Picture fuzzy ARAS (Chowdhury, Chatterjee and Chakraborty, 2025)

Five things change; the decision logic does not.

Cells. In classical ARAS every cell is a single number. Here every cell consists of three degrees: yes (μ), abstain (η) and no (ν); their sum cannot exceed 1, and the remaining share is the refusal (π) share. Weights, too, are not crisp numbers but are given in the same three-degree structure; this is a point specific to this extension, unlike other ARAS members. DecisionMind also reduces weights to a single crisp number for reporting purposes, but the calculation itself is carried out with the three-degree weight.

Scale equalisation and direction. In other ARAS members, a cost criterion is turned to direction during the calculation, by taking a complement or reversing it. Here direction is turned before the calculation, in the data itself: if a criterion is "lower is better," low raw values are coded from the start to correspond to a higher yes degree. The algorithm itself therefore treats every criterion as "higher is better." If the user enters raw data as "lower is better," DecisionMind swaps the yes and no degree on that criterion, converting the data to "higher is better" before the calculation.

Weighting. Every cell is weighted with its own criterion's three-degree weight through the picture fuzzy multiplication rule: yes degrees are multiplied, while abstain and no degrees are combined by their own aggregation rules. This is a different algebra from classical ARAS's "multiply the value by the weight" operation.

Optimal alternative and sum. The optimal alternative is built from the weighted cells: the highest yes, the lowest abstain and the lowest no on every criterion. Every row's cells, including the real alternatives and the optimal alternative, are then combined across the criteria with the picture fuzzy summation rule.

Result and defuzzification. Every row's three-degree sum descends to a single number through a defuzzification rule of the form yes plus half of abstain plus a set share of the refusal share. The degree of utility B carries the same meaning as K in classical ARAS: the ratio of a real alternative's defuzzified score to the optimal alternative's defuzzified score. DecisionMind keeps this defuzzification rule and the conversion of direction at the data stage fixed; the three-degree cells are never ranked directly at any stage, ranking happens only on the defuzzified B.

The degree of utility B carries the same meaning here. The best alternative is treated as 100, and the others receive a percentage relative to it; this percentage is valid only for this particular set of alternatives and these weights. The difference is this: beneath B, three independent degrees, namely yes, abstain and no, are compressed into a single number. A cell with a high abstain share produces a different defuzzified value from a cell with the same yes degree but a low abstain share; this difference should be made visible in the report.

Thus instead of writing:

"Picture fuzzy ARAS shows T1 to be the best alternative"

the report should read:

"With the given weights, T1 holds the highest degree of utility (B=0.968); once the weight is concentrated on the first criterion (MRR), T16 moves into first place and T1 falls to seventh, so the weight distribution should be justified separately in the report"

This method is suitable when an assessment can be counted separately as yes, abstain and no, that is, when it comes from a vote, a survey or a distribution of opinion. Examples: a board's voting result on an alternative, a customer survey's support-undecided-oppose distribution, an expert panel's distribution of opinion.

If the assessment answers only "how suitable" and the abstain share is not separately measured, moving to the picture fuzzy structure offers no contribution; intuitionistic fuzzy or fuzzy ARAS is enough in that case. Classical ARAS's exit condition still applies here: if no compromise is acceptable on one criterion, this extension too is fully compensatory and does not screen out anything below a threshold.

Violating the value space. In every cell, the sum of yes, abstain and no must not exceed 1 (Cuong, 2013). Carrying intuitionistic fuzzy (two-degree) or Pythagorean fuzzy (sum-of-squares constrained) input directly into this structure is invalid.

Trying to reverse direction during the calculation. In this extension, direction is reversed by DecisionMind at the data stage (by swapping the yes and no degree) when the user marks the raw data as "lower is better." In the source paper's own example, this reversal was already done while the survey data was collected; reversing direction in both places at once flips it twice and leads to a wrong result.

Giving crisp weights while cells are picture fuzzy. The method expects the criterion weight to be three-degree as well; if a crisp weight is to be given, it must be converted to a three-degree equivalent using the manifest's defuzzification rule (Equation 16), and this conversion must be stated in the report.

Mistaking the weighting operation for "multiplying by a crisp weight." The source paper's own text describes "multiplying by a crisp weight" in places, but reproducing the paper's own tables requires weighting through the three-degree multiplication rule. DecisionMind adopts this second reading, the one that matches the paper's tables, as the standard.

The governing principle is this:

Picture fuzzy ARAS exists to carry the yes, abstain and no shares of a distribution of opinion through to a ratio against the optimal alternative; any use that reverses direction twice, or applies the weights with the wrong algebra, corrupts the one contribution the three-degree structure makes.