Hesitant fuzzy TOPSIS (Xu and Zhang, 2013)

Four things change; the decision logic stays the same.

Cells. In crisp TOPSIS every cell is a single number. Here every cell is a set. The set holds several plausible values for the same criterion together, for example {0.3; 0.3; 0.3; 0.4; 0.5}. Within the same column, different alternatives' sets can be different lengths: one expert might give two values while another gives four. In that case DecisionMind extends every set in the column to the length of the column's longest set. This is called length harmonisation. The missing places are filled using a coefficient called η: the set's largest and smallest values are blended in the proportion η, and this value is added to the set. The default has η equal to zero, which fills the missing place with the set's smallest value; this is called cautious completion. If η is one, it is filled with the largest value instead; this is optimistic completion. Weights can also arrive differently. Base TOPSIS does not generate weights; it only takes them from outside. HF-TOPSIS offers one of three routes instead: the weight can be supplied exactly from outside; it can be assumed entirely unknown and derived by the method itself from how far the sets diverge from one another; or, where it is partially known, the method can find the most discriminating weight within the bounds the user supplies.

Scale equalisation. Crisp TOPSIS equalises columns according to their magnitude. Hesitant sets have no such problem, because the values are already degrees between 0 and 1. This is why HF-TOPSIS has no scale-equalisation step. Length harmonisation, described above, operates in its place.

Distance. The distance between two sets is found as follows: both are first extended to the same length and sorted from smallest to largest, then the differences between values at matching positions are taken, squared, averaged, and the square root is taken. The ideal and anti-ideal points are also no longer single numbers. For each criterion, the ideal set is built from the largest value at each position across the observed sets, and the anti-ideal set from the smallest. DecisionMind combines every alternative's total distance to these two references using the same Euclidean logic as crisp TOPSIS: the squared distance on each criterion is multiplied by its weight, the products are summed, and the square root is taken.

Result and defuzzification. The output is again a single closeness score between 0 and 1. This score is the ratio of the distance to the anti-ideal over the total distance. The sets descend to a single number only at this final step. Throughout the rest of the calculation, the hesitancy is preserved.

For classical HF-TOPSIS, DecisionMind fixes cautious completion (η = 0), the sort-and-compare rule, and the squared, weighted combination. The user chooses the weight mode: fully known, partially known, or entirely unknown.

The closeness score is read exactly as in crisp TOPSIS. It only ranks the alternatives in this set relative to one another. What differs is this: when weights are derived from the data, which weight mode is chosen directly affects the result. The same table can give a different ranking under the entirely-unknown weight mode than under the partially-known weight mode. This is an additional source of uncertainty that crisp TOPSIS does not have.

Thus instead of writing:

"HF-TOPSIS gives a more reliable ranking because it takes uncertainty into account"

the report should read:

"The weights have been derived from the data; the choice of weight-information mode can change the ranking, and which mode was used must be stated in the report"

This extension is appropriate when more than one plausible, defensible degree exists for the same criterion-alternative pair — for example, when several experts have scored independently and you want to keep these scores together without collapsing them into an average. The distinction on the data-type card applies here too: where a single measured value exists, a single number is written into the cell, and no set is built. DecisionMind requires the table to hold a single data type. Base TOPSIS's compensatory nature, and the condition that no compromise is possible on one criterion, apply here exactly as before.

Not declaring the length-harmonisation coefficient (η). η = 0 and η = 1 can give the same table a different distance, and hence a different closeness score. The value used must be stated in the report.

Choosing the weight-information mode without justification and without stating it in the report. As Case 1 shows, the ranking can change between the entirely-unknown mode and the partially-known mode.

Entering partially-known weight constraints that contradict one another. In that case DecisionMind reports that the constraint set is empty and does not fabricate a number.

The "more sophisticated" fallacy. Without a genuine source supporting more than one value, arbitrarily expanding a crisp number into several nearby values adds no information.

The governing principle is this:

A set must genuinely come from more than one defensible source for the same criterion-alternative pair. Leaving the length-harmonisation coefficient and the weight mode unjustified means the same data can produce a different ranking.