Grey-VIKOR - VIKOR yönteminin Grey uzantısı

Formül adımları

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

1. Adım 1 — Gri karar matrisini normalize et; beyazlat: v̂_ij = ½(v_ij^L + v_ij^U). En iyi/kötü beyazlatılmış değerler.

   $$\begin{gathered} \otimes v_{ij}=[v_{ij}^L,v_{ij}^U]; \\ {\widehat{v}}_{ij}=\frac{1}{2}(v_{ij}^L+v_{ij}^U); \\ {\widehat{v}}_j^{*}={max}_i{\widehat{v}}_{ij}(\text{benefit}); \\ {\widehat{v}}_j^{-}={min}_i{\widehat{v}}_{ij} \end{gathered}$$
   LaTeX

   \otimes v_{ij}=[v_{ij}^L,v_{ij}^U];\\ \hat{v}_{ij}=\tfrac{1}{2}(v_{ij}^L+v_{ij}^U);\\ \hat{v}_j^*=\max_i\hat{v}_{ij}\;(\text{benefit});\\ \hat{v}_j^-=\min_i\hat{v}_{ij}

2. Adım 2-3 — Normalleştirilmiş boşluk: f_ij = (v̂_j* - v̂_ij)/(v̂_j* - v̂_j-). Fayda S_i ve pişmanlık R_i.

   $$\begin{gathered} f_{ij}=\frac{{\widehat{v}}_j^{*}-{\widehat{v}}_{ij}}{{\widehat{v}}_j^{*}-{\widehat{v}}_j^{-}}; \\ {S}_i=\sum _{j=1}^n{w}_j{f}_{ij}; \\ {R}_i={max}_j(w_j{f}_{ij}) \end{gathered}$$
   LaTeX

   f_{ij}=\frac{\hat{v}_j^*-\hat{v}_{ij}}{\hat{v}_j^*-\hat{v}_j^-};\\ S_i=\sum_{j=1}^n w_j f_{ij};\\ R_i=\max_j(w_j f_{ij})

3. Adım 4-5 — Uzlaşı indeksi Q_i = v·(S_i-S*)/(S⁻-S*) + (1-v)·(R_i-R*)/(R⁻-R*), v=0.5. Artan sırayla sırala; C1 ve C2 koşullarını kontrol et.

   $$\begin{gathered} S^{*}={min}_i{S}_i,S^{-}={max}_i{S}_i;R^{*}={min}_i{R}_i,R^{-}={max}_i{R}_i; \\ {Q}_i=v\frac{S_i-S^{*}}{S^{-}-S^{*}}+(1-v)\frac{R_i-R^{*}}{R^{-}-R^{*}}; \\ \text{rank ascending by }{Q}_i;𝒞=\{A^{(1)}\}\text{ if C1 and C2; }\{A^{(1)},A^{(2)}\}\text{ if only C2 fails; otherwise include ordered }{A}^{(k)}\text{ while }Q(A^{(k)})-Q(A^{(1)})<DQ \end{gathered}$$
   LaTeX

   S^*=\min_i S_i,\ S^-=\max_i S_i;\ R^*=\min_i R_i,\ R^-=\max_i R_i;\\ Q_i=v\frac{S_i-S^*}{S^--S^*}+(1-v)\frac{R_i-R^*}{R^--R^*};\\ \text{rank ascending by }Q_i;\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

Grey outranking/ranking - Grey Interval Number (GIN: [x̲, x̄]). Output typically utility (higher value = preferred).

Sonucu okuma: grey-vikor extends VIKOR to handle Grey uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Grey Interval Number (GIN: [x̲, x̄]) algebra. The final scores are defuzzified via whitenisation: (x̲ + x̄)/2 before ranking.

Varsayımlar

• Decision matrix entries are valid Grey 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

Sınırlılıklar

• Rank reversal known on alternative-set changes (ref: inherited from crisp base; cf. Belton-Gear 1983, Wang-Luo 2009)
• Assumes: Decision matrix entries are valid Grey 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 GIN: x̲ ≤ x̄ (lower and upper bounds of interval) koşulunu sağladığından emin olun.
• Defuzzifikasyon yöntemi sıralamayı etkiler: whitenisation: (x̲ + x̄)/2 kanonik seçimdir.