MACBETH Method
Measuring Attrness by a Categorical Based Evaluation Technique
MACBETH (Measuring Attractiveness by a Categorical Based Evaluation Technique) is an MCDM approach for constructing cardinal value functions and criterion weights from qualitative judgments. The MACBETH method is appropriate for examining and ranking alternatives with respect to a wide range of qualitative and quantitative attributes.
Video Demonstration
Step-by-Step Algorithm
Form the decision matrix
Create the decision matrix with $m$ alternatives and $n$ criteria :
\[ X=\begin{bmatrix} r_{11} & r_{12} & \cdots & r_{1n}\\ r_{21} & r_{22} & \cdots & r_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ r_{m1} & r_{m2} & \cdots & r_{mn} \end{bmatrix} \]
$r_{ij}$ presents the performance value of ith alternative on jth criterion, $m$ and $n$ are the numbers of alternatives and criteria respectively.
Converting the semantic scale into numerical scale
MACBETH uses qualitative judgments to express the difference in attractiveness between two options. The decision-maker selects one of the following semantic categories:
| Semantic scale | Benefit attribute | Cost attribute |
|---|---|---|
| No difference | 0 | 6 |
| Very weak | 1 | 5 |
| Weak | 2 | 4 |
| Moderate | 3 | 3 |
| Strong | 4 | 2 |
| Very strong | 5 | 1 |
| Extreme | 6 | 0 |
Reference levels
For each criterion \(C_j\), determine the worst and best reference levels :
Where :
- \(r_j^{-}\): worst (negative) reference level
- \(r_j^{+}\): best (positive) reference level
- \(r_{ij}\): performance of alternative \(A_i\) on criterion \(C_j\).
| Metric | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 1 | 3 | 2 |
| $A_2$ | 3 | 1 | 5 |
| $A_3$ | 6 | 6 | 5 |
| $r_j^{-}$ | 1 | 1 | 2 |
| $r_j^{+}$ | 6 | 6 | 5 |
Calculate the MACBETH scores
The MACBETH score of alternative \(A_i\) for criterion \(C_j\) is calculated as :
where the reference-level scores are :
Therefore, the MACBETH score can be simplified to :
where \(r_j^-\) is the negative (worst) reference level and \(r_j^+\) is the positive (best) reference level.
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.00 | 40.00 | 0.00 |
| $A_2$ | 40.00 | 0.00 | 100.00 |
| $A_3$ | 100.00 | 100.00 | 100.00 |
Overall scores
The overall score of alternative \(A_i\) is calculated by combining its MACBETH scores with the corresponding criterion weights :
The criterion weights satisfy :
Using weights: $w_1=0.40$, $w_2=0.35$, $w_3=0.25$
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.00 | 14.00 | 0.00 |
| $A_2$ | 16.00 | 0.00 | 25.00 |
| $A_3$ | 40.00 | 35.00 | 25.00 |
Ranking
The final ranking score of alternative \(A_i\) is calculated by summing its weighted MACBETH scores across all criteria:
Where \(V_{ij}\) is the weighted MACBETH score of alternative \(A_i\) for criterion \(C_j\).
The alternatives are ranked in descending order :
| Alternative | Total Score ($V_i$) |
|---|---|
| $A_1$ | 14.00 |
| $A_2$ | 41.00 |
| $A_3$ | 100.00 |
Final ranking: $A_3 \succ A_2 \succ A_1$