Hesitant fuzzy linguistic PROMETHEE (Liang, Wang and Zhang, 2018)

Three things change; the logic of the preference function and the flows stays the same.

Cells. In crisp PROMETHEE every cell is a single number. Here every cell is more than one term given together, chosen from a linguistic term set announced in advance (from "very bad" to "very good," terms ordered s₀ to s₆). This honestly carries situations such as the decision-maker being unable to decide between "good" and "very good," or several experts proposing different terms; it does not reduce them to a single term too early. Criterion weights remain crisp; this extension carries more than one decision-maker's judgement not through a separate aggregation step, but directly through the multi-term set in the cell itself.

Reduction to a closeness degree. In crisp PROMETHEE, the difference between two alternatives on a criterion is a direct subtraction. Here, the set carrying the highest term among all alternatives is first declared the "positive ideal" for each criterion. The average of every cell's terms is then scaled against this ideal; this ratio is combined with a parameter called "risk attitude" (risk-seeking, risk-averse or neutral) and converted into a single closeness degree (RC) between 0 and 1. The difference is taken between these closeness degrees and passed through the preference function; everything after this, that is the positive and negative flows and the net flow, is identical, step for step, to crisp PROMETHEE.

Cost direction. In crisp PROMETHEE, the difference is reversed on a cost criterion. Here the term set itself is reversed: every term in the sequence is moved to its mirror position in the term order (on a seven-term scale, for example, s₁ becomes s₅). This preserves the criterion's lower-is-better meaning at the linguistic-term level.

For this entry, DecisionMind fixes the derivation of the positive ideal from all alternatives, and the closeness-degree formula. The default preference function and risk attitude (neutral, 0.5) are the engine's own defaults; neither is currently offered as a setting adjustable from the interface, and both can only be changed through advanced input.

The net flow is read as in crisp PROMETHEE: it is assessed together with the positive and negative flows, and it cannot be compared with another method's score. See the PROMETHEE card.

The difference is this: the net flow has already reduced both the term uncertainty and the risk attitude to a single closeness degree at the first step. The default preference function ("usual") looks only at the sign of the difference between two alternatives, not its magnitude. So if an alternative is already ahead by a clear margin, widening that margin further does not change the net flow; the method gives no extra credit to an alternative that is "already winning." This is confirmed in the illustrative example below: if A1's advantage on one criterion is widened further, the net flow stays the same, but when A3, already behind, is improved on the same criterion, its net flow genuinely rises.

Thus instead of writing:

"According to HFL-PROMETHEE, A1 is by far the best alternative"

the report should read:

"With the 'usual' preference function chosen, A1 ranks first with a net flow of 0.80; this superiority shows only which alternative is ahead on which criterion, not the size of the gap"

Use this extension where the decision-maker expresses a judgement not as a single verbal term but with several terms, as in "between this term and that term," or where several experts propose different terms. A typical setting is a board decision where experts cannot agree on a single term for an unmeasurable criterion such as service quality or organisational fit.

Where criteria are already measured or can be comfortably expressed with a single term, this layered structure adds no information; a single term can be written as a one-element set, but this only repeats crisp or linguistic PROMETHEE in a more complicated form. Crisp PROMETHEE's exit conditions apply here too: if the number of alternatives exceeds fifty, the pairwise-comparison burden grows; where an honest partial relation rather than a full ranking is needed, PROMETHEE I's reading should be considered.

Changing the term-set size from expert to expert. If one expert uses a seven-term scale and another a five-term scale, the same term ("good," say) corresponds to a different numerical value, and the cost-direction reversal is thrown off too. The scale must be declared once, before the analysis.

Leaving the risk attitude at its default (neutral) without ever questioning it. This attitude only changes the result together with a magnitude-sensitive preference function (other than usual); when working with the usual function, the attitude has no effect on the result at all, and this should be reported.

Assuming that widening an "already winning" alternative's gap will also widen its net flow. This is not true under the usual preference function; the net flow looks only at who outranks whom, not by how much.

Not equalising hesitant sets of different sizes (two terms in one cell, four in its neighbour, say) before comparing them. The averaging step does this implicitly, but if the sets were built inconsistently (not drawn from the same scale), the average becomes meaningless.

The governing principle is this:

Hesitant fuzzy linguistic PROMETHEE's contribution is to carry a judgement set honestly, without asking too early which term is exactly right. If which scale the set was drawn from, and which preference function processed it, are not stated, the net flow is left unjustified.