EAMR Method
Evaluation by Area-Based Method of Ranking
EAMR (Evaluation based on Average Ranking) is an MCDM method developed for ranking alternatives according to their performance across multiple criteria. The basic idea is to evaluate alternatives using their average ranking across the considered criteria, producing a final score that can be used to establish the overall ranking. EAMR provides a relatively simple way to obtain a balanced ranking of alternatives, particularly when several criteria and different weighting scenarios are involved.
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
Create the normalized decision matrix in the following way :
$$ r_{ij}=\frac{x_{ij}} {\displaystyle\max_i(x_{ij})}, \qquad i=1,2,\ldots,m,\; j=1,2,\ldots,n. $$The normalized decision matrix is
$$ R=[r_{ij}]=\begin{bmatrix} r_{11} & r_{12} & \cdots & r_{1n}\\ r_{21} & r_{22} & \cdots & r_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ r_{m1} & r_{m2} & \cdots & r_{mn} \end{bmatrix}. $$Column maxima: $\max_i(x_{i1})=25$, $\max_i(x_{i2})=25$, $\max_i(x_{i3})=35$.
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.4000 | 0.8000 | 0.8571 |
| $A_2$ | 0.6000 | 1.0000 | 1.0000 |
| $A_3$ | 1.0000 | 0.6000 | 1.0000 |
Construct the weighted normalized decision matrix
The weighted normalized decision matrix is obtained by multiplying each normalized value by the corresponding criterion weight.
$$ v_{ij}=w_j\,r_{ij}, \qquad i=1,2,\ldots,m, \quad j=1,2,\ldots,n, $$Where
- \(w_j\) is the weight of criterion \(j\),
- \(r_{ij}\) is the normalized value of alternative \(i\) under criterion \(j\).
The weighted normalized decision matrix is
$$ V=[v_{ij}]=\begin{bmatrix} v_{11} & v_{12} & \cdots & v_{1n}\\ v_{21} & v_{22} & \cdots & v_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ v_{m1} & v_{m2} & \cdots & v_{mn} \end{bmatrix}. $$| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.2912 | 0.1607 | 0.0610 |
| $A_2$ | 0.4367 | 0.2009 | 0.0712 |
| $A_3$ | 0.7279 | 0.1205 | 0.0712 |
Comprehensive distances and ranking
The score values are calculated by separating the beneficial and non-beneficial criteria.
Sum of Beneficial Criteria :
\(J^{+}\) represents the set of beneficial criteria.
Sum of Non-Beneficial Criteria :
\(J^{-}\) represents the set of non-beneficial criteria.
The score value of alternative \(i\) is calculated as :
A higher \(S_i\) indicates a better overall performance.
The alternatives are ranked according to their score values :
Benefit criteria: $C_1, C_3$. Cost criterion: $C_2$.
| Alternative | $G_i^{+}$ | $G_i^{-}$ | $S_i$ |
|---|---|---|---|
| $A_1$ | 0.3522 | 0.1607 | 2.1917 |
| $A_2$ | 0.5079 | 0.2009 | 2.5281 |
| $A_3$ | 0.7991 | 0.1205 | 6.6311 |
Final ranking: $A_3 \succ A_2 \succ A_1$