Picture fuzzy MARCOS (Tatar & Ayvaz, 2024)

Four things change. The decision logic stays the same.

Cells. In crisp MARCOS every cell holds a single number. Here every cell holds three degrees: support, abstention, rejection; their sum cannot exceed 1. This triple comes from a board's or a survey's vote distribution on the same judgement. Weights come from outside, as single numbers.

Scale equalisation. Crisp MARCOS normalises every cell by ratioing it to the criterion's ideal value. Here, the cost-direction criteria are completed first: in every cell of a cost criterion, support and rejection swap places while abstention stays the same. Then, in every criterion column, the row with the highest support-minus-rejection score is chosen as the ideal, and the row with the lowest score as the anti-ideal. Unlike crisp MARCOS's imagined extreme, this is a real row chosen from within the set itself; it follows the same logic as the ideal and anti-ideal selection in PIF-VIKOR.

Utility ratios. In crisp MARCOS, every alternative's weighted sum is divided separately by the ideal's and the anti-ideal's sums, giving two ratios. The same logic works here, but the summation runs through the picture fuzzy weighted aggregation rule: an alternative's triples across the criteria, the ideal row's triples, and the anti-ideal row's triples are each merged separately into three separate summary triples. These three triples are each reduced to a number with the support-minus-rejection score, and ratioed against the ideal's and the anti-ideal's scores.

Result and defuzzification. As in crisp MARCOS, these two ratios are each turned into a utility function and combined into a single final utility degree. The result is again a single number. The uncertainty is not defuzzified beforehand; all three degrees are used both while choosing the ideal and anti-ideal and while building the summary triples.

DecisionMind fixes, for this member, the selection of the ideal and anti-ideal rows from the set's real rows, through the score function.

The final utility degree is read as in crisp MARCOS: it says where the alternative stands relative to this set's ideal and anti-ideal references, and cannot be compared with a different analysis. What differs is this: the ideal and anti-ideal rows themselves now carry an uncertainty coming from the vote distribution. If the row with the highest support on a criterion also has a large abstention share, the ideal reference itself carries uncertainty too.

Thus instead of writing:

"This proposal is closest to the ideal"

the report should read:

"This proposal has the most balanced utility ratio relative to the set's own ideal and anti-ideal references; these references also carry the abstention share in the board's vote distribution"

Use this extension when criterion assessments come from a board's, a survey's, or a panel's yes, abstain, no vote distribution on the same judgement, and you want both how much of the ideal has been reached and how far the anti-ideal has been left behind to appear together. As the Picture Fuzzy data-type card explains, the abstention share must be counted separately; it must not be invented from a single support percentage.

There is no need to expand a measured criterion into a picture fuzzy triple. DecisionMind requires a single data type; if some criteria in the matrix rest on votes and others on measurement, all of them must be written in the same type. The q-rung orthopair fuzzy data type also uses a similar pair structure, but there the constraint depends on a power (q), and q is not chosen by the user but determined by the method; here the constraint is the direct sum of the degrees. The two should not be confused.

The exit condition is the same as for crisp MARCOS. If no concession is acceptable on a criterion, this extension is compensatory too.

Value-space violation. In every cell, the sum of support, abstention and rejection must not exceed 1. This check must be made before the ideal and anti-ideal rows are chosen.

Forgetting to complete a cost criterion. If support and rejection are left unswapped in a cost-direction criterion, the alternative with high support on that criterion is wrongly counted as close to the ideal.

Changing the score function midway through. The score function used to choose the ideal and anti-ideal rows must stay the same when reducing the summary triples to numbers.

Defuzzifying first and then running crisp MARCOS. Reducing the three degrees to a single number at the outset and then applying the crisp method is not this extension. The abstention and opposing-vote information is erased in the first step.

Inventing a triple from a single proportion. Saying "there is 60 per cent support, so 40 per cent is opposed" sets the abstention share to zero. As the Picture Fuzzy data-type card explains, every degree must be counted separately.

The governing principle is this:

Picture fuzzy MARCOS exists to carry a board's or a survey's abstention share through to the ideal and anti-ideal references. Any application that invents a degree or crispens the input from the outset erases the method's one contribution.

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