Rough-TOPSIS - TOPSIS yönteminin Rough uzantısı

Formül adımları

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

1. Adım 1 — Ham matris sınır uç noktalarından crisp skaler olarak PIS x⁺(j) ve NIS x⁻(j) (normalizasyon yok, ağırlıklandırma yok).

   $$\begin{gathered} x^{+}(j)=\{\begin{array}{cc}{max}_i(x_{ij}^U)\hfill & j\in B\hfill \\ {min}_i(x_{ij}^L)\hfill & j\in C\hfill \end{array},x^{-}(j)=\{\begin{array}{cc}{min}_i(x_{ij}^L)\hfill & j\in B\hfill \\ {max}_i(x_{ij}^U)\hfill & j\in C\hfill \end{array} \\ [\mathrm{4pt}]\text{(Eq.23; }{x}^{+},x^{-}\text{ are crisp scalars; B=benefit, C=cost)} \end{gathered}$$
   LaTeX

   x^+(j)=\begin{cases}\max_i(x_{ij}^U) & j\in B\\\min_i(x_{ij}^L) & j\in C\end{cases},\quad x^-(j)=\begin{cases}\min_i(x_{ij}^L) & j\in B\\\max_i(x_{ij}^U) & j\in C\end{cases}\\[4pt]\text{(Eq.23; }x^+,x^-\text{ are crisp scalars; B=benefit, C=cost)}

2. Adım 2 — Rough sapma katsayıları d⁺ ve d⁻ (crisp PIS/NIS'ten lineer mesafe, Öklid değil).

   $$\begin{gathered} \text{Benefit }j:d_{ij}^{+L}=x^{+}(j)-x_{ij}^U,d_{ij}^{+U}=x^{+}(j)-x_{ij}^L \\ {d}_{ij}^{-L}=x_{ij}^L-x^{-}(j),d_{ij}^{-U}=x_{ij}^U-x^{-}(j) \\ [\mathrm{6pt}]\text{Cost }j:d_{ij}^{+L}=x_{ij}^L-x^{+}(j),d_{ij}^{+U}=x_{ij}^U-x^{+}(j) \\ {d}_{ij}^{-L}=x^{-}(j)-x_{ij}^U,d_{ij}^{-U}=x^{-}(j)-x_{ij}^L \\ [\mathrm{4pt}]\text{(Eq.24–27; all components}\ge 0\text{; }{d}^{\pm L}\le d^{\pm U}) \end{gathered}$$
   LaTeX

   \text{Benefit }j:\;d^{+L}_{ij}=x^+(j)-x^U_{ij},\;d^{+U}_{ij}=x^+(j)-x^L_{ij}\\d^{-L}_{ij}=x^L_{ij}-x^-(j),\;d^{-U}_{ij}=x^U_{ij}-x^-(j)\\[6pt]\text{Cost }j:\;d^{+L}_{ij}=x^L_{ij}-x^+(j),\;d^{+U}_{ij}=x^U_{ij}-x^+(j)\\d^{-L}_{ij}=x^-(j)-x^U_{ij},\;d^{-U}_{ij}=x^-(j)-x^L_{ij}\\[4pt]\text{(Eq.24–27; all components}\ge 0\text{; }d^{\pm L}\le d^{\pm U})

3. Adım 3 — Rough sapma katsayılarını kriter bazında sütun maksimum üst sınırına göre normalize et.

   $$\begin{gathered} d_{ij}^{+\prime L}={\displaystyle \frac{d_{ij}^{+L}}{{max}_i(d_{ij}^{+U})}},d_{ij}^{+\prime U}={\displaystyle \frac{d_{ij}^{+U}}{{max}_i(d_{ij}^{+U})}} \\ [\mathrm{6pt}]d_{ij}^{-\prime L}={\displaystyle \frac{d_{ij}^{-L}}{{max}_i(d_{ij}^{-U})}},d_{ij}^{-\prime U}={\displaystyle \frac{d_{ij}^{-U}}{{max}_i(d_{ij}^{-U})}} \\ [\mathrm{4pt}]\text{(Eq.30–31; denominator = }{max}_i\{max(d^{\pm L},d^{\pm U})\}={max}_i(d^{\pm U})\text{ since }{d}^{\pm U}\ge d^{\pm L}) \end{gathered}$$
   LaTeX

   d^{+\prime L}_{ij}=\dfrac{d^{+L}_{ij}}{\max_i(d^{+U}_{ij})},\quad d^{+\prime U}_{ij}=\dfrac{d^{+U}_{ij}}{\max_i(d^{+U}_{ij})}\\[6pt]d^{-\prime L}_{ij}=\dfrac{d^{-L}_{ij}}{\max_i(d^{-U}_{ij})},\quad d^{-\prime U}_{ij}=\dfrac{d^{-U}_{ij}}{\max_i(d^{-U}_{ij})}\\[4pt]\text{(Eq.30–31; denominator = }\max_i\{\max(d^{\pm L},d^{\pm U})\}=\max_i(d^{\pm U})\text{ since }d^{\pm U}\ge d^{\pm L})

