m-Polar Hesitant TOPSIS (Akram, Adeel and Alcantud, 2019)

Four things change; the decision logic stays the same.

Cells. In crisp TOPSIS every cell is a single number. Here every cell carries m independent poles, and each pole is itself a hesitant set, that is, it holds more than one plausible membership value. For example, a brand-name proposal's "clarifying identity" criterion is treated with four sub-features (vision, mission, values, direction) as separate poles, and each sub-feature is written with its own hesitant set, holding together the values several assessors have given. Different poles within the same cell can carry hesitant sets of different lengths. DecisionMind completes these to a common length; the completion can be optimistic (repeating the largest value) or conservative (repeating the smallest value), and the user chooses which. Criterion weights are crisp numbers, taken from outside.

Scale equalisation. The values in the hesitant sets are already degrees between 0 and 1, so crisp TOPSIS's step of dividing by column magnitude is absent here. In its place runs the length-completion just described; every pole's every hesitant set is brought to the same length.

Weighting. In crisp TOPSIS, weighting is applied to the whole column. Here every hesitant value in every pole is multiplied separately by its own criterion's weight; the pole and hesitancy structure is preserved, only the values shrink.

Distance and the result. The ideal and anti-ideal are built, for every criterion, from the best and worst value at every hesitancy rank of every pole separately. Distance is an averaged Euclidean distance taken over every pole and every hesitancy rank. The closeness coefficient is again a single number between 0 and 1.

DecisionMind fixes, for this extension, the number of poles, the rule for equalising hesitancy length, and the averaged distance. The direction of completion is chosen by the user and must be stated in the report.

The closeness coefficient is read as in crisp TOPSIS, only it ranks this particular set of alternatives. The difference is here: the result can also carry the opinion of more than one independent decision-maker, because the poles can be different assessors. Where two alternatives' scores are close, looking at which pole carries the disagreement gives more information than looking at the score alone.

Thus instead of writing:

"m-Polar hesitant TOPSIS gives the most balanced result because it uses several viewpoints together"

the report should read:

"The score melts the agreement or disagreement between the poles into a single number; which pole carries the disagreement should be examined separately, and this disagreement should be reported for close-scoring alternatives"

This extension is appropriate where a criterion genuinely comes from several independent viewpoints, for instance several assessor groups or sub-features, and hesitancy exists among experts within every viewpoint as well. Where there is more than one viewpoint but each viewpoint carries a single crisp value, the pole layer can still be used, and the hesitancy layer is then unnecessary. Where there is only a single viewpoint but hesitancy exists, HF-TOPSIS is sufficient. Where a criterion is measured, the base method should be kept; DecisionMind asks that the table hold a single data type.

Changing the number of poles from cell to cell. Every cell must hold the same number of poles; three poles in one cell and four in another leaves the construction of the ideal and anti-ideal undefined.

Copying a single-source score across the poles. Poles represent independent viewpoints; writing a single assessor's score into all four poles in the same way looks like using the pole structure without actually using it.

Weighting through a different operation. In this kernel, weighting is the direct multiplication of every hesitant value by the criterion's weight; using a different combination rule does not reproduce the paper's figures.

Choosing the weight distribution without justification. As Case 1 shows, how the weights are distributed across criteria can change the order of close-scoring alternatives.

The governing principle is this:

Every pole must carry the data of its own independent viewpoint, the weight distribution must not be chosen without justification, and the sensitivity of close-scoring alternatives to this choice must be shown in the report.