CCSD - Kriter Korelasyonu ve Standart Sapma Nesnel Ağırlıklandırması

Formül adımları

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

1. Adım 1 — Karar matrisini yön duyarlı min-maks ile [0,1]'e normalleştir. Fayda: z_{ij}=(x_{ij}-x_j^min)/(x_j^max-x_j^min); maliyet: z_{ij}=(x_j^max-x_{ij})/(x_j^max-x_j^min).

   $$z_{ij}=\frac{x_{ij}-x_j^{min}}{x_j^{max}-x_j^{min}}\text{ (benefit, }j\in {\Omega }_b\text{)};z_{ij}=\frac{x_j^{max}-x_{ij}}{x_j^{max}-x_j^{min}}\text{ (cost, }j\in {\Omega }_c\text{)}$$
   LaTeX

   z_{ij} = \frac{x_{ij}-x_j^{\min}}{x_j^{\max}-x_j^{\min}} \text{ (benefit, }j \in \Omega_b\text{)};\quad z_{ij} = \frac{x_j^{\max}-x_{ij}}{x_j^{\max}-x_j^{\min}} \text{ (cost, }j \in \Omega_c\text{)}

2. Adım 2 — Her normalleştirilmiş sütunun popülasyon standart sapmasını σ_j hesapla (ddof=0). Payda 1/n, 1/(n-1) DEĞİL.

   $${\sigma }_j=\sqrt{\frac{1}{n}{\sum }_{i=1}^n(z_{ij}-{\overline{z}}_j{)}^2},{\overline{z}}_j=\frac{1}{n}\sum _{i=1}^n{z}_{ij}$$
   LaTeX

   \sigma_j = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(z_{ij}-\bar{z}_j)^2}, \quad \bar{z}_j = \frac{1}{n}\sum_{i=1}^{n}z_{ij}

3. Adım 3 — Aday ağırlık vektörü w için, kriter j olmadan genel değerlendirme: d_{ij} = Σ_{k≠j} z_{ik}·w_k. Ardından R_j = z_j sütunu ile d_j vektörü arasındaki Pearson korelasyonu. NOT: R_j, w'ya bağlıdır (döngüsel bağımlılık — F4'e bak).

   $$d_{ij}=\sum _{k=1,k\ne j}^m{z}_{ik}{w}_k,R_j=\frac{{\sum }_{i=1}^n(z_{ij}-{\overline{z}}_j)(d_{ij}-{\overline{d}}_j)}{\sqrt{{\sum }_i(z_{ij}-{\overline{z}}_j{)}^2·{\sum }_i(d_{ij}-{\overline{d}}_j{)}^2}}$$
   LaTeX

   d_{ij} = \sum_{k=1,k\ne j}^{m} z_{ik}\,w_k, \qquad R_j = \frac{\sum_{i=1}^{n}(z_{ij}-\bar{z}_j)(d_{ij}-\bar{d}_j)}{\sqrt{\sum_{i}(z_{ij}-\bar{z}_j)^2 \cdot \sum_{i}(d_{ij}-\bar{d}_j)^2}}

4. Adım 4 — İdeal ağırlık formülü: w_j = σ_j·√(1−R_j) / Σ_k σ_k·√(1−R_k). R_j(w) ağırlıklara bağlı olduğundan doğrusal olmayan bir sistemdir. Doğrusal olmayan optimizasyon ile çöz: J = Σ_j (w_j − ...) karesi minimize et, kısıt: Σw_j=1, w_j≥0. Optimumda J*=0.

   $$w_j=\frac{{\sigma }_j\sqrt{1-R_j}}{{\sum }_{k=1}^m{\sigma }_k\sqrt{1-R_k}},{min}_{w}J=\sum _{j=1}^m{(w_j-\frac{{\sigma }_j\sqrt{1-R_j(w)}}{{\sum }_k{\sigma }_k\sqrt{1-R_k(w)}})}^2\text{ s.t. }\sum _j{w}_j=1,w_j\ge 0$$
   LaTeX

   w_j = \frac{\sigma_j\sqrt{1-R_j}}{\sum_{k=1}^{m}\sigma_k\sqrt{1-R_k}}, \quad \min_w J = \sum_{j=1}^{m}\!\left(w_j - \frac{\sigma_j\sqrt{1-R_j(w)}}{\sum_k \sigma_k\sqrt{1-R_k(w)}}\right)^{\!2} \text{ s.t. } \textstyle\sum_j w_j = 1,\; w_j \ge 0

Sezgi

Correlation-penalised standard-deviation weighting. Output typically weight (higher value = preferred).

Sonucu okuma: CCSD uses a nonlinear weight formula w_j = σ_j·√(1−R_j)/Σσ_k·√(1−R_k) where R_j is the Pearson correlation of criterion j with the overall assessment computed WITHOUT criterion j. This circular dependency makes it a nonlinear system requiring an iterative solver (e.g. SLSQP). CCSD is DIFFERENT from CRITIC: CRITIC uses pairwise inter-criterion correlations ρ_{jk}, while CCSD uses R_j (correlation with the partial aggregate). The paper itself (Wang & Luo 2010, p.2) states 'similar to but different from the CRITIC method'.

Varsayımlar

• Decision matrix exists with measurable criteria
• Sufficient inter-alternative variation per criterion

Ne zaman kullanılmaz

• No data variation (constant criterion) → weight degenerates
• Expert judgment is the actual driver → use subjective weighting

Sınırlılıklar

• Assumes: Decision matrix exists with measurable criteria
• Assumes: Sufficient inter-alternative variation per criterion

Sık yapılan hatalar

• Sabit kriter: σ_j = 0 → w_j = 0 (kriter hiçbir ayrım sağlamaz).