m-Polar Linguistic TOPSIS (Adeel, Akram and Koam, 2019)

Four things change; the decision logic stays the same.

Cells. In crisp TOPSIS every cell is a single number. Here every cell is the set of degrees a linguistic term receives from m separate viewpoints (poles); every pole is a number between 0 and 1. In a model's "appearance" assessment, for instance, the value of the term "quite pretty" is scored separately on the poles of facial features, posture and body proportions. Where several decision-makers are involved, each supplies their own matrix; DecisionMind merges these into a single matrix by simple arithmetic mean, and this merging is done BEFORE weighting. Criterion weights, too, are taken separately from every decision-maker and merged the same way, by averaging.

Scale equalisation. There is no column normalisation here, unlike in crisp TOPSIS. Every pole value is already entered as a degree between 0 and 1. A raw score, for instance a mark out of 100, is not scaled by the engine; the user must first bring it into the 0-to-1 range themselves.

Weighting. In crisp TOPSIS, weighting is applied to the whole column. Here the merged weight is multiplied directly, as a scalar, into every pole of the merged matrix. This multiplication is not exponential; because there is no hesitancy layer, it differs from the exponential weighting used in MHF-TOPSIS and MPF-HF-TOPSIS.

Distance and the result. The ideal and anti-ideal are built, for every linguistic term, from the best and worst value at every pole separately. Distance is an equally weighted Euclidean distance taken over the poles; because the criterion weight is already embedded into the values at the previous step, it is not multiplied a second time in the distance formula. The closeness coefficient is again a single number between 0 and 1.

DecisionMind fixes, for this extension, merging the group before weighting and taking an equally weighted distance over the poles.

The closeness coefficient is read as in crisp TOPSIS, only it ranks this particular set of alternatives. The difference is here: the decision-makers' matrices are merged by averaging, before weighting. Because of this, a single outlying decision-maker can shift the ideal and anti-ideal point for every alternative.

Thus instead of writing:

"m-polar linguistic TOPSIS is free of subjectivity because it uses a group decision"

the report should read:

"The decision-makers' matrices have been merged by simple averaging; a single outlying decision-maker can shift the ideal and anti-ideal point, and this risk should be assessed in the report"

This extension is suitable where several decision-makers assess a criterion with linguistic terms from m separate viewpoints. Even with a single decision-maker, the pole structure can still be used. Where there is only a single viewpoint and the decision-maker chooses a single linguistic term, 2-tuple linguistic TOPSIS is sufficient; where the term is distributed with a probability, Probabilistic linguistic TOPSIS is sufficient; the pole layer adds unnecessary complexity in these cases. Where a criterion is measured, the base method should be kept; DecisionMind asks that the table hold a single data type.

Giving a raw score without scaling it. Writing a mark out of 100 directly into a cell breaks the 0-to-1 range constraint; the method does not perform this scaling itself.

Not keeping the number and meaning of poles fixed. Every pole, for instance facial features or posture, must carry the same meaning for all decision-makers, alternatives and linguistic terms; otherwise building the column-wise best and worst value is inconsistent.

Giving linguistic weight labels without converting them to numbers. Where weights are given only as labels ("medium," "high"), a numerical equivalent must first be assigned, and these must sum to 1; the method does not perform this conversion itself.

Multiplying the criterion weight a second time in the distance formula. The weight is already embedded in the weighted matrix; multiplying the deviation by the weight again in the distance calculation distorts the paper's figures.

The governing principle is this:

The decision-makers' matrices are merged before weighting; once merged, a single outlying opinion's effect becomes permanent for every alternative, and this risk must be assessed separately.