ENTROPY Method
Entropy Weighting Method
Entropy Weighting Method is an objective weighting method in MCDM used to determine the relative importance of criteria based on the amount of information and variability contained in the decision matrix. A criterion with greater variation among alternatives contains more useful information and receives a higher weight, while a criterion with less variation receives a lower weight.
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
The normalized value is calculated as :
\(r_{ij}\) indicates the normalized value of the decision matrix of ith alternative in jth criteria.
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.2000 | 0.3333 | 0.3000 |
| $A_2$ | 0.3000 | 0.4167 | 0.3500 |
| $A_3$ | 0.5000 | 0.2500 | 0.3500 |
Calculate the Entropy value
The entropy value of criterion \(j\) is calculated as :
where:
If \(r_{ij}=0\), the term \(r_{ij}\ln(r_{ij})\) is considered equal to \(0\).
The entropy value generally satisfies :
| Metric | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| Entropy ($e_j$) | 0.9366 | 0.9825 | 0.9938 |
Weights of criteria
The degree of diversification of criterion \(j\) is :
The entropy weight of criterion \(j\) is calculated as :
The resulting weights satisfy :
| Metric | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| Diversification ($d_j$) | 0.0634 | 0.0175 | 0.0062 |
| Weight ($w_j$) | 0.7279 | 0.2009 | 0.0712 |