MOOSRA Method
Multi-Objective Optimization on the Basis of Simple Ratio Analysis
MOOSRA (Multi-Objective Optimization on the basis of Simple Ratio Analysis) is an MCDM method used to rank alternatives by calculating the ratio between the weighted performance of benefit criteria and the weighted performance of cost criteria. It is similar to MOORA but uses a simple ratio, which avoids negative performance scores.
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
MOOSRA 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 |
MOOSRA scores and ranking
$Y_i$ is the overall performance score of alternative $A_i$ :
$g$ is the number of benefit criteria. The remaining criteria $g+1,\ldots,n$ are non-benefit (cost) criteria.
Rank the alternatives according to their MOOSRA performance scores in descending order :
| Alternative | MOOSRA Score ($Y_i$) |
|---|---|
| $A_1$ | 2.4032 |
| $A_2$ | 2.7966 |
| $A_3$ | 7.4353 |
Final ranking: $A_3 \succ A_2 \succ A_1$