Picture fuzzy TOPSIS (Sindhu, Rashid and Kashif, 2019)

Four things change. The decision logic stays the same.

Cells. In crisp TOPSIS every cell holds a single number. Here every cell holds three degrees: support (yes), abstention and rejection (no). The sum of these three degrees cannot exceed 1. This triple does not come from an expert score; it comes from a committee's or a survey's vote distribution on the same judgement. For example, for the judgement "this supplier is reliable," what percentage said yes, what percentage stayed undecided and what percentage said no are counted separately. This member in DecisionMind does not itself merge the votes of several experts; the committee vote is already entered as a single pooled triple. Weights come from outside as single numbers and must sum to 1.

Scale equalisation. Crisp TOPSIS divides every column by the square root of the sum of its squares, because criteria are expressed in different units: lira, days and points cannot be summed in the same column. Picture fuzzy triples already lie between 0 and 1 and all share the same structure, so no such division is needed; this step is skipped entirely. Instead, two hypothetical references are built for every criterion column. The positive ideal carries the highest support, the lowest abstention and the lowest rejection seen in that column. The negative ideal carries the lowest support, the lowest abstention and the highest rejection. For a cost criterion these two references are built differently, which is exactly the point most often confused; this is explained further below.

Distance. In crisp TOPSIS the gap between two points is the difference of a single number, and Euclidean distance is computed. Here the similarity between two picture fuzzy triples is measured differently: the sum of the differences of the three components is taken, and to this is added the largest of these differences. So it is not only the total difference that is penalised, but also the single component that deviates most among the three degrees. An alternative's similarity to the positive ideal and to the negative ideal is computed with this measure, weighted across the criteria.

Result and defuzzification. The closeness ratio is found by dividing the similarity to the negative ideal by the sum of the two similarities. This is the same definition as in crisp TOPSIS. The result is again a single number between 0 and 1. Uncertainty is not defuzzified; it dissolves inside the similarity calculation.

DecisionMind fixes, for this member, that weights come from outside as fixed numbers. This member does not implement the founding paper's option of deriving weights through linear programming; the user must supply the weight.

The closeness ratio is read as in crisp TOPSIS: it is not a percentage, it is not compared with a different analysis, and the reference points change when the alternative set changes. What differs is this: beneath the score there now sits an uncertainty coming from a vote distribution. If the three degrees sum to less than 1, there is a portion of opinion that did not commit either way. A large abstention share feeds into the score, but the score itself does not show this.

Thus instead of writing:

"This supplier's closeness score is high because the committee trusts it"

the report should read:

"This supplier's closeness score is high because its support share is high while its abstention and opposing-vote shares are low; the uncertainty in the vote distribution has also entered the calculation"

An alternative with a large abstention share can reach the same closeness score as one with a lower but decisive support share. The report keeps these two apart.

Use this extension when criterion evaluations come from a committee's, survey's or panel's yes, abstain, no vote distribution on the same judgement, and these three shares can be counted separately. The picture fuzzy data-type card explains that the abstention share must be a separately counted share, distinct from indecision. Deriving the other two degrees from a single support percentage erases this extension's sole contribution.

There is no need to expand a measured criterion, such as price or duration, into a picture fuzzy triple. DecisionMind requires a single data type. If a measured value is nonetheless to be carried as a triple, this does not mean the value is "approximate"; it means it is the voted form of a measured judgement, such as "is this price acceptable." Pythagorean fuzzy and spherical fuzzy data types use similar triple or pair structures. The difference lies in the constraint form: on those cards the sum of squares cannot exceed 1, here the sum of the degrees themselves cannot exceed 1. The two must not be confused with one another.

The exit condition is the same as for crisp TOPSIS. If no compromise is acceptable on one criterion, this extension too is compensatory and will not eliminate anything below a threshold.

Violating the value space. In every cell the sum of support, abstention and rejection must not exceed 1. This check must be made before the positive and negative ideals are built. If one cell exceeds it, the references built from it are also invalid.

Getting the abstention degree wrong for the negative ideal on a cost criterion. For a benefit criterion, the negative ideal takes the lowest abstention. For a cost criterion, the negative ideal also takes the lowest abstention. If intuition says "it should be reversed for cost" and the highest abstention is taken instead, the abstention polarity is inverted. This is the most easily misunderstood step in the founding paper.

Skipping the largest-difference term in the similarity measure. If similarity is computed only from the sum of the three components' differences, it collapses into an ordinary Hamming distance and the ranking can change. The single largest deviation term must also be included.

Defuzzifying first and then running crisp TOPSIS. Reducing the three degrees to a single number from the outset, for example to the support share alone, and then applying the crisp method is not this extension. The abstention and opposing-vote information is erased at the first step.

Deriving a triple from a single ratio. Saying "there is 60 per cent support, so 40 per cent opposition" zeroes out the abstention share. As the picture fuzzy data-type card explains, every degree must be counted separately.

The governing principle is this:

Picture fuzzy TOPSIS exists to carry the abstention share of a committee's or survey's vote distribution through to the final score. Any implementation that fabricates a degree, skips the abstention, or crisps the input from the outset destroys the method's sole contribution.

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