ARAS Method
Additive Ratio Assessment
The Additive Ratio Assessment (ARAS) method is a multi-criteria decision-making (MCDM) technique developed in 2010 by Zavadskas and Turskis . The Additive Ratio Assessment (ARAS) method is a widely used multi-criteria decision-making (MCDM) approach that assesses alternatives relative to an ideal (optimal hypothetical) alternative. The method computes a utility degree for each alternative, and alternatives with higher utility degrees are considered more preferable and receive higher rankings.
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 |
Determine the optimal values
Construct the optimal alternative $A_0$ by selecting the best value for each criterion.
For benefit criteria:
For cost criteria:
| Metric | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| Optimal ($x_{0j}$) | 25 | 15 | 35 |
Normalize the decision matrix
Normalization depends on criterion type. The optimal alternative $A_0$ is included in the summation.
For benefit criteria:
For cost criteria, first transform :
Then normalize :
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_0$ | 0.3333 | 0.2985 | 0.2593 |
| $A_1$ | 0.1333 | 0.2239 | 0.2222 |
| $A_2$ | 0.2000 | 0.1791 | 0.2593 |
| $A_3$ | 0.3333 | 0.2985 | 0.2593 |
Weighted normalized decision matrix
Multiply each normalized value by its criterion weight.
Where :
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_0$ | 0.2426 | 0.0599 | 0.0185 |
| $A_1$ | 0.0971 | 0.0450 | 0.0158 |
| $A_2$ | 0.1456 | 0.0360 | 0.0185 |
| $A_3$ | 0.2426 | 0.0599 | 0.0185 |
Optimality function
The optimality score of each alternative is obtained by summing the weighted normalized values.
The value $S_0$ corresponds to the ideal (optimal) alternative.
| Alternative | Optimality ($S_i$) |
|---|---|
| $A_0$ | 0.3210 |
| $A_1$ | 0.1579 |
| $A_2$ | 0.2001 |
| $A_3$ | 0.3210 |
Degree utility and ranking
Compute the utility degree for each alternative.
Ranking
The alternatives are ranked in descending order of $K_i$. The alternative with the highest utility degree is considered the best.
| Alternative | Utility Degree ($K_i$) |
|---|---|
| $A_0$ | 1.0000 |
| $A_1$ | 0.4919 |
| $A_2$ | 0.6237 |
| $A_3$ | 1.0000 |
Final ranking: $A_3 \succ A_2 \succ A_1$