Complex fuzzy Z-number WASPAS (Shahid et al., 2026)

Four things change; the WASPAS logic of blending a sum and a product with a single coefficient does not.

Cells. In crisp WASPAS every cell is a single number. Here every cell consists of four numbers: an amplitude (σ) and a phase ratio (τ), accompanied by a reliability amplitude (ϖ) and a reliability phase ratio (R). σ and τ carry the judgement itself; ϖ and R carry how reliable that judgement is, that is, the B component of the Z-number. All four lie between 0 and 1. Weights are taken from outside as crisp numbers; the method does not generate weights.

Scale equalisation. Crisp WASPAS equalises every column by dividing by its best value. Here that division is carried out separately on each of the four components (σ, τ, ϖ, R). For a benefit column, every component is divided by its own column maximum; for a cost column, by its own column minimum. Value and reliability thus pass through two scalings that are independent of one another.

The WSM and WPM components. Crisp WASPAS's weighted sum is matched here by summing the four equalised components separately across the criteria; its weighted product is matched by multiplying the four components separately. Neither step yields a single number: both produce a four-number result, and the calculation is not yet defuzzified at this stage.

Combination and defuzzification. In crisp WASPAS the WSM and WPM scores are already single numbers; they are blended with λ and the calculation ends there. Here the four-number WSM result and the four-number WPM result are first blended with the same λ coefficient, which DecisionMind names ϒ; this blend, too, remains four numbers. A single score is only reached at the final step, by averaging the product of amplitude and reliability amplitude with the product of phase ratio and reliability phase ratio. Crisp WASPAS therefore has no defuzzification step, because it is already a single number; here defuzzification happens right at the end, in one step, and up to that point both value and reliability are carried within four numbers.

ϒ (the WASPAS coefficient) is closed to the user here. The founding article (Shahid et al., 2026) produces Table 7 with ϒ=1, a choice of pure weighted summation in which the product component contributes nothing at all, and the article states this openly in its text. When this card's author read the DM3 manifest, ϒ was listed as one of six parameters, but all six are flagged as not user-selectable (user_selectable: false); the interface carries no slider or input box for it. The kernel code, however, defaults ϒ to 0.5 when no external value is supplied. The value the article uses (1) and the default written into the engine's code (0.5) therefore differ, and which one is used in which calculation is invisible to the user. This has been verified by this card's author running the kernel directly (see Cases 1 and 2).

DecisionMind fixes, for this extension, the scale equalisation, the four-component WSM/WPM aggregation and the score function in the final step. Weights come from outside.

The combined score is read as in crisp WASPAS: it is a position relative to the other alternatives in this set, not a percentage or a probability (see the WASPAS card). The difference lies here: this score carries both the judgement itself and the reliability of whoever gave it. Two alternatives sharing the same judgement values (σ, τ) does not mean they share the same score; a difference in reliability between the two comes into play whenever it exists.

Thus instead of writing:

"In complex fuzzy Z-number WASPAS, reliability is purely indicative and does not change the result"

the report should read:

"Reliability has no effect only when it is identical across alternatives; where different alternatives rest on sources of different reliability, that difference genuinely feeds through into the combined score and the ranking"

This extension is worth considering when both a judgement's amplitude-phase pair and that pair's reliability are known separately, and reliability varies across the alternatives. The typical situation is one where judgements combine, in the same table, values partly grounded in verified measurement and partly in expert estimation. If every cell carries the same reliability, because all of them come from the same source, this extra dimension adds nothing; the sibling extension Complex fuzzy Z-number MARCOS handles the same data structure with a different aggregation. Crisp WASPAS's exit condition applies here too: where one criterion cannot be traded away, this extension remains compensatory as well.

Assuming ϒ is fixed in the interface and not asking which value was used. The value the literature uses (1, pure summation) and the default the kernel falls back to when no external value is supplied (0.5) differ; the report must state plainly which one was computed.

Writing the same reliability into every cell and calling it "using a Z-number." If reliability never varies across alternatives, scale equalisation and aggregation carry this constant through every alternative in the same proportion; this has been verified by this card's author with the kernel (see Case 1).

Feeding value and reliability into a sum separately. The score function at the final step multiplies amplitude by reliability amplitude and phase ratio by reliability phase ratio; summing or averaging the two separately gives a different result that is not defined in the manifest.

Forgetting that the intermediate sums can exceed the 0 to 1 range. The manifest reports that, in the article's own Table 5, the phase ratio rises as high as 1.35 after the WSM step. The stated [0,1] bound is therefore exceeded at this step. This is a natural consequence of summing without normalising; the intermediate steps should not be assumed, and read, as still lying within [0,1].

Defuzzifying too early. Reducing the four-number WSM and WPM results to a single normalised component and summing them as a single number, in the manner of crisp WASPAS, hides how reliability is multiplied against value.

The governing principle is this:

Reliability only changes the combined score when it genuinely varies across the alternatives; whichever value ϒ was run with must be stated plainly in the report, because the interface does not show it to the user.