Rough-ARAS - ARAS yönteminin Rough uzantısı

Formül adımları

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

1. Adım 1 — Optimal alternatif A_0 (her kriterin en iyisi) eklenmiş matris.

   $$X_{0j}^{IRN}=\{\begin{array}{cc}{max}_i\mathrm{I}\mathrm{R}\mathrm{N}(x_{ij})\hfill & j\in {\Omega }_{+}\hfill \\ {min}_i\mathrm{I}\mathrm{R}\mathrm{N}(x_{ij})\hfill & j\in {\Omega }_{-}\hfill \end{array},\mathrm{I}\mathrm{R}\mathrm{N}(x_{ij})=[RN(I_{li}^{\prime }),RN(I_{ui}^{\prime })]$$
   LaTeX

   X_{0j}^{IRN} = \begin{cases}\max_i \mathrm{IRN}(x_{ij}) & j\in\Omega_{+}\\\min_i \mathrm{IRN}(x_{ij}) & j\in\Omega_{-}\end{cases},\quad \mathrm{IRN}(x_{ij}) = [RN(I'_{li}), RN(I'_{ui})]

2. Adım 2 — Doğrusal-toplam normalleştirme (maliyet kriterleri önce ters çevrilir).

   $$\begin{gathered} {\overline{x}}_{ij}^{IRN}=\mathrm{I}\mathrm{R}\mathrm{N}(x_{ij})\oslash \sum _{k=0}^m\mathrm{I}\mathrm{R}\mathrm{N}(x_{kj}),j\in {\Omega }_{+} \\ [\mathrm{4pt}]j\in {\Omega }_{-}:\mathrm{I}\mathrm{R}\mathrm{N}(x_{ij})\leftarrow [\mathrm{R}\mathrm{N}(I{\backslash }_{li}^{\prime }{)}^{-1}{]}^{\mathrm{I}\mathrm{R}\mathrm{N}}=[1/\mathrm{R}\mathrm{N}(I{\backslash }_{ui}^{\prime }),1/\mathrm{R}\mathrm{N}(I{\backslash }_{li}^{\prime })]\text{(IRN inverse: swap bounds to maintain }L\le U\text{; then normalize as benefit)} \end{gathered}$$
   LaTeX

   \bar{x}_{ij}^{IRN} = \mathrm{IRN}(x_{ij}) \oslash \sum_{k=0}^{m} \mathrm{IRN}(x_{kj}),\quad j\in\Omega_{+}\\[4pt]j\in\Omega_{-}:\;\mathrm{IRN}(x_{ij})\leftarrow\bigl[\mathrm{RN}(I\'_{li})^{-1}\bigr]^{\mathrm{IRN}}=\bigl[1/\mathrm{RN}(I\'_{ui}),\;1/\mathrm{RN}(I\'_{li})\bigr]\quad\text{(IRN inverse: swap bounds to maintain }L\le U\text{; then normalize as benefit)}

3. Adım 3 — Ağırlıklı normalleştirilmiş matris d_ij = w_j · x̄_ij.

   $$d_{ij}^{IRN}=w_j\otimes {\overline{x}}_{ij}^{IRN}=[w_j·\mathrm{R}\mathrm{N}(I_{li}^{\prime }),w_j·\mathrm{R}\mathrm{N}(I_{ui}^{\prime })]$$
   LaTeX

   d_{ij}^{IRN} = w_j \otimes \bar{x}_{ij}^{IRN} = [w_j \cdot \mathrm{RN}(I'_{li}), w_j \cdot \mathrm{RN}(I'_{ui})]

4. Adım 4 — Optimallık fonksiyonu S_i = Σ d_ij (S_0 satırı dahil).

   $$S_i^{IRN}={⨁}_{j=1}^n{d}_{ij}^{IRN}=[{\sum }_{j}R{N}_l(d_{ij}),{\sum }_{j}R{N}_u(d_{ij})],i=0,1,\dots ,m$$
   LaTeX

   S_i^{IRN} = \bigoplus_{j=1}^{n} d_{ij}^{IRN} = \left[\sum_j RN_l(d_{ij}), \sum_j RN_u(d_{ij})\right],\quad i=0,1,\ldots,m

5. Adım 5 — Fayda K_i = S_i / S_0 ve azalan sıralama.

   $$K_i=\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{u}\mathrm{z}\mathrm{z}(S_i^{IRN}\oslash S_0^{IRN}),\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{u}\mathrm{z}\mathrm{z}([R{N}_l,R{N}_u])=({\overline{RN}}_l+{\overline{RN}}_u)/2;\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\mathrm{b}\mathrm{y}{K}_i\downarrow$$
   LaTeX

   K_i = \mathrm{defuzz}(S_i^{IRN} \oslash S_0^{IRN}),\quad \mathrm{defuzz}([RN_l, RN_u]) = (\bar{RN}_l + \bar{RN}_u)/2;\quad \mathrm{rank\ by\ }K_i\downarrow

Sezgi

Rough outranking/ranking - Rough number (lower approximation L, upper approximation U). Output typically utility (higher value = preferred).

Sonucu okuma: ROUGH-ARAS (IRN-ELH-ARAS) extends ARAS to IRN uncertainty. Step 1 (F1): define optimal baseline X_0 via element-wise max/min. Step 2 (F2): normalise using ARAS sum-based formula; for cost criteria invert IRN as [1/U,1/L] (swap bounds) before summing. Step 3 (F3): weight normalised values with crisp or rough weights. Step 4 (F4): compute rough row sums S_i^{IRN}. Step 5 (F5): utility degree K_i = defuzz(S_i^{IRN} ⊘ S_0^{IRN}) - rough IRN division first, then midpoint defuzz. Rank descending by K_i.

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 ARAS directly (avoid unnecessary uncertainty layer)
• Aggregation operator (PFWA/PFOWA/etc.) not specified - output ambiguous

Sınırlılıklar

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

Sık yapılan hatalar

• Değer-uzayı ihlali: hesaplamadan önce tüm girişlerin Rough: L ≤ U; approximations defined by equivalence classes koşulunu sağladığından emin olun.
• Defuzzifikasyon yöntemi sıralamayı etkiler: midpoint (L+U)/2 kanonik seçimdir.