Rough-VIKOR - VIKOR yönteminin Rough uzantısı

Formül adımları

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

1. Adım 1 — Kriter başına en iyi ve en kötü crisp değerler, rough sınır uç noktalarından türetilir (defuzzifikasyon yok).

   $$\begin{gathered} f_j^{*}=\{\begin{array}{cc}{max}_i(x_{ij}^U)\hfill & j\in {\Omega }_{+}\hfill \\ {min}_i(x_{ij}^L)\hfill & j\in {\Omega }_{-}\hfill \end{array},f_j^{-}=\{\begin{array}{cc}{min}_i(x_{ij}^L)\hfill & j\in {\Omega }_{+}\hfill \\ {max}_i(x_{ij}^U)\hfill & j\in {\Omega }_{-}\hfill \end{array} \\ [\mathrm{4pt}]\text{(}{f}_j^{*},f_j^{-}\text{ are crisp scalars — extremes of rough bounds)} \end{gathered}$$
   LaTeX

   f_j^{*}=\begin{cases}\max_i(x_{ij}^{U}) & j\in\Omega_{+}\\\min_i(x_{ij}^{L}) & j\in\Omega_{-}\end{cases},\quad f_j^{-}=\begin{cases}\min_i(x_{ij}^{L}) & j\in\Omega_{+}\\\max_i(x_{ij}^{U}) & j\in\Omega_{-}\end{cases}\\[4pt]\text{(}f_j^{*},f_j^{-}\text{ are crisp scalars — extremes of rough bounds)}

2. Adım 2 — Sınır-özgü oran aritmetiği ile rough grup faydası S_i ve rough bireysel pişmanlık R_i.

   $$\begin{gathered} {\Delta }_j=|f_j^{-}-f_j^{*}|(\text{unsigned span}); \\ [\mathrm{4pt}]\text{benefit }j:{\rho }_{ij}^L=\frac{f_j^{*}-x_{ij}^U}{{\Delta }_j},{\rho }_{ij}^U=\frac{f_j^{*}-x_{ij}^L}{{\Delta }_j};\text{cost }j:{\rho }_{ij}^L=\frac{x_{ij}^L-f_j^{*}}{{\Delta }_j},{\rho }_{ij}^U=\frac{x_{ij}^U-f_j^{*}}{{\Delta }_j} \\ [\mathrm{4pt}]S_i^L=\sum _j{w}_j^L{\rho }_{ij}^L,S_i^U=\sum _j{w}_j^U{\rho }_{ij}^U \\ [\mathrm{4pt}]R_i^L={max}_j\left(w_j^L{\rho }_{ij}^L\right),R_i^U={max}_j\left(w_j^U{\rho }_{ij}^U\right) \end{gathered}$$
   LaTeX

   \Delta_j=\bigl|f_j^{-}-f_j^{*}\bigr|\;(\text{unsigned span});\\[4pt]\text{benefit }j:\;\rho_{ij}^{L}=\frac{f_j^{*}-x_{ij}^{U}}{\Delta_j},\;\rho_{ij}^{U}=\frac{f_j^{*}-x_{ij}^{L}}{\Delta_j};\quad\text{cost }j:\;\rho_{ij}^{L}=\frac{x_{ij}^{L}-f_j^{*}}{\Delta_j},\;\rho_{ij}^{U}=\frac{x_{ij}^{U}-f_j^{*}}{\Delta_j}\\[4pt]S_i^{L}=\sum_j w_j^{L}\rho_{ij}^{L},\quad S_i^{U}=\sum_j w_j^{U}\rho_{ij}^{U}\\[4pt]R_i^{L}=\max_j\bigl(w_j^{L}\rho_{ij}^{L}\bigr),\quad R_i^{U}=\max_j\bigl(w_j^{U}\rho_{ij}^{U}\bigr)

3. Adım 3 — Crisp uç noktalar S*/S^-/R*/R^- ve v parametresi (0.5) ile rough VIKOR uzlaşma indeksi Q_i^[L,U].

   $$\begin{gathered} S^{*}={min}_i{S}_i^L,S^{-}={max}_i{S}_i^U,R^{*}={min}_i{R}_i^L,R^{-}={max}_i{R}_i^U(\text{all crisp}) \\ [\mathrm{4pt}]Q_i^L=v\frac{S_i^L-S^{*}}{S^{-}-S^{*}}+(1-v)\frac{R_i^L-R^{*}}{R^{-}-R^{*}} \\ [\mathrm{4pt}]Q_i^U=v\frac{S_i^U-S^{*}}{S^{-}-S^{*}}+(1-v)\frac{R_i^U-R^{*}}{R^{-}-R^{*}};v=0.5 \end{gathered}$$
   LaTeX

   S^{*}=\min_i S_i^{L},\;S^{-}=\max_i S_i^{U},\;R^{*}=\min_i R_i^{L},\;R^{-}=\max_i R_i^{U}\;(\text{all crisp})\\[4pt]Q_i^{L}=v\,\frac{S_i^{L}-S^{*}}{S^{-}-S^{*}}+(1-v)\,\frac{R_i^{L}-R^{*}}{R^{-}-R^{*}}\\[4pt]Q_i^{U}=v\,\frac{S_i^{U}-S^{*}}{S^{-}-S^{*}}+(1-v)\,\frac{R_i^{U}-R^{*}}{R^{-}-R^{*}};\quad v=0.5

4. Adım 4 — VIKOR uzlaşma sıralaması: kabul edilebilir avantaj (C1) ve kararlılık (C2) kontrolü.

   $${\overline{Q}}_i=\frac{Q_i^L+Q_i^U}{2};DQ=\frac{1}{m-1};C1:\overline{Q}(A^{(2)})-\overline{Q}(A^{(1)})\ge DQ;C2:A^{(1)}\text{ best in }\overline{S}\text{ or }\overline{R};\text{rank by }{\overline{Q}}_i\uparrow \text{ (lower}=\text{better)};𝒞=\{A^{(1)}\}\text{ if C1 and C2; }\{A^{(1)},A^{(2)}\}\text{ if only C2 fails; otherwise include ordered }{A}^{(k)}\text{ while }Q(A^{(k)})-Q(A^{(1)})<DQ$$
   LaTeX

   \bar{Q}_i=\frac{Q_i^{L}+Q_i^{U}}{2};\quad DQ=\frac{1}{m-1};\quad C1:\bar{Q}(A^{(2)})-\bar{Q}(A^{(1)})\ge DQ;\quad C2:A^{(1)}\text{ best in }\bar{S}\text{ or }\bar{R};\quad\text{rank by }\bar{Q}_i\uparrow\text{ (lower}=\text{better)};\quad \mathcal{C}=\{A^{(1)}\}\text{ if C1 and C2; }\{A^{(1)},A^{(2)}\}\text{ if only C2 fails; otherwise include ordered }A^{(k)}\text{ while }Q(A^{(k)})-Q(A^{(1)})<DQ

Sezgi

Sonucu okuma: ROUGH-VIKOR (Zhu et al. 2015) extends VIKOR to rough [L,U] intervals. PIS/NIS (f*/f-) are crisp scalars derived from bound extremes - no defuzzification at this step. S and R are rough intervals computed via bound-specific ratio arithmetic (w^L with ratio^L, w^U with ratio^U). Rough Q [L,U] is computed component-wise using crisp extremes S*/S^-/R*/R^-. Final ranking is by Q midpoint (L+U)/2 ascending - lower Q means better rank.

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 VIKOR 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.