COPRAS Method
Complex Proportional Assessment Assessment
COPRAS (COmplex PRoportional ASsessment) is an MCDM method introduced by Zavadskas et al. in 1994. It evaluates alternatives by considering the maximizing and minimizing criteria separately and determining their relative significance.
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.
Column sums: $\sum x_{i1}=50$, $\sum x_{i2}=60$, $\sum x_{i3}=100$.
| 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 |
Construct the weighted normalized decision matrix
The normalized decision matrix is multiplied by the corresponding criterion weights.
$$ v_{ij}=w_j r_{ij} $$Where \(w_j\) represents the weight of criterion \(j\), and
$$ \sum_{j=1}^{n}w_j=1. $$| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.1456 | 0.0670 | 0.0214 |
| $A_2$ | 0.2184 | 0.0837 | 0.0249 |
| $A_3$ | 0.3640 | 0.0502 | 0.0249 |
Sums of weighted normalized values
For beneficial criteria, calculate :
$$ S_+i=\sum_{j=1}^{k}v_{ij} $$where \(k\) represents the number of beneficial criteria.
For non-beneficial (cost) criteria, calculate :
$$ S_-i=\sum_{j=k+1}^{n}v_{ij} $$\(n-k\) represents the number of non-beneficial criteria.
Benefit criteria: $C_1, C_3$. Cost criterion: $C_2$.
| Alternative | $S_{+i}$ | $S_{-i}$ |
|---|---|---|
| $A_1$ | 0.1670 | 0.0670 |
| $A_2$ | 0.2433 | 0.0837 |
| $A_3$ | 0.3889 | 0.0502 |
Relative significances of the alternatives, quantitative utility and ranking
The relative significance \(Q_i\) is calculated as :
$$ Q_i=S_+i + \frac{ \displaystyle\sum_{i=1}^{m}S_-i}{ S_-i \displaystyle\sum_{i=1}^{m}\frac{1}{S_-i}} $$The utility degree of each alternative is calculated relative to the best alternative :
$$ U_i=\frac{Q_i}{Q_{\max}}\times100\% $$where \(Q_{\max}=\max_i Q_i\).
The alternatives are ranked according to their relative significance \(Q_i\) or utility degree \(U_i\).
$$ Q_i>Q_k \quad\Longrightarrow\quad A_i >A_k $$| Alternative | $Q_i$ | $U_i$ |
|---|---|---|
| $A_1$ | 0.2311 | 48.72% |
| $A_2$ | 0.2946 | 62.11% |
| $A_3$ | 0.4744 | 100.00% |
Utility degrees: $U_3=100.00\% >U_2=62.11\% >U_1=48.72\%$
Final ranking: $A_3 \succ A_2 \succ A_1$