CCSD - Criteria Correlation and Standard Deviation objective weighting

Overview

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'.

Output

Weight, higher is better

Data

Crisp, complete numeric matrix

Weights

Derived internally, no weight source needed

Size

2+ alternatives, 3-12 criteria works best

Used for

Any (objective weighting)

How it works

1. 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).

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

   Wang & Luo 2010, p.2 Eqs.(1)-(2)

2. 2

   Compute population standard deviation σ_j of each normalised column (ddof=0). This uses 1/n, NOT 1/(n-1).

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

   Wang & Luo 2010, p.3 Eq.(11)

3. 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).

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

   Wang & Luo 2010, p.3 Eqs.(6)-(9)

4. 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.

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

   Wang & Luo 2010, p.3 Eqs.(10)-(12)

Look elsewhere when

Assumptions to verify

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

Edge cases and pitfalls

• Constant criterion: σ_j = 0 → w_j = 0 (criterion provides no discrimination - safely excludes it from weighting).

Works with

How to cite