Picture fuzzy COPRAS (Lu, Zhang, Wu & Wei, 2021)

Four things change. The decision logic stays the same.

Cells. In crisp COPRAS 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. This member can also merge the votes of more than one expert; if more than one expert matrix is entered, DecisionMind reduces them to a single board matrix with the picture fuzzy addition rule. Weights come from outside, as single numbers.

Scale equalisation. Crisp COPRAS converts every column into a proportion by dividing it by its own sum. Picture fuzzy triples are already a structure between 0 and 1, and the division by the column sum is not applied here. Instead, every cell is weighted through the picture fuzzy exponentiation-multiplication rule with the criterion's weight. This operation rescales, to the power of the weight, not the number itself, but the triple the three degrees jointly form.

Benefit and cost sums. In crisp COPRAS the weighted values on benefit criteria are summed, and those on cost criteria are summed separately. The same split is made here, but the summation runs through the picture fuzzy addition rule: an alternative's triples on the benefit criteria are merged into a single benefit triple, and its triples on the cost criteria are merged into a single cost triple. These two triples are each reduced to a number with the support-minus-rejection score, and combined with a correction term as in crisp COPRAS.

Result and defuzzification. The relative significance value is divided by its highest value and converted into a percentage; the best alternative is again 100. The result is again a single number between 0 and 100. The uncertainty is not defuzzified beforehand; all three degrees are used while building the benefit and cost sums, and only the last step reduces them to a single number with the score function.

DecisionMind fixes, for this member, the direct use of the weights when the user supplies them; when no weight is given, the founding paper's objective weight-derivation route (a method based on correlation between criteria) takes over.

The benefit degree is read as in crisp COPRAS: it says what percentage of the best alternative's total benefit has been reached, not an absolute success percentage. What differs is this: the benefit and cost sums themselves now carry an uncertainty coming from the vote distribution. If an alternative's abstention share on the cost side is large, its apparent cost advantage is less certain than it looks.

Thus instead of writing:

"This proposal is 88 per cent successful"

the report should read:

"This proposal reaches 88 per cent of the benefit delivered by the proposal with the highest benefit degree; this calculation also carries 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 to report the result directly as "a percentage of the best." 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 intuitionistic fuzzy data type also uses degrees of support and rejection, but there the abstention share is not counted separately; it is derived as what remains after support and rejection. Here abstention is an independent degree, separately counted. The two types should not be confused.

The exit condition is the same as for crisp COPRAS. If no concession is acceptable on a criterion, this extension is compensatory too, and will let a low cost mask a high benefit.

Value-space violation. In every cell, the sum of support, abstention and rejection must not exceed 1. This check must be made before the weighting step.

Marking the criterion direction incorrectly. If a cost criterion is marked as a benefit criterion, it is included in the benefit sum, and the alternative with high support on that criterion, that is, the expensive one, is rewarded.

Reading the benefit degree as an absolute quality percentage. A value of 88 per cent is only a share relative to the best in this set; it cannot be compared with a different analysis.

Defuzzifying first and then running crisp COPRAS. 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 COPRAS exists to carry a board's or a survey's abstention share through to the benefit and cost sums. Any application that invents a degree or crispens the input from the outset erases the method's one contribution.

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