Fuzzy RAFSI

Cells. In crisp RAFSI every cell is a single number. Here every cell carries three numbers: lowest, most likely, highest. Weights, too, are requested as triangles. If a crisp weight is to be given, its three components are written identically. As noted above, the ideal and anti-ideal points are not separately requested from the decision-maker here. These points come from the alternative set itself, from the column's most extreme triangle components. The base RAFSI card's principle that "reference points must be independent of the alternative set" does not apply in this fuzzy version.

Scale equalisation. Each triangle's three components are mapped linearly, separately, onto the range 1 to 6, according to the end values of their own criterion's column. On a benefit criterion, the column's largest component maps to 6, the smallest to 1. On a cost criterion, the direction is reversed: the smallest value moves to 6, the largest to 1. This mapping follows the same logic as crisp RAFSI's. The only difference is that the three components are mapped separately, by the same rule.

Weighting and aggregation. The mapped triangle is multiplied, component by component, by the same criterion's triangular weight. Then the weighted values on the benefit criteria are averaged arithmetically. Those on the cost criteria are averaged harmonically. The average of these two gives each alternative a triangular score. Crisp RAFSI defines this step as dividing by the fixed scale interval's own averages. Here, instead of division, the arithmetic and harmonic average is taken directly over the weighted values.

Result and defuzzification. This triangular score is reduced to a single crisp number by the average of its three components, that is, by the centroid. Ranking is done on this number. In crisp RAFSI the score is generally kept between 0 and 1 because it is divided by the fixed scale interval. Here, because there is no division step, the score can move outside this range to visibly larger values. What should be read is not the magnitude of the score, but the ranking and gap among the alternatives.

DecisionMind holds fixed, in this fuzzy RAFSI, the 1-to-6 scale and the weighted arithmetic-harmonic combination.

The score states where the alternative stands relative to the others in this table. Crisp RAFSI's most important promise is this: because the reference points are fixed, different analyses can be compared, and adding a new alternative does not disturb the existing ranking. This promise does not hold for the fuzzy version, because the reference points derive from the alternative set itself. In this respect the score's interpretation is closer to Fuzzy TOPSIS's. It is a ranking only within this alternative set, and must be recalculated whenever an alternative is added or removed.

So instead of:

"Fuzzy RAFSI was used, so the ranking will not be disturbed even if a new supplier is added, because the reference points are fixed"

the report should read:

"The ideal and anti-ideal ends for these three alternatives have been taken from the best and worst values among themselves. If a fourth alternative is added, these ends may change too, and the ranking must be recalculated"

Fuzzy RAFSI can be used if the criterion scores are not measured but come from expert judgement or approximate estimation. To use it, you need to accept, knowingly, that this version of DecisionMind derives the ideal and anti-ideal points from the alternative set. Mapping onto the 1-to-6 range and the weighted arithmetic-harmonic combination offer a calculation route different from TOPSIS. If you are expecting extra robustness against rank reversal, that expectation is not currently met by this fuzzy version. The robustness on the base RAFSI card holds only with fixed, user-defined reference points.

If no compromise is acceptable on a criterion, the exit condition on the base RAFSI card applies just the same: the method is compensatory and is not suitable in that case.

Breaking the triangle's l ≤ m ≤ u order. The mapping onto 1 to 6 reverses the direction on a cost criterion. If this step is skipped, the triangle falls into an undefined shape.

Claiming robustness against rank reversal. As explained above, in this fuzzy version the ideal and anti-ideal derive from the alternative set. The claim "the reference points are fixed" does not hold here.

Assuming the score is a ratio between 0 and 1. Because there is no scale division before defuzzification, the score can exceed 1.

Giving crisp weights but fuzzy scores. The method expects weights to be triangular too. If a crisp weight is to be given, its three components are written identically, and this is stated in the report.

The governing principle is this:

Fuzzy RAFSI carries crisp RAFSI's idea of mapping onto a 1-6 scale into fuzzy numbers. But because the reference points derive from the alternative set in this version, crisp RAFSI's promise of robustness against rank reversal does not hold here. The report must state this explicitly.