Fuzzy ARAS (Turskis and Zavadskas, 2010)

Four things change; the decision logic does not.

Cells. In crisp ARAS every cell is a single, strictly positive number. Here every cell consists of three positive numbers: the lowest, most likely and highest value, a triangular fuzzy number. Weights, too, can be given as triangles. Where a crisp weight is supplied, DecisionMind embeds it in a triangle whose three components are identical (w; w; w). This is not an approximation; it is a neutral way of fitting a crisp number into the fuzzy matrix, and it adds no width. The method does not combine group decisions on its own; where multiple experts' triangles exist, combining them into a single matrix is a step taken before the analysis.

Scale equalisation. Crisp ARAS first builds the optimal alternative (the best value on every criterion), reverses cost criteria, then divides every column by its own sum. The same two steps are applied here to the triangles, separately at each corner. On a cost criterion, every triangle (lowest; most likely; highest) is reversed to (1/highest; 1/most likely; 1/lowest) once inverted. The triangle's ordered structure is preserved this way. Every column, including the optimal-alternative row, is then divided by its own triangle sum. The optimal alternative's score thus remains a single reference, but it is now a triangle.

Distance/score/aggregation. ARAS has no distance to two reference points; it is a direct weighted sum. The same holds here, except the summation is carried out on the triangles, separately at each corner. Every equalised triangle is multiplied, corner by corner, by the criterion's weight (triangular or crisp). Every row's triangles, including the optimal alternative's, are then summed, corner by corner, across the criteria. This produces a single triangular "optimality score" for every row.

Result and defuzzification. Every row's triangle sum is reduced to a single number by averaging its three corners, the centre of area. This is defuzzification, and it is carried out only at this final step. The utility degree K carries the same meaning as in crisp ARAS: the ratio of the real alternative's defuzzified score to the optimal alternative's defuzzified score. The output is again a single number and a rank. Uncertainty is carried through the calculation right up to the final step, and reduced to a single number only at the end.

DecisionMind fixes the centre-of-area defuzzification and the corner-by-corner additive aggregation. Triangles are never ranked directly at any stage; ranking is done only on the defuzzified K.

The utility degree K carries the same meaning here. The best alternative is taken as 100, and the others receive a percentage relative to it; this percentage is valid only for this set of alternatives and these weights. The difference lies here: although K appears as a single number, a three-cornered uncertainty lies beneath it. Because this uncertainty is reduced to a single number at the final step, it does not show in the report. Two alternatives' K gap can easily swap places if the input triangles are wide. The report should therefore show not only the K values but also whether the ranking holds once the weight distribution changes.

Thus instead of writing:

"According to Fuzzy ARAS, A3 is the best alternative"

the report should read:

"Relative to the optimal alternative derived from this alternative set, A3 has the highest utility degree (K=0.892); had the weights been distributed equally across the four criteria, this lead would pass to A2, so the justification for the given weight distribution must be shown separately in the report"

This extension is preferable where criterion values come from an expert's judgement or an estimate, and reducing that judgement to a single number would create an artificial precision. Examples: an estimate of investment return, criteria scored verbally such as a supplier's "service quality," and situations where multiple experts' numerical opinions need to be combined.

Opening a measured criterion (price, distance, capacity) into a triangle afterwards manufactures uncertainty rather than modelling it. DecisionMind requires a single data type; where the table is mixed, the measured criterion is also written as a triangle, with its three components equal. No information is added, only conformity to the format. Crisp ARAS's own exit condition applies just the same. If no compromise is acceptable on one criterion, this extension too is fully compensatory and will not eliminate anything below a threshold.

Breaking the triangle's order. Every cell must satisfy lowest ≤ most likely ≤ highest, and all components must be positive. The reversal operation on a cost criterion preserves this order automatically. But if the input triangle is already malformed (the most likely value entered larger than the highest, say), the result becomes silently meaningless.

Changing the defuzzification method without saying so. The centre of area ((lowest + most likely + highest) / 3) is the canonical choice but not the only one. A different defuzzification rule can change the ranking. Which rule was used must be stated in the report.

Defuzzifying first and then running crisp ARAS. In the illustrative example below, this route also puts A3 first (K=0.921), but it conceals the gap in second place. In the real fuzzy calculation, A4 (0.851) clearly overtakes A2 (0.844). But in the table defuzzified first, A2 (0.894) and A4 (0.894) become almost indistinguishable. Once the uncertainty is erased at the first step, a genuine difference buried in the middle of the calculation is lost.

Giving crisp weights and fuzzy cells without stating it. The method embeds a crisp weight as (w; w; w). This is legitimate, but the report must state that "the weights are crisp, the criterion values are fuzzy." Otherwise the reader may assume the weights, too, came from an expert judgement.

The governing principle is this:

Fuzzy ARAS exists to carry the approximation in criterion evaluation through to the ratio against the optimal alternative; any application that defuzzifies early, or does not state which defuzzification was used, conceals the method's one genuine contribution.