}, a s_{i}^{k} = \frac{n s p_{i}^{k} + n s n_{i}^{k}}{2} (Eq.38) [6pt] R B_{i}^{u} = a s_{i}^{l U} - a s_{i}^{l L}, R B_{i}^{l} = a s_{i}^{u U} - a s_{i}^{u L}, μ_{i} = \frac{R B_{i}^{u}}{R B_{i}^{u} + R B_{i}^{l}} (Eq.39--42) [4pt] I_{i} = μ_{i} a s_{i}^{u L} + (1 - μ_{i}) a s_{i}^{l U} (Eq.43--46) [4pt] rank by I_{i} ↓; higher I_{i} \Rightarrow better rank$

LaTeX
IRN(AS_i)=\frac{IRN(NSP_i)\oplus IRN(NSN_i)}{2},\quad as_i^k=\frac{nsp_i^k+nsn_i^k}{2}\quad\text{(Eq.38)}\\[6pt]RB_i^u=as_i^{lU}-as_i^{lL},\quad RB_i^l=as_i^{uU}-as_i^{uL},\quad \mu_i=\frac{RB_i^u}{RB_i^u+RB_i^l}\quad\text{(Eq.39--42)}\\[4pt]I_i=\mu_i as_i^{uL}+(1-\mu_i)as_i^{lU}\quad\text{(Eq.43--46)}\\[4pt]\text{rank by }I_i\downarrow;\quad\text{higher }I_i\Rightarrow\text{better rank}

Sezgi

Sonucu okuma: rough-edas (IRN-EDAS) extends EDAS to handle two-interval rough numbers [[lL,lU],[uL,uU]]. ALL intermediate quantities - AV, PDA, NDA, VP, VN, SP, SN, NSP, NSN, AS - are maintained as full IRN throughout. Two distinct subtraction rules apply: cross-paired for (x−av) and same-level for (av−x). Division: [[a,b],[c,d]]/[[e,f],[g,h]]=[[a/h,b/g],[c/f,d/e]]. Final ranking converts IRN(AS_i) to crisp via boundary intersection points (Eq.39-46), not midpoint.

Varsayımlar

Decision matrix entries are valid Rough numbers/tuples
Underlying crisp method's compensation assumption holds in uncertain space
All decision-maker(s) and experts use the same linguistic/uncertainty scale

Ne zaman kullanılmaz

Classical data sufficient - use base EDAS directly (avoid unnecessary uncertainty layer)
Aggregation operator (PFWA/PFOWA/etc.) not specified - output ambiguous

Sık yapılan hatalar

Değer-uzayı ihlali: hesaplamadan önce tüm girişlerin Rough: L ≤ U; approximations defined by equivalence classes koşulunu sağladığından emin olun.
Bu yöntem sıralama için midpoint (L+U)/2 KULLANMAZ. Sıralama sınır kesişim noktaları I_i=(1-θ)·RB^li+θ·RB^ui (Denklem 39-46) ile yapılır.