CRITIC - CRiteria Importance Through Intercriteria Correlation

Overview

CRITIC assigns higher weights to criteria that have high variance (contrast) AND low correlation with other criteria (unique information). Criteria with redundant information (high positive correlation) get lower weights. It is fully data-driven - no DM input needed beyond criterion direction.

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

Objective weighting, criteria correlation analysis

How it works

1. 1

   Min-max normalisation per criterion direction. Benefit: (x−min)/(max−min); cost: (max−x)/(max−min). Result x̄_ij ∈ [0,1].

   $${\bar{x}}_{ij}=\frac{x_{ij}-{\min }_i{x}_{ij}}{{\max }_i{x}_{ij}-{\min }_i{x}_{ij}}(J^{+});\frac{{\max }_i{x}_{ij}-x_{ij}}{{\max }_i{x}_{ij}-{\min }_i{x}_{ij}}(J^{-})$$

   Diakoulaki 1995, p.765 Eq.(1)

2. 2

   Standard deviation σ_j per criterion (population variant, divide by m). Relative weights are identical with sample std (divide by m-1) because σ enters uniformly across criteria.

   $${\sigma }_j=\sqrt{\frac{1}{m}\sum _{i=1}^m({\bar{x}}_{ij}-{\overline{\stackrel{ˉ}{x}}}_j{)}^2},{\overline{\stackrel{ˉ}{x}}}_j=\frac{1}{m}\sum _i{\bar{x}}_{ij}$$

   Diakoulaki 1995, p.766 Eq.(2)

3. 3

   Pearson correlation ρ_jk between every criterion pair, computed on the normalised matrix N (NOT the raw X).

   $${\rho }_{jk}=\frac{{\sum }_i({\bar{x}}_{ij}-{\overline{\stackrel{ˉ}{x}}}_j)({\bar{x}}_{ik}-{\overline{\stackrel{ˉ}{x}}}_k)}{\sqrt{{\sum }_i({\bar{x}}_{ij}-{\overline{\stackrel{ˉ}{x}}}_j{)}^2{\sum }_i({\bar{x}}_{ik}-{\overline{\stackrel{ˉ}{x}}}_k{)}^2}}$$

   Diakoulaki 1995, p.766 Eq.(3)

4. 4

   Information amount C_j = σ_j · Σ_k (1−ρ_jk). High contrast (σ_j large) AND low correlation with other criteria (Σ(1-ρ) large) yields higher C_j.

   $$C_j={\sigma }_j\sum _{k=1}^n(1-{\rho }_{jk})$$

   Diakoulaki 1995, p.766 Eq.(4)

5. 5

   Normalised CRITIC weights w_j = C_j / Σ_k C_k; Σ w_j = 1.

   $$w_j=\frac{C_j}{{\sum }_{k=1}^n{C}_k},\sum _{j=1}^n{w}_j=1$$

   Diakoulaki 1995, p.766 Eq.(5)

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, C_j = 0, w_j = 0 - exclude constant criteria before applying CRITIC.

• Two perfectly correlated criteria: combined C reduces - CRITIC penalises redundancy.

• Small sample size (m<5): Pearson ρ becomes unreliable - interpret weights cautiously.

• Direction-aware normalisation is critical: applying CRITIC on RAW (non-normalised) matrix gives σ_j scaled by absolute units, producing scale-dependent weights.

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