Rough-SAW - SAW yönteminin Rough uzantısı

Formül adımları

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

1. Adım 1 — IRN girişlerinin maksimum tabanlı lineer normalleştirilmesi (maliyet kriterleri ters çevrilir).

   $$\begin{gathered} j\in {\Omega }_{+}:r_{ij}^{IRN}=x_{ij}^{IRN}\oslash [{max}_i{l}_{ij},{max}_i{u}_{ij}]=[l_{ij}/{max}_i{u}_{ij},u_{ij}/{max}_i{l}_{ij}]\text{(Eq.30--32)} \\ [\mathrm{4pt}]j\in {\Omega }_{-}:r_{ij}^{IRN}=[{min}_i{l}_{ij},{min}_i{u}_{ij}]\oslash x_{ij}^{IRN}=[{min}_i{l}_{ij}/u_{ij},{min}_i{u}_{ij}/l_{ij}]\text{(Eq.31,33)} \end{gathered}$$
   LaTeX

   j\in\Omega_{+}: r_{ij}^{IRN} = x_{ij}^{IRN}\oslash[\max_i l_{ij},\,\max_i u_{ij}] = \bigl[l_{ij}/\max_i u_{ij},\;u_{ij}/\max_i l_{ij}\bigr]\quad\text{(Eq.30--32)}\\[4pt]j\in\Omega_{-}: r_{ij}^{IRN} = [\min_i l_{ij},\,\min_i u_{ij}]\oslash x_{ij}^{IRN} = \bigl[\min_i l_{ij}/u_{ij},\;\min_i u_{ij}/l_{ij}\bigr]\quad\text{(Eq.31,33)}

2. Adım 2 — IRN alanında ağırlıklı toplam: S_i^IRN = ⊕_j w_j ⊗ r_ij^IRN.

   $$v_{ij}=[w_{jL}·r_{ijL},w_{jU}·r_{ijU}];S_i^{IRN}={⨁}_{j=1}^n{v}_{ij}=[\sum _j{w}_{jL}·r_{ijL},\sum _j{w}_{jU}·r_{ijU}]$$
   LaTeX

   v_{ij} = [w_{jL}\cdot r_{ijL},\;w_{jU}\cdot r_{ijU}];\quad S_i^{IRN} = \bigoplus_{j=1}^{n} v_{ij} = \Bigl[\sum_j w_{jL}\cdot r_{ijL},\;\sum_j w_{jU}\cdot r_{ijU}\Bigr]

3. Adım 3 — Bulanıklıktan arındırma ve azalan sıralama: S_i = (S_l + S_u)/2.

   $$\begin{gathered} S_i^L=\sum _j{w}_j^L·r_{ij}^L,S_i^U=\sum _j{w}_j^U·r_{ij}^U\text{(Eq.35 sums)} \\ [\mathrm{4pt}]A{V}_i={\displaystyle \frac{S_i^L+S_i^U}{2}};\text{rank by }A{V}_i\downarrow \text{(Table 10)} \end{gathered}$$
   LaTeX

   S_i^{L}=\sum_j w_j^{L}\cdot r_{ij}^{L},\quad S_i^{U}=\sum_j w_j^{U}\cdot r_{ij}^{U}\quad\text{(Eq.35 sums)}\\[4pt]AV_i=\dfrac{S_i^{L}+S_i^{U}}{2};\quad\text{rank by }AV_i\downarrow\quad\text{(Table 10)}

Sezgi

Sonucu okuma: rough-saw extends SAW to handle Rough uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Rough number (lower approximation L, upper approximation U) algebra. The final scores are defuzzified via midpoint (L+U)/2 before ranking.

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 SAW 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.
• Defuzzifikasyon yöntemi sıralamayı etkiler: midpoint (L+U)/2 kanonik seçimdir.