Fuzzy WASPAS (Turskis, Zavadskas, Antuchevičienė and Kosareva, 2015)

Four things change; the logic of blending the sum and the product does not.

Cells. In crisp WASPAS every cell is a single number. Here every cell is a triangular fuzzy number (lowest, most likely, highest). Criterion weights, too, may be entered as triangles; if a crisp weight is given instead, DecisionMind expands it into a triangle with all three components identical (w, w, w) — this does not add width to a measured weight, it only fits it to the structure.

Scale equalisation. Crisp WASPAS divides every column by its own best crisp value. Here, for a benefit criterion, all three components of each triangle are divided by the largest upper value in the column. For a cost criterion, the smallest lower value in the column is taken and divided into the triangle's three components in reverse order: the lowest component divides this constant by the upper value, the highest component divides it by the lower value. Without this reversal the division would break the lowest-most likely-highest order; Turskis's equalisation deliberately preserves it.

Sum and product component. The weighted-sum (WSM) component multiplies the equalised triangle by the weight triangle component by component and sums this as a triangle across criteria; this is the crisp WASPAS summation carried out separately on the three components. The weighted-product (WPM) component is built by raising the equalised triangle to the weight's power; here too, exponentiation reverses the corners (the triangle's lowest component is raised using the weight's highest plausible form, and its highest component using the weight's lowest form), because fuzzy exponentiation produces a narrowing interval as the exponent grows, and this reversal is needed to preserve the order.

Result and defuzzification. For each alternative, the sum component's three corners are added across criteria and the product component's three corners are multiplied; each triangle is then reduced to a single number by the average of its three corners (the centroid). In crisp WASPAS, λ is a fixed coefficient supplied from outside (0.5 if not stated); here DecisionMind does not treat λ as fixed. Instead it computes λ adaptively from the two defuzzified components' (sum and product) share of the total across all alternatives. This is the only extension in the WASPAS family where λ stops being a user decision and becomes a data-driven output.

In Fuzzy WASPAS, DecisionMind fixes this scale equalisation, the corner-wise sum/product rules, and the adaptive λ calculation; weights are taken from outside, the method does not generate weights.

The output is a combined score and a ranking, as in crisp WASPAS; it is read the same way: it is not a percentage or a probability, and it cannot be compared with a different analysis.

The difference is here: λ is no longer a decision to be explained in the report as a chosen value, it is an outcome that emerges from the data itself. The report should therefore state what value λ settled at, and whether it reflects that the sum component or the product component dominated. Beneath the score also lies a chain of triangles, and the score conceals this. The score difference between two alternatives may or may not be meaningful, depending on the width of the inputs.

Thus instead of writing:

"According to Fuzzy WASPAS, A2 is the best alternative"

the report should read:

"The adaptive λ came out at 0.49, meaning the sum and product components contributed almost equally; with these weights, A2 has the highest combined score, and this does not change if the two criteria's weights are swapped"

Use this extension when criteria are scored through verbal or approximate expert judgement, and reducing that judgement to a single number would create artificial precision. Stay with the base method when criteria are measured: opening a measured number into a triangle is not modelling uncertainty, it is manufacturing it. If the table is mixed and DecisionMind requires a single data type, a measured criterion is also written as a triangle with all three components equal; this embedding adds no uncertainty.

WASPAS's exit condition applies unchanged: this extension too is compensatory and is unsuitable when no compromise is acceptable on one criterion; if a weight, not a ranking, is being sought, fuzzy or crisp weighting methods such as AHP, BWM, CRITIC or Entropy should be used instead.

Breaking the triangle's order. Every cell's lowest, most likely and highest component must appear in this order; if data entry breaks this order (for example, entering a most-likely value greater than the highest), the scale-equalisation and exponentiation steps produce a meaningless result.

Changing the defuzzification method arbitrarily. The centroid (the average of the three components) is the sole, fixed rule here; a different defuzzification (such as taking only the most likely value) can produce a different ranking and becomes inconsistent with Turskis's steps.

Defuzzifying first and then running crisp WASPAS. Reducing the triangles to a single number at the outset and applying the crisp method is not Fuzzy WASPAS; how the sum and product components grow corner by corner is then invisible.

Fixing λ by hand. In this extension, λ is an output of the data; fixing it at 0.5, as in other WASPAS extensions, or entering it manually, cancels the method's adaptive-blending contribution.

Giving crisp weights alongside fuzzy scores. This is not itself a problem; DecisionMind expands a crisp weight into a degenerate triangle. But the report should note that this expansion adds no width.

The governing principle is this:

In Fuzzy WASPAS, the triangle's order, the defuzzification rule and the adaptive calculation of λ form the method's fixed skeleton; changing any one of them by hand means the result is no longer Turskis's WASPAS-F.

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