MAIRCA Method
Multi-Attributive Ideal-Real Comparative Analysis
MAIRCA (Multi-Attributive Ideal-Real Comparative Analysis) is an MCDM method that ranks alternatives by comparing their theoretical/ideal ratings with their real/observed ratings. The differences (gaps) are aggregated for each alternative; the alternative with the smallest total gap is preferred because it is closest to the ideal assessment.
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 |
Calculate the initial preference of alternatives
Assuming that all alternatives have equal priority, the initial preference of each alternative is :
where \(m\) is the number of alternatives.
| Alternative | Initial Preference ($P_{A_i}$) |
|---|---|
| $A_1$ | 0.3333 |
| $A_2$ | 0.3333 |
| $A_3$ | 0.3333 |
Calculate the theoretical evaluation matrix
The theoretical evaluation of alternative \(A_i\) for criterion \(C_j\) is calculated as :
Where \(w_j\) is the weight of criterion \(j\) :
The matrix \(T=[T_{ij}]\) represents the theoretical distribution of the criterion weights among the alternatives.
Since $P_{A_i}=0.3333$ for all alternatives, all rows of $T$ are identical: $T_{ij}=0.3333\times w_j$.
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.2426 | 0.0670 | 0.0237 |
| $A_2$ | 0.2426 | 0.0670 | 0.0237 |
| $A_3$ | 0.2426 | 0.0670 | 0.0237 |
Calculate the real evaluation matrix
For a benefit criterion :
For a cost criterion :
The real evaluation matrix is calculated by multiplying the theoretical rating matrix by the normalized decision matrix :
where \(R_{ij}\) is the real rating, \(T_{ij}\) is the theoretical rating, and \(r_{ij}\) is the normalized value of alternative \(A_i\) for criterion \(C_j\).
C₁: $x_1^{\max}=25,\;x_1^{\min}=10$ (benefit) · C₂: $x_2^{\max}=25,\;x_2^{\min}=15$ (cost) · C₃: $x_3^{\max}=35,\;x_3^{\min}=30$ (benefit)
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.0000 | 0.0335 | 0.0000 |
| $A_2$ | 0.0809 | 0.0000 | 0.0237 |
| $A_3$ | 0.2426 | 0.0670 | 0.0237 |
Total gap matrix
Calculate the difference between the theoretical and real ratings :
A smaller gap indicates that the real performance is closer to the theoretical ideal distribution.
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.2426 | 0.0335 | 0.0237 |
| $A_2$ | 0.1617 | 0.0670 | 0.0000 |
| $A_3$ | 0.0000 | 0.0000 | 0.0000 |
$Q_i$ values and ranking
For each alternative, sum the gaps across all criteria :
$$ Q_i=\sum_{j=1}^{n}G_{ij} $$Rank the alternatives according to their total gaps :
$$ Q_i < Q_k \quad\Longrightarrow\quad A_i >A_k $$A smaller $Q_i$ indicates the alternative is closer to the theoretical ideal. The best alternative has the minimum total gap.
| Alternative | Total Gap ($Q_i$) |
|---|---|
| $A_1$ | 0.2998 |
| $A_2$ | 0.2287 |
| $A_3$ | 0.0000 |
Final ranking: $A_3 \succ A_2 \succ A_1$
MAIRCA Algorithm Flowchart
Qi values and ranking