CRITIC Method
Criteria Importance Through Intercriteria Correlation
CRITIC (CRiteria Importance Through Intercriteria Correlation) is an objective weighting method for MCDM, introduced by Diakoulaki et al. in 1995. CRITIC determines criterion weights from the information contained in the decision matrix by considering both the standard deviation of each criterion and the correlation between criteria. A criterion receives a higher weight when it has high variability and low correlation with the other criteria.
Video Demonstration
Step-by-Step Algorithm
Form the decision matrix
Create the decision matrix with $m$ alternatives and $n$ criteria :
\[ X=\begin{bmatrix} x_{11} & x_{12} & \cdots & x_{1n}\\ x_{21} & x_{22} & \cdots & x_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ x_{m1} & x_{m2} & \cdots & x_{mn} \end{bmatrix} \]
\(x_{ij}\) presents the performance value of ith alternative on jth criterion, m and n are the numbers of alternatives and criteria respectively.
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 10 | 20 | 30 |
| $A_2$ | 15 | 25 | 35 |
| $A_3$ | 25 | 15 | 35 |
Normalize the decision matrix
For a benefit criterion , the normalized value is :
For a cost criterion :
\(r_{ij}\) indicates the normalized value of the decision matrix of ith alternative in jth criteria.
C₁, C₃: benefit (range 15, 5). C₂: cost (range 10).
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.0000 | 0.5000 | 0.0000 |
| $A_2$ | 0.3333 | 0.0000 | 1.0000 |
| $A_3$ | 1.0000 | 1.0000 | 1.0000 |
Construct the correlation matrix
The correlation matrix shows the relationship between each pair of criteria. It is used to measure the degree of conflict or similarity between the criteria.
The correlation matrix is :
The diagonal elements satisfy :
Values close to \(1\) indicate a strong positive correlation, while values close to \(-1\) indicate a strong negative correlation.
| Metric | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| Mean ($\overline{r}_j$) | 0.4444 | 0.5000 | 0.6667 |
Each $\tilde{r}_{jk}$ uses the means of criteria $j$ and $k$.
| $\tilde{R}$ | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $C_1$ | 1.0000 | 0.6547 | 0.7559 |
| $C_2$ | 0.6547 | 1.0000 | 0.0000 |
| $C_3$ | 0.7559 | 0.0000 | 1.0000 |
Calculate the standard deviation
The standard deviation of each criterion is calculated as :
where \(\overline{r}_j\) is the mean value of criterion \(j\).
With $m=3$, the divisor is $m-1=2$. Each $\sigma_j$ uses the criterion mean and all normalized values in that column.
| Criterion | $\sigma_j$ |
|---|---|
| $C_1$ | 0.5092 |
| $C_2$ | 0.5000 |
| $C_3$ | 0.5774 |
Calculate information quantity (\(C_{j}\))
The information quantity of criterion \(j\) is calculated by combining its standard deviation with its correlation with the other criteria:
$C_j=\sigma_j \times \Sigma(1-\tilde{r}_{jk})$. A criterion with low correlation to others carries more unique information.
| Criterion | $\Sigma(1-\tilde{r}_{jk})$ | $C_j$ |
|---|---|---|
| $C_1$ | 0.5894 | 0.3001 |
| $C_2$ | 1.3453 | 0.6727 |
| $C_3$ | 1.2441 | 0.7183 |
Weights of criteria
The weights satisfy :
The resulting objective criterion-weight vector is :
A higher \(w_j\) indicates that criterion \(j\) provides more information for distinguishing between the alternatives.
| Criterion | Weight ($w_j$) |
|---|---|
| $C_1$ | 0.1775 |
| $C_2$ | 0.3978 |
| $C_3$ | 0.4247 |
Verification: $w_1 + w_2 + w_3=0.1775 + 0.3978 + 0.4247=1.0000$ ✓