Grey-TOPSIS - TOPSIS yönteminin Grey uzantısı

Formül adımları

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

1. Adım 1-2 — Gri karar matrisi ⊗a_ij=[a_ij^L, a_ij^U] oluştur ve normalleştir: fayda ⊗v_ij=[a_ij^L/max(a_ij^U), a_ij^U/max(a_ij^U)]; maliyet ⊗v_ij=[min(a_ij^L)/a_ij^U, min(a_ij^L)/a_ij^L].

   $$\begin{gathered} \otimes a_{ij}=[a_{ij}^L,a_{ij}^U]; \\ \text{benefit: }\otimes v_{ij}=[\frac{a_{ij}^L}{{max}_k{a}_{kj}^U},\frac{a_{ij}^U}{{max}_k{a}_{kj}^U}];\text{cost: }\otimes v_{ij}=[\frac{{min}_k{a}_{kj}^L}{a_{ij}^U},\frac{{min}_k{a}_{kj}^L}{a_{ij}^L}] \end{gathered}$$
   LaTeX

   \otimes a_{ij}=[a_{ij}^L,a_{ij}^U];\\ \text{benefit: } \otimes v_{ij}=\Bigl[\frac{a_{ij}^L}{\max_k a_{kj}^U},\frac{a_{ij}^U}{\max_k a_{kj}^U}\Bigr]; \quad \text{cost: } \otimes v_{ij}=\Bigl[\frac{\min_k a_{kj}^L}{a_{ij}^U},\frac{\min_k a_{kj}^L}{a_{ij}^L}\Bigr]

2. Adım 3 — Ağırlıklı normalleştirilmiş gri matris: ⊗ṽ_ij = w_j·⊗v_ij = [w_j·v_ij^L, w_j·v_ij^U]. Sonra beyazlatma: v̂_ij = ½(ṽ_ij^L + ṽ_ij^U).

   $$\begin{gathered} \otimes {\tilde{v}}_{ij}=w_j·\otimes v_{ij}=[w_j{v}_{ij}^L,w_j{v}_{ij}^U]; \\ {\widehat{\tilde{v}}}_{ij}=\frac{1}{2}(w_j{v}_{ij}^L+w_j{v}_{ij}^U) \end{gathered}$$
   LaTeX

   \otimes\tilde{v}_{ij}=w_j\cdot\otimes v_{ij}=[w_j v_{ij}^L,\; w_j v_{ij}^U];\\ \hat{\tilde{v}}_{ij}=\tfrac{1}{2}(w_j v_{ij}^L+w_j v_{ij}^U)

3. Adım 4 — Gri PIS/NIS: ⊗A+ = {max(ṽ_ij^L), max(ṽ_ij^U)}; ⊗A- = {min(ṽ_ij^L), min(ṽ_ij^U)}. Beyazlatılmış: v̂_j+ = max(v̂_ij); v̂_j- = min(v̂_ij).

   $$\begin{gathered} {\widehat{v}}_j^{+}={max}_i{\widehat{\tilde{v}}}_{ij}(\text{benefit}); \\ {\widehat{v}}_j^{-}={min}_i{\widehat{\tilde{v}}}_{ij}(\text{benefit}) \end{gathered}$$
   LaTeX

   \hat{v}_j^+=\max_i\hat{\tilde{v}}_{ij}\;(\text{benefit});\\ \hat{v}_j^-=\min_i\hat{\tilde{v}}_{ij}\;(\text{benefit})

4. Adım 5 — Beyazlatılmış mesafelerle gri Öklid ayrım ölçüleri: d_i+ = √Σ_j w_j(v̂_ij - v̂_j+)²; d_i- = √Σ_j w_j(v̂_ij - v̂_j-)².

   $$\begin{gathered} d_i^{+}=\sqrt{{\sum }_{j=1}^n{w}_j({\widehat{\tilde{v}}}_{ij}-{\widehat{v}}_j^{+}{)}^2}; \\ {d}_i^{-}=\sqrt{{\sum }_{j=1}^n{w}_j({\widehat{\tilde{v}}}_{ij}-{\widehat{v}}_j^{-}{)}^2} \end{gathered}$$
   LaTeX

   d_i^+=\sqrt{\sum_{j=1}^{n} w_j(\hat{\tilde{v}}_{ij}-\hat{v}_j^+)^2};\\ d_i^-=\sqrt{\sum_{j=1}^{n} w_j(\hat{\tilde{v}}_{ij}-\hat{v}_j^-)^2}

5. Adım 6 — Yakınlık katsayısı CC_i = d_i- / (d_i+ + d_i-) ∈ [0,1]. Azalan sırayla sırala.

   $$\begin{gathered} C{C}_i=\frac{d_i^{-}}{d_i^{+}+d_i^{-}},C{C}_i\in [0,1]; \\ \text{rank descending} \end{gathered}$$
   LaTeX

   CC_i=\frac{d_i^-}{d_i^++d_i^-},\quad CC_i\in[0,1];\\ \text{rank descending}

Sezgi

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

Sonucu okuma: grey-topsis extends TOPSIS 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 TOPSIS 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.