Cubic fuzzy WASPAS (Jun, Kim & Yang, 2012)

Three things change; the four-step skeleton that blends the weighted sum and the weighted product with a λ does not.

Cells. In crisp WASPAS every cell is a single number. Here every cell consists of three numbers: an interval's lower end, that same interval's upper end, and a confidence degree (λ) attached to that interval. The interval carries the information "this assessment is plausible within this band"; the confidence degree carries "how much this band itself is trusted." For example, if a quality test says "performance falls between 0.65 and 0.75, with a confidence of 0.7 in this," the cell is written as (0.65; 0.75; 0.7). Criterion weights are taken from outside as crisp numbers.

Defuzzification comes right at the start. Every three-part cell is reduced to a single number in the very first step of the calculation, with Jun, Kim and Yang's score function. This number adds a quarter of the interval's width to the interval's midpoint, and subtracts half the complement of the confidence degree. For a cost criterion, the cell's complement is taken first: the lower and upper ends swap places, and the confidence degree is also complemented. The same score function is then applied. This extension's "score" is therefore not a reversible interval-point pair; it is only a number.

Scale equalisation, WSM and WPM. Once cells have been reduced to scores, they are equalised into the 0-to-1 range by dividing by the column's largest value, in the same way as crisp WASPAS. The rest is crisp WASPAS itself: the equalised scores are multiplied by the weight and summed (WSM), and raised to a power equal to the weight and multiplied (WPM).

λ is closed to the user here. In crisp WASPAS, λ is an input that DecisionMind defaults to 0.5 when unspecified, but which can be changed if needed. When this card's author read the DM3 manifest, CUBIC-WASPAS's D.parameters list held only weights and criterion directions; λ is not defined as a parameter at all. The kernel code confirms this: λ is never read from external input, and is hard-coded at the top of the file as LAM = 0.5. In other words, this coefficient, which can be changed from the interface in crisp WASPAS, is fixed here so that it cannot even be changed from the code.

DecisionMind keeps the score function, the complementing rule and the λ=0.5 constant unchangeable in this extension. Weights come from outside; the method does not generate weights.

The combined score is read as in crisp WASPAS: it is a position relative to the other alternatives in this set (see the WASPAS card). The difference lies here: this score is not a reversible interval-point pair, but comes from reducing the two to a single score at an early stage. Two alternatives sharing the same score does not mean they share the same interval or the same confidence degree; a narrow interval with high confidence and a wide interval with low confidence can both give the same score.

Thus instead of writing:

"Cubic fuzzy WASPAS gives a more robust result because it uses interval and confidence information together"

the report should read:

"This score comes from reducing the interval and the confidence degree to a single score. Which alternative has a wide interval and which has low confidence does not show up in the score; this must be reported separately"

This extension is worth considering when experts give a criterion both as a band (at least this, at most that) and as a confidence degree held in that band. The aim is to combine this pair of pieces of information into a single structure, rather than keeping them apart. Converting a measured criterion into an interval-confidence pair models uncertainty rather than manufacturing it. Crisp WASPAS's exit condition applies here too: where one criterion cannot be traded away, this extension remains compensatory as well; where weights rather than a ranking are needed, methods such as AHP, BWM, CRITIC or Entropy should be considered instead.

Writing a cell without verifying its constraint. The interval's lower end cannot exceed its upper end, and the confidence degree must lie between 0 and 1; a cell entered without this check renders the score function meaningless.

Assuming λ can be changed. In crisp WASPAS, λ is an input parameter; here it is hard-coded (0.5) and cannot be changed from any interface field. A sentence such as "λ=0.3 was also tried" in the report is invalid for this extension.

Computing the score for a cost criterion without complementing. The direction correction here is not division but complementing (the lower and upper ends swap, the confidence degree is complemented); skipping this step can mistakenly reward a wide, low-confidence interval on a "lower is better" criterion.

Assuming the score is reversible. Once the score is reduced to a single number, which part, interval or confidence, is driving the result stops being visible; this has been verified by this card's author by reading the kernel code.

The governing principle is this:

An interval and a confidence degree should only be written into this structure when they genuinely come from two separate questions, how plausible is the value and how much is this assessment trusted; λ is fixed here and must not be presented as a choice in the report.