MARCOS Method
Measurement Alternatives and Ranking According to Compromise Solution
MARCOS (Measurement of Alternatives and Ranking according to COmpromise Solution) is an MCDM method that ranks alternatives by evaluating their relationship with the ideal and anti-ideal solutions. It calculates a utility degree for each alternative and obtains a compromise ranking.
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 |
Ideal and anti-ideal solutions
Add the ideal alternative \(AI\) and anti-ideal alternative \(AAI\) to the decision matrix :
For a benefit criterion :
For a cost criterion :
C₁ and C₃ are benefit criteria, C₂ is a cost criterion.
| Solution | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $AAI$ | 10 | 25 | 30 |
| $AI$ | 25 | 15 | 35 |
Normalize the extended decision matrix
For a benefit criterion :
For a cost criterion :
where \(x_{AI,j}\) is the ideal value of criterion \(j\).
C₁ and C₃ use benefit normalization. C₂ uses cost normalization (divides ideal by value). The matrix includes $AAI$ and $AI$ rows.
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $AAI$ | 0.4000 | 0.6000 | 0.8571 |
| $A_1$ | 0.4000 | 0.7500 | 0.8571 |
| $A_2$ | 0.6000 | 0.6000 | 1.0000 |
| $A_3$ | 1.0000 | 1.0000 | 1.0000 |
| $AI$ | 1.0000 | 1.0000 | 1.0000 |
Calculate the weighted normalized matrix
Multiply each normalized value by the corresponding criterion weight :
Where :
Using weights from the Entropy method: $w_1=0.7279$, $w_2=0.2009$, $w_3=0.0712$
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $AAI$ | 0.2912 | 0.1205 | 0.0610 |
| $A_1$ | 0.2912 | 0.1507 | 0.0610 |
| $A_2$ | 0.4367 | 0.1205 | 0.0712 |
| $A_3$ | 0.7279 | 0.2009 | 0.0712 |
| $AI$ | 0.7279 | 0.2009 | 0.0712 |
Calculate Si values
\(S_i\) is the sum of the weighted normalized values :
| Alternative | $S_i$ |
|---|---|
| $AAI$ | 0.4727 |
| $A_1$ | 0.5029 |
| $A_2$ | 0.6284 |
| $A_3$ | 1.0000 |
| $AI$ | 1.0000 |
Calculate the utility degree
Calculate the utility degree of each alternative with respect to the anti-ideal and ideal solutions :
| Alternative | $K_i^-$ | $K_i^+$ |
|---|---|---|
| $AAI$ | 1.0000 | 0.4727 |
| $A_1$ | 1.0639 | 0.5029 |
| $A_2$ | 1.3295 | 0.6284 |
| $A_3$ | 2.1155 | 1.0000 |
| $AI$ | 2.1155 | 1.0000 |
Utility function and ranking
The utility function of alternative \(A_i\) is :
Where the utility functions related to the ideal and anti-ideal solutions are :
Rank the alternatives according to their final utility function values :
| Alternative | $f(K_i^+)$ | $f(K_i^-)$ | $f(K_i)$ |
|---|---|---|---|
| $AAI$ | 0.3210 | 0.6790 | 0.4104 |
| $A_1$ | 0.3210 | 0.6790 | 0.4366 |
| $A_2$ | 0.3210 | 0.6790 | 0.5456 |
| $A_3$ | 0.3210 | 0.6790 | 0.8682 |
| $AI$ | 0.3210 | 0.6790 | 0.8682 |
Final ranking: $A_3 \succ A_2 \succ A_1$
MARCOS Algorithm Flowchart
Calculate Si values