MABAC Method
Multi-Attributive Border Approximation Area Comparison
MABAC (Multi-Attributive Border Approximation Area Comparison) is an MCDM ranking method that evaluates alternatives by measuring their distance from the Border Approximation Area (BAA). Alternatives positioned above the border are considered more favorable, while those below it are less favorable.
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
For a benefit criterion , where a higher value is preferred :
For a cost criterion , where a lower value is preferred :
Where \(x_j^{+}\) and \(x_j^{-}\) are respectively the maximum and minimum values of criterion \(j\).
C₁: $x_1^{+}=25,\;x_1^{-}=10$ (benefit) · C₂: $x_2^{+}=25,\;x_2^{-}=15$ (cost) · C₃: $x_3^{+}=35,\;x_3^{-}=30$ (benefit)
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.0000 | 0.5000 | 0.0000 |
| $A_2$ | 0.3333 | 0.0000 | 1.0000 |
| $A_3$ | 1.0000 | 1.0000 | 1.0000 |
Weighted normalized decision matrix
Using the criterion weights \(w_j\), calculate :
Where :
The weighted matrix incorporates the relative importance of each criterion.
Using weights: $w_1=0.7279$, $w_2=0.2009$, $w_3=0.0712$
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.7279 | 0.3014 | 0.0712 |
| $A_2$ | 0.9705 | 0.2009 | 0.1424 |
| $A_3$ | 1.4558 | 0.4018 | 0.1424 |
Calculate the border approximation area (BAA)
For each criterion \(j\), calculate the geometric mean of the weighted normalized values :
The Border Approximation Area matrix is :
The BAA represents the boundary against which the alternatives are compared.
| Metric | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| BAA ($g_j$) | 1.0094 | 0.2897 | 0.1130 |
Calculate the distance from the BAA
Calculate the distance of each alternative from the border approximation area :
The matrix of distances is :
A positive \(q_{ij}\) means that the alternative is above the BAA for criterion \(j\), while a negative value means that it is below the BAA.
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | -0.2815 | 0.0117 | -0.0418 |
| $A_2$ | -0.0389 | -0.0888 | 0.0294 |
| $A_3$ | 0.4464 | 0.1121 | 0.0294 |
Total distances from the border approximation area and ranking
For each alternative, sum its distances from the BAA :
Where \(S_i\) is the overall MABAC score of alternative \(A_i\).
Rank the alternatives according to their MABAC scores :
A positive $S_i$ means the alternative is above the BAA (desirable). A negative $S_i$ means it is below (undesirable).
| Alternative | MABAC Score ($S_i$) |
|---|---|
| $A_1$ | -0.3116 |
| $A_2$ | -0.0983 |
| $A_3$ | 0.5879 |
Final ranking: $A_3 \succ A_2 \succ A_1$