CCSD: Criteria Correlation and Standard Deviation objective weighting

Overview

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

Strengths

• •Method-specific: Correlation-penalised standard-deviation weighting

Limitations

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

Assumptions to verify

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

When not to use

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

Edge cases

• •See F.steps and D.parameters for CCSD-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'CCSD bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix exists with measurable criteria
• •Hatalı: 'CCSD bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Sufficient inter-alternative variation per criterion
• •Hatalı: CCSD'yi 'No data variation (constant criterion) → weight degenerates' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: CCSD'yi 'Expert judgment is the actual driver → use subjective weighting' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1: Min-max normalise decision matrix to [0,1] (direction-aware). Benefit criteria: z_{ij}=(x_{ij}-x_j^min)/(x_j^max-x_j^min); cost criteria: z_{ij}=(x_j^max-x_{ij})/(x_j^max-x_j^min). Formül: 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{)} Anchor: Wang & Luo 2010, p.2 Eqs.(1)-(2)
2. 2.Adım 2 (F2): Step 2: Compute population standard deviation σ_j of each normalised column (ddof=0). This uses 1/n, NOT 1/(n-1). Formül: \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} Anchor: Wang & Luo 2010, p.3 Eq.(11)
3. 3.Adım 3 (F3): Step 3: For candidate weight vector w, compute overall assessment without criterion j: d_{ij} = Σ_{k≠j} z_{ik}·w_k. Then R_j = Pearson correlation between column z_j and vector d_j. NOTE: R_j depends on w (circular dependency: see F4). Formül: 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}} Anchor: Wang & Luo 2010, p.3 Eqs.(6)-(9)
4. 4.Adım 4 (F4): Step 4: The ideal weight formula is w_j = σ_j·√(1−R_j) / Σ_k σ_k·√(1−R_k). Because R_j(w) depends on the weights themselves, this is a nonlinear system. Solve via nonlinear optimisation: minimise J = Σ_j (w_j − σ_j·√(1−R_j(w)) / Σ_k σ_k·√(1−R_k(w)))² subject to Σw_j=1, w_j≥0. At optimum J*=0. Formül: 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 Anchor: Wang & Luo 2010, p.3 Eqs.(10)-(12)

Commonly paired with

• •CCSD + TOPSIS (high)
• •CCSD + VIKOR (high)
• •CCSD + EDAS (high)
• •CCSD + WASPAS (high)
• •CCSD + MARCOS (high)

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