4. Adım 4 — Ağırlıklı rough ayrım ölçüleri S⁺ ve S⁻ (rough ağırlık × normalleştirilmiş rough sapma, bileşen bazlı).

   $$\begin{gathered} S_i^{+}=[S_i^{+L},S_i^{+U}]=\sum _j[w_j^L·d_{ij}^{+\prime L},w_j^U·d_{ij}^{+\prime U}] \\ [\mathrm{6pt}]S_i^{-}=[S_i^{-L},S_i^{-U}]=\sum _j[w_j^L·d_{ij}^{-\prime L},w_j^U·d_{ij}^{-\prime U}] \\ [\mathrm{4pt}]\text{(Eq.34–35; rough weight }{w}_j=[w_j^L,w_j^U]\text{ from rough AHP)} \end{gathered}$$
   LaTeX

   S_i^+=\bigl[S_i^{+L},\,S_i^{+U}\bigr]=\sum_j\bigl[w_j^L\cdot d^{+\prime L}_{ij},\;w_j^U\cdot d^{+\prime U}_{ij}\bigr]\\[6pt]S_i^-=\bigl[S_i^{-L},\,S_i^{-U}\bigr]=\sum_j\bigl[w_j^L\cdot d^{-\prime L}_{ij},\;w_j^U\cdot d^{-\prime U}_{ij}\bigr]\\[4pt]\text{(Eq.34–35; rough weight }w_j=[w_j^L,w_j^U]\text{ from rough AHP)}

5. Adım 5 — S⁺/S⁻'nin α-defuzzifikasyonu, yakınlık katsayısı CCᵢ ∈ [0,1], azalan sıralama.

   $$\begin{gathered} S_i^{+*}=(1-\alpha )S_i^{+L}+\alpha S_i^{+U};S_i^{-*}=\alpha S_i^{-L}+(1-\alpha )S_i^{-U} \\ [\mathrm{6pt}]C{C}_i={\displaystyle \frac{S_i^{-*}}{S_i^{-*}+S_i^{+*}}},0\le C{C}_i\le 1;\text{rank by }C{C}_i\downarrow \\ [\mathrm{4pt}]\alpha =0.5\text{ (moderate; }\alpha >0.5\text{ optimistic, }\alpha <0.5\text{ pessimistic)} \\ [\mathrm{4pt}]\text{(Eq.36–38)} \end{gathered}$$
   LaTeX

   S_i^{+*}=(1-\alpha)S_i^{+L}+\alpha S_i^{+U};\quad S_i^{-*}=\alpha S_i^{-L}+(1-\alpha)S_i^{-U}\\[6pt]CC_i=\dfrac{S_i^{-*}}{S_i^{-*}+S_i^{+*}},\quad 0\le CC_i\le 1;\quad\text{rank by }CC_i\downarrow\\[4pt]\alpha=0.5\text{ (moderate; }\alpha>0.5\text{ optimistic, }\alpha<0.5\text{ pessimistic)}\\[4pt]\text{(Eq.36–38)}

Sezgi

Sonucu okuma: ROUGH-TOPSIS (Song, Ming & Wu 2013) is NOT standard TOPSIS adapted to rough numbers. It uses a deviation-coefficient approach: (1) PIS/NIS are crisp scalars from raw matrix bound extremes (no normalization); (2) rough deviation coefficients d+/d− are linear distances (not Euclidean); (3) deviations are normalized per criterion by column max upper bound; (4) separation measures S+/S− are rough intervals from weighted normalized deviations; (5) final CC uses α-defuzzification: S+*=(1-α)S+L+αS+U, S−*=αS−L+(1-α)S−U; CC=S−*/(S−*+S+*); rank descending. α=0.5 = moderate risk propensity.

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 TOPSIS 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.
• α parametresi sıralamayı etkiler: S⁺*=(1−α)S⁺L+αS⁺U ve S⁻*=αS⁻L+(1−α)S⁻U - asimetri dikkat (S⁺ için α üst sınıra, S⁻ için alt sınıra). Varsayılan α=0.5 (ılımlı risk); α>0.5 = iyimser, α<0.5 = kötümser.