GRA Method
Grey Relational Analysis
Grey Relational Analysis (GRA) is a prominent Multi-Criteria Decision-Making (MCDM) method rooted in Grey System Theory, originally developed by Deng Julong . It evaluates and ranks alternatives by measuring the degree of similarity and relationship between actual data sequences and an ideal reference sequence.
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 , where a higher value is better :
For a non-benefit criterion , where a lower value is better :
After normalization, larger values indicate better performance.
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 |
Deviation sequence
The ideal reference sequence is defined as :
The deviation between alternative \(A_i\) and the reference sequence for criterion \(j\) is :
Since \(x_{0j}=1\), this can be written as :
The minimum and maximum deviations are :
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 1.0000 | 0.5000 | 1.0000 |
| $A_2$ | 0.6667 | 1.0000 | 0.0000 |
| $A_3$ | 0.0000 | 0.0000 | 0.0000 |
Grey relational coefficient
The grey relational coefficient measures the closeness of each alternative to the reference sequence :
where \(\delta\) is the distinguishing coefficient, usually :
A commonly used value is:
A higher \(\xi_{ij}\) indicates that alternative \(A_i\) is closer to the ideal reference for criterion \(j\).
Using $\delta=0.5$, $\Delta_{\min}=0$, $\Delta_{\max}=1$. Each $\xi_{ij}$ relates to its deviation cell.
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.3333 | 0.5000 | 0.3333 |
| $A_2$ | 0.4286 | 0.3333 | 1.0000 |
| $A_3$ | 1.0000 | 1.0000 | 1.0000 |
Grey relational grade and ranking
The grey relational grade aggregates the grey relational coefficients of all criteria :
where \(w_j\) is the weight of criterion \(j\), satisfying :
If all criteria have equal weights, the grey relational grade becomes :
The alternatives are ranked according to their grey relational grades :
| Alternative | $\Gamma_i$ |
|---|---|
| $A_1$ | 0.3668 |
| $A_2$ | 0.4502 |
| $A_3$ | 1.0000 |
Final ranking: $A_3 \succ A_2 \succ A_1$