DEMATEL (Weights) Method
DEMATEL-Based Criteria Weights
DEMATEL (Weights) is an MCDM weighting method that determines the relative importance of criteria by considering their direct and indirect influence relationships. The method calculates the total influence of each criterion using the total-relation matrix, then uses its prominence to obtain normalized criterion weights.
Video Demonstration
Step-by-Step Algorithm
Construct the direct-relation matrix
Experts evaluate the degree of direct influence of criterion \(i\) on criterion \(j\). The direct-relation matrix is :
\[ X=\begin{bmatrix} 0 & x_{12} & \cdots & x_{1n}\\ x_{21} & 0 & \cdots & x_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ x_{n1} & x_{n2} & \cdots & 0 \end{bmatrix} \]
Where \(x_{ij}\) represents the direct influence of criterion \(i\) on criterion \(j\), and \(x_{ii}=0\).
0=no influence, 1=low influence, 2=medium influence, 3=high influence, 4=very high influence .
| Criterion | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $C_1$ | 0 | 3 | 2 |
| $C_2$ | 2 | 0 | 1 |
| $C_3$ | 1 | 2 | 0 |
Normalize the direct-relation matrix
The normalization factor is calculated as:
The normalized direct-relation matrix is then obtained as :
Where \(X\) is the direct-relation matrix and \(s\) is the normalization factor.
Normalization Factor Calculation:
| Metric | $C_1$ | $C_2$ | $C_3$ | Max |
|---|---|---|---|---|
| Row Sums | 5 | 3 | 3 | 5 |
| Col Sums | 3 | 5 | 3 | 5 |
Normalized Matrix ($N$):
| Criterion | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $C_1$ | 0 | 0.60 | 0.40 |
| $C_2$ | 0.40 | 0 | 0.20 |
| $C_3$ | 0.20 | 0.40 | 0 |
Calculate the total-relation matrix
The total-relation matrix includes both direct and indirect influences :
Where \(I\) is the identity matrix.
Identity Matrix ($I$):
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
Normalized Matrix ($N$):
| 0 | 0.60 | 0.40 |
| 0.40 | 0 | 0.20 |
| 0.20 | 0.40 | 0 |
Inverse Matrix $(I-N)^{-1}$:
| Criterion | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $C_1$ | 1.7969 | 1.4844 | 1.0156 |
| $C_2$ | 0.8594 | 1.7969 | 0.7031 |
| $C_3$ | 0.7031 | 1.0156 | 1.4844 |
Total-Relation Matrix ($T=N \times (I-N)^{-1}$):
| Criterion | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $C_1$ | 0.7969 | 1.4844 | 1.0156 |
| $C_2$ | 0.8594 | 0.7969 | 0.7031 |
| $C_3$ | 0.7031 | 1.0156 | 0.4844 |
Calculate ri , ci , ri + ci and ri - ci values
The sum of the columns gives \(r_i\) :
The sum of the rows of the total-relation matrix gives \(c_i\) :
The prominence of criterion \(i\) is :
The causal relation of criterion \(i\) is :
| Criterion | $r_i$ | $c_i$ | $r_i+c_i$ | $r_i-c_i$ |
|---|---|---|---|---|
| $C_1$ | 3.2969 | 2.3594 | 5.6563 | 0.9375 |
| $C_2$ | 2.3594 | 3.2969 | 5.6563 | -0.9375 |
| $C_3$ | 2.2031 | 2.2031 | 4.4063 | 0.0000 |
Calculate DEMATEL weights
First, calculate the initial weight of each criterion :
Then, normalize the weights :
The final weights satisfy :
Initial Weights ($w_i'$):
| Criterion | $w_i'$ |
|---|---|
| $C_1$ | 3.2969 |
| $C_2$ | 2.3594 |
| $C_3$ | 2.2032 |
| Sum | 7.8595 |
Final Normalized Weights ($w_i$):
| Criterion | Weight ($w_i$) |
|---|---|
| $C_1$ | 0.4194 |
| $C_2$ | 0.3002 |
| $C_3$ | 0.2803 |
| Sum | 1.0000 |
DEMATEL (Weights) Algorithm Flowchart
Calculate ri , ci , ri + ci and ri - ci values