Dual hesitant fuzzy VIKOR (An, Zhang, Liu and Zuo, 2025)

Five things change; the most important is the direction of ranking in the last item.

Cells and the number of decision-makers. In crisp VIKOR every cell is a single number. Here every cell carries a support set and a rejection set; every element of each set belongs to one of K decision-makers (K = 3 in this extension). If a decision-maker has no opinion on a cell, that slot is marked "/" and left blank; the gaps in the support and rejection sets are independent of one another, so one decision-maker's support may be missing while their rejection is recorded, or the reverse. The decision-makers' own weights (λ) are normalised to sum to 1.

Decision-maker aggregation. DecisionMind first scales every cell by decision-maker weight (Ē = λ⊗E): each decision-maker's given support and rejection value is multiplied by their own λ, and blank slots are dropped after this scaling. With a single decision-maker this step changes nothing.

A direction-sensitive reference point. Crisp VIKOR reads the genuine best and worst value on every criterion from the column. Here the reference point is not read from the data; it is defined directly: for a benefit criterion the ideal is ⟨{1},{0}⟩ (full support, no rejection at all) and the anti-ideal is ⟨{0},{1}⟩; for a cost criterion the two swap. Every option's distance to these two fixed points is measured with a hybrid hesitant distance (the sets' most extreme elements plus the difference between the sets' means).

From partly known weight to full weight. Crisp VIKOR takes the weight fully from outside. Here the senior manager knows the weight of only some criteria (in this example, only one criterion); the rest are unknown. DecisionMind completes the unknown weights through an optimisation that combines each decision-maker's own subjective weight estimate with a data-based objective that best separates the options, balanced by a coefficient α. α = 1 rests on the data alone, α = 0 on the decision-makers' subjective estimate alone; the default of 0.5 weighs the two equally. The user may also enter all weights directly, in which case the optimisation does not run at all.

Result and ranking direction: the reverse of crisp VIKOR. S (group utility) and R (individual regret) are defined as in crisp VIKOR: S is the sum of weighted distances, R is the largest of them. Q is a compromise coefficient β's normalised combination of S and R; β plays the same role as v in classical VIKOR. But An and colleagues (2025) scale S and R here, in their own normalisation, against the largest value. They rank in DESCENDING order of Q: the HIGHEST Q is the best option. This is the exact opposite of the classical Opricovic-Tzeng rule, where the lowest Q is best. DecisionMind applies this reversal exactly as given, and does not silently convert it back to the classical rule. Neither of crisp VIKOR's two compromise conditions (acceptable advantage, acceptable stability) appears in this extension; the engine does not compute them and the manifest does not define them. An and colleagues' own method rests on the Q ranking alone.

In this extension, a large Q is good, not a small one. This is the exact opposite of every other VIKOR extension in the family (and of crisp VIKOR itself) and must be stated explicitly in the report every time; otherwise a reader will bring the "lowest Q wins" habit and pick the wrong option. S and R must be read separately: a small S shows overall closeness to the ideal, a small R shows little loss on the worst criterion, exactly as in crisp VIKOR; the difference is that these two values feed into Q in the DESCENDING direction. Classical VIKOR's two compromise conditions are absent here; the distinction between "a single compromise solution" and "a compromise set" is never drawn in this extension, there is only a ranking.

Thus instead of writing:

"According to dual hesitant fuzzy VIKOR, the option with the lowest Q wins"

the report should read:

"In this extension, by An and colleagues' (2025) own definition, the HIGHEST Q is the best option; this runs opposite to classical VIKOR, and the difference must be stated explicitly in the report"

This extension is appropriate when several decision-makers' opinions have been collected separately, some decision-makers have not given an opinion on some criteria, and part of the criterion weights has been fixed by management while the rest needs to be derived from the data and from the decision-makers' subjective estimates. If a single decision-maker has filled in a table with complete information, this complexity is not necessary; HF-VIKOR or crisp VIKOR is sufficient. Crisp VIKOR's exit condition applies here too: if no compromise at all is acceptable on one criterion, this extension limits regret but does not eliminate it.

Reversing the ranking direction. In this extension the highest Q is best; taking the engine's output and saying, out of habit, "take the lowest Q" reads the result the wrong way round from the start. This is a warning the manifest states of its own accord.

Treating a blank "/" slot as zero. A slot where a decision-maker gave no opinion is marked "/" and never enters the calculation; reading it as 0 adds a false signal, as if that decision-maker had said "I do not support this at all".

Assuming that gaps in support and rejection travel together. One decision-maker's support may be missing while their rejection is recorded, or the reverse; Case 1's table contains exactly such disjoint examples. Treating the two as present or absent together in the same slot causes a loss of information.

Mistaking the optimisation-derived weight for a simple average. The partly known weights are combined with the decision-makers' subjective estimates and a data-based objective through a balance coefficient (α); this is not a plain average of the decision-makers' subjective weights. Saying "the weights are an average" without changing α is incorrect.

Still looking here for classical VIKOR's two conditions. Acceptable advantage and acceptable stability are neither computed nor defined in this extension; the term "compromise set" is not used here.

The governing principle is this:

In this extension a large Q is good; this reversal is An and colleagues' (2025) own definition, not an error, but if it is not stated explicitly every time, the reader is misled towards the wrong option.