RAFSI Method
Ranking of Alternatives Through Functional Mapping of Criterion Sub-Intervals
RAFSI (Ranking of Alternatives through Functional mapping of criterion sub-intervals into a Single Interval) is a multi-criteria is an MCDM method used to normalize and rank alternatives by mapping criterion values onto a common scale while considering the relative importance of the criteria.
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.
Determine ideal values (AI) and anti-ideal values (AAI) for each criterion.
For cost criteria , the ideal value must be less than the minimum value, while the anti-ideal value must be greater than the maximum value :
For benefit criteria , the ideal value must be greater than the maximum value, while the anti-ideal value must be less than the minimum value :
\(x_{AI}\)=ideal value, and \(x_{AAI}\)=anti-ideal value.
| 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 values chosen by the decision-maker:
| Metric | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| Ideal ($x_{AIj}$) | 30 | 10 | 40 |
| Anti-ideal ($x_{AAIj}$) | 5 | 30 | 25 |
Standardized decision matrix
Transform every performance value into a common standardized interval \([n_1,n_{2k}]\) using a linear mapping function. The values \(n_1\) and \(n_{2k}\) are chosen by the decision-maker. Commonly, \(n_1=1\) and \(n_{2k}=6\).
This produces the standardized matrix \(S=[s_{ij}]\).
Using interval parameters: $n_1=1$, $n_{2k}=6$
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 2.0000 | 3.5000 | 2.6667 |
| $A_2$ | 3.0000 | 2.2500 | 4.3333 |
| $A_3$ | 5.0000 | 4.7500 | 4.3333 |
Arithmetic and harmonic means
Compute the arithmetic mean \(A\) and harmonic mean \(H\) of the endpoints of the standardized interval.
Computed from interval endpoints $n_1=1$ and $n_{2k}=6$ only.
| Metric | Value |
|---|---|
| Arithmetic mean ($A$) | 3.5000 |
| Harmonic mean ($H$) | 1.7143 |
Normalized decision matrix
Convert the standardized values into the interval \([0,1]\) using the arithmetic and harmonic means, treating benefit and cost criteria differently.
For benefit criteria :
For cost criteria :
Normalized matrix \(\hat{S}=[\hat{s}_{ij}]\).
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.2857 | 0.2449 | 0.3810 |
| $A_2$ | 0.4286 | 0.3810 | 0.6190 |
| $A_3$ | 0.7143 | 0.1805 | 0.6190 |
Criteria functions V(Ai ) and ranking
Compute the overall score of each alternative by multiplying the normalized values by the criterion weights and summing them.
\(w_j\)=weight of criterion \(j\), where \[ \sum_{j=1}^{n}w_j=1 \]
Rank the alternatives in descending order of \(V(A_i)\). The alternative with the highest score is the best.
Using given weights: $w_1=0.7279$, $w_2=0.2009$, $w_3=0.0712$
| Alternative | Score ($V(A_i)$) |
|---|---|
| $A_1$ | 0.2843 |
| $A_2$ | 0.4326 |
| $A_3$ | 0.6003 |
Final ranking: $A_3 \succ A_2 \succ A_1$