Dual hesitant fuzzy TOPSIS (Wang, Li, Zhang and Han, 2020)

Three things change; the idea of closeness to the ideal does not.

Cells. In plain hesitant TOPSIS every cell carries a single set, several possible degrees of support. In dual hesitant TOPSIS the cell consists of two sets: a support set (h) and an independent rejection set (g). Just as "how strongly do I support this airline's check-in speed" can have more than one plausible answer, "how strongly do I reject it" can have more than one plausible, independent answer as well. The sum of the two sets' largest elements cannot exceed 1. If the rejection set is empty (g = ∅), the cell reduces to the same information as in the plain hesitant structure; the dual structure only adds something where independent hesitation genuinely exists on the rejection side too. Weights are given from outside; if they are not known, DecisionMind can derive them from the set itself with a distance-based maximisation model.

No normalisation. Crisp TOPSIS's first step is column normalisation. There is no separate normalisation step for dual hesitant sets, because the support and rejection degrees already lie between 0 and 1; the cells are processed directly as they stand.

Ideal and anti-ideal point, distance. For every criterion, the ideal point is built from the largest element of the column's support sets and the smallest element of its rejection sets (the roles swap for a cost criterion); the anti-ideal is the exact opposite. The distance between two sets is measured with a generalised distance that uses both the sets' means (G_m, G_n) and their spread (V_m, V_n, how "wide" the set is). This distance can compare support and rejection sets of different lengths directly, without equalising them (without adding a repeated element to either set); crisp TOPSIS has no such equalising problem because its cells are already single numbers.

DecisionMind uses fixed values, for this family, for the coefficients α and β that set the relative weight of the mean and the spread (summing to 1, default 0.5-0.5) and for λ, which sets the degree of the distance; these three parameters can be changed by the user and must be stated in the report.

The closeness coefficient is read as in crisp TOPSIS: it states how close one is to the ideal, given this set of options and these weights. The difference lies here: this coefficient comes from both the sets' means and their width (weighted by α and β), and the parameter λ determines how the distance is computed. The same data, with a different choice of α, β and λ, can give a different coefficient, even a different ranking.

Thus instead of writing:

"According to dual hesitant fuzzy TOPSIS, this option is the best"

the report should read:

"This coefficient has been computed with the given values of α, β and λ; if the coefficient gap between two options is small, this gap may be sensitive to the choice of parameters, and the report should show this"

This extension is appropriate when both support and rejection on a criterion are expressed, independently of one another, with more than one plausible degree. Example: part of an expert panel strongly supports an option, another part holds independent reservations about the same option for separate reasons, and the two groups' views are to be preserved without being collapsed into one shared number. If there is no separate hesitation on the rejection side, that is, if rejection is always computed as "1 minus support", plain hesitant TOPSIS is sufficient and the dual structure carries no extra information. Crisp TOPSIS's exit condition applies here too: if no compromise is acceptable on one criterion, this extension is also compensatory.

Deriving the rejection set from the support set. Filling the g set as "1 minus h" effectively collapses the dual structure into the plain hesitant structure and erases the independent information on the rejection side; the condition γ⁺+η⁺≤1 (γ⁺ the support set's largest element, η⁺ the rejection set's) must be checked in every cell.

Reporting without stating α, β and λ. Wang and colleagues' (2020) own paper shows that these three parameters can change the ranking; the values used must always be stated in the report.

Confusing the direction reversal with normalisation. For a cost criterion the ideal and anti-ideal roles swap; this is not a normalisation step, because this family has no separate normalisation step.

Comparing sets by equalising their length. The engine compares sets of different length directly; equalising length by repeating a set's missing element is unnecessary and misleading.

The governing principle is this:

Dual hesitant fuzzy TOPSIS exists to hold support and rejection as two independent sources; if the α, β, λ parameters are not stated in the report, the same data can produce a different ranking.