Fuzzy SWARA (Vrtagić et al., 2021)

Three things change; the decision logic does not.

Cells. In crisp SWARA, each criterion's relative importance difference from the one before it (s_j) is a single number. Here this difference is a triangular fuzzy number (lower, middle, upper); an expert can say "20-25 per cent less important, most likely 22." The scale is converted into a triangle beforehand from pre-declared verbal terms; for example, "equally important" = (0;0;0), "moderately less important," "weakly less important." The most important criterion's difference value is, as in crisp SWARA, always zero (here (0;0;0)).

Calculating the coefficient and the intermediate weight. In crisp SWARA, the coefficient (s_j+1) and the intermediate weight (the previous one divided by the coefficient) proceed with single numbers. Here the same operations run with triangular fuzzy arithmetic: the coefficient triangle is found by adding 1 to each component of s̃_j, and the intermediate weight by dividing the previous intermediate weight by this coefficient in triangular form. Triangular division, unlike crisp division, requires dividing the lower bound by the upper bound and the upper bound by the lower bound; this explains why the width grows.

Normalisation and defuzzification. In crisp SWARA, the intermediate weights are divided by their sum to give the final weight directly. Here, every intermediate weight triangle is first divided by its own sum to give a fuzzy final weight, and this triangle is then reduced to a single number by a weighted average (a graded mean, dividing l+4m+u by 6); the total is then renormalised to 1. Uncertainty is consumed at the last step, not the first.

DecisionMind fixes this defuzzification rule (the graded mean) and the triangular-division definition for classical Fuzzy SWARA.

The weight, as in base SWARA, shows a criterion's relative importance to the others within the given ranking and set of successive comparisons; the weights always sum to 1. No version of SWARA carries an independent consistency indicator; the fuzzy version does not fill this gap either.

The difference is here. The weight has been defuzzified from a triangular fuzzy chain of successive comparisons, and the defuzzified single number conceals this approximation. The weight gap between the most important criterion and the second depends directly on "how much" less important the second criterion was judged than the first, that is, on the width of the triangle.

Thus instead of writing:

"Since the Fuzzy SWARA weight came out this way, the criterion's importance is this precise"

the report should read:

"This weight rests on the expert's given ranking and on the successive fuzzy importance differences; had the chosen verbal term differed (for instance 'weakly less important' instead of 'moderately less important'), the size of the weight would also have differed"

This extension is used when an expert states a successive importance difference as "approximately" or "around" rather than a precise percentage, and reducing this approximation to a single number would create an artificial precision. Opening a measured difference, for instance when two budget items' ratio is already known, into a triangle is not modelling uncertainty but manufacturing it.

SWARA's exit condition applies here too. If the expert is not confident in the correct ranking of the criteria, this extension does not test this uncertainty either; a ranking error is not corrected in fuzzy form, it is only carried forward in fuzzified form. If an independent consistency check is considered essential, fuzzy BWM or fuzzy AHP should be preferred instead.

Entering the most important criterion's reference difference (s̃_1) as a triangle other than zero. Because no criterion ranks above the most important one, this value must always be (0;0;0).

Not declaring the verbal scale, or changing it from expert to expert. The triangular equivalent of a term such as "moderately less important" is fixed before the analysis; using a different triangle for the same term makes results incomparable.

Not rebuilding the chain when the ranking changes. If the criterion order changes, the coefficient-intermediate weight-final weight chain must be recalculated from scratch; reporting old fuzzy weights alongside a new ranking is inconsistent.

Defuzzifying at the first step. Reducing the successive differences to a single number first and then running crisp SWARA is not Fuzzy SWARA; the fuzziness is erased at the first step, and the information carried in the width of the difference is lost.

The governing principle is this:

Fuzzy SWARA's weights are a defuzzified summary of the expert's given ranking and successive fuzzy importance differences; no version of SWARA tests the correctness of this ranking, so if the initial ranking is contested, the weights are contested too.