Neutrosophic WASPAS

Several things change, but the logic of blending the sum and the product stays the same.

Cells. In crisp WASPAS every cell is a single number. Here every cell is a judgement's degree of truth (T), indeterminacy (I) and falsity (F); the three are independent of one another and their sum cannot exceed 3. Weights remain crisp; neutrosophic data is used only in the decision-matrix cells.

Complementation on a cost criterion. On a "lower is better" criterion, every cell's truth and falsity components swap places, and the indeterminacy component is subtracted from 1. This is more involved than IF-WASPAS's simple (μ, ν) swap, because a third component (I) also has to be transformed; DecisionMind fixes this transformation.

Score (early defuzzification). Similarly to Grey WASPAS, N-WASPAS reduces indeterminacy to a single number before scale equalisation: every cell's three components are converted to a single number by a score function that rewards truth and penalises indeterminacy twice as heavily as falsity. In Fuzzy WASPAS and IF-WASPAS, the sum and product components are built in their own native form, as triangular numbers or support/rejection pairs, and are defuzzified only at the very end. In N-WASPAS, defuzzification is carried out cell by cell, before the sum and product are built.

Sum and product. Once the matrix has been reduced to scores, the Weighted Sum Model (WSM) and the Weighted Product Model (WPM) are computed with exactly the formulas used in crisp WASPAS. Any score in the product component that could come out negative is clipped to zero, because raising a negative base to a power can become undefined.

Result. The combined score blends the sum and product components with λ. Unlike Fuzzy WASPAS, λ is not computed adaptively; the crisp WASPAS default of 0.5 is used. But unlike IF-WASPAS, λ here is a separate, named user parameter in the manifest; it can be changed directly.

For N-WASPAS, DecisionMind fixes the complementation rule and the score function; weights are taken from outside as crisp numbers, and the method does not generate weights.

The output is a combined score and a ranking, as in crisp WASPAS, and it is read the same way: it is not a percentage or a probability, and the ranking can shift when the alternative set changes.

The difference is this. Every cell's truth-indeterminacy-falsity triple has already been compressed into a single score before the sum and product are computed. The output therefore does not show how large indeterminacy (I) is, only how far indeterminacy has pulled the score down. When two alternatives' score gap is small, which component this gap comes from, low truth, high indeterminacy or high falsity, must be shown separately; otherwise the question "why did this come out low" cannot be answered.

Thus instead of writing:

"According to N-WASPAS, A3 is the most reliable alternative"

the report should read:

"A3 has the highest combined score; this lead holds up even when the weights of the two heaviest criteria are swapped, but the order between A1 and A2 reverses under the same weight swap, and the gap between these two is not robust"

Use it when the information about a criterion assessment is incomplete, contradictory or inconsistent, and this matters for the decision. It suits evaluating alternatives with no track record, expert opinions drawn from conflicting sources, and situations where "there is both strong evidence for and strong evidence against this." Where the indeterminacy component is computed as 1 minus truth minus falsity, that is, it has no source of its own, this extension is unnecessary; an intuitionistic fuzzy or crisp structure carries the same information. Where a value is measured, move to the base method.

WASPAS's exit condition applies exactly as it does there: the sum component is compensatory, and the method is unsuited to a case where no compromise is acceptable on one criterion; where weights rather than a ranking are needed, methods such as AHP, BWM, CRITIC or Entropy should be used instead.

Letting the sum of T+I+F exceed 3. This is the validity limit stated on the neutrosophic data-type card; a triple that exceeds it renders the score function, and every step that follows, meaningless.

Writing indeterminacy as 1 minus truth minus falsity. The core warning on the neutrosophic data-type card applies here too; in that case I carries no independent information, and the extension's one contribution, keeping contradictory or incomplete evidence separate, disappears.

Swapping only T and F on a cost criterion and leaving I unchanged. Complementation transforms all three components at once (T and F swap, I is subtracted from 1); converting only two of them leaves the direction correction half-done.

Changing or skipping the score function. The score that rewards truth and penalises indeterminacy twice as heavily as falsity is the fixed rule here; a different score function (for instance, just the T-F difference) can produce a different ranking and must not be changed without stating this in the report.

Concealing the early defuzzification and giving the impression that "all three components are fully reflected in the calculation." Because the score is computed before the sum and product, the three components' separate effects do not show in the final score; which component pulled the score down must be shown separately.

The governing principle is this:

Truth, indeterminacy and falsity must come from separate sources and their sum must not be forced to equal 1; from the moment the score function reduces the three to a single number, which component is driving the result must be reported separately.

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