MOORA Method
Multi-Objective Optimization on the Basis of Ratio Analysis
MOORA (Multi-Objective Optimization on the basis of Ratio Analysis) is an MCDM method used to evaluate and rank alternatives with multiple objectives/criteria. It converts different criteria into dimensionless normalized ratios, then combines benefit and cost criteria to obtain an overall assessment.
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
MOORA uses vector normalization. Each decision-matrix value is divided by the square root of the sum of squared values for its criterion :
Where $x_{ij}^{*}$ is the normalized value.
This produces dimensionless values that can be compared across criteria with different units.
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.3244 | 0.5657 | 0.5185 |
| $A_2$ | 0.4867 | 0.7071 | 0.6049 |
| $A_3$ | 0.8111 | 0.4243 | 0.6049 |
Calculate the weighted normalized decision matrix
Multiply each normalized value by the corresponding criterion weight :
Where $w_j$ is the weight of criterion $C_j$.
The resulting matrix is the weighted normalized decision matrix :
Using weights: $w_1=0.7279$, $w_2=0.2009$, $w_3=0.0712$
| Alternative | $C_1$ (B) | $C_2$ (C) | $C_3$ (B) |
|---|---|---|---|
| $A_1$ | 0.2361 | 0.1136 | 0.0369 |
| $A_2$ | 0.3543 | 0.1421 | 0.0431 |
| $A_3$ | 0.5904 | 0.0852 | 0.0431 |
MOORA scores and ranking
Add the weighted normalized values of benefit criteria and subtract those of cost criteria :
Where $g$ is the number of benefit criteria and $n-g$ is the number of cost criteria.
Alternatives are ranked according to their MOORA scores :
| Alternative | MOORA Score ($y_i$) |
|---|---|
| $A_1$ | 0.1594 |
| $A_2$ | 0.2553 |
| $A_3$ | 0.5483 |
Final ranking: $A_3 \succ A_2 \succ A_1$