Fermatean Fuzzy PROMETHEE (Akram and Bibi, 2023)

Four things change; the logic of the preference function and the flows does not.

Cells. In crisp PROMETHEE every cell is a single number. Here every cell consists of four parts: a term and a small shift for the support side, and a separate term and shift for the rejection side. The term is chosen from a pre-declared, nine-term linguistic scale (from weakest to best); the shift is the same 2-tuple idea found on the linguistic data-type card, and it prevents information loss. The sum of the cubes of the numerical counterparts of the support and rejection terms cannot exceed 1; this admits a wider region than intuitionistic fuzzy data, so a judgement can be both strongly supported and strongly rejected. This extension directly supports group decisions: judgements given in this form by multiple decision-makers are combined, each with its own weight, into a single shared judgement. Criterion weights can likewise be collected from multiple decision-makers in the same form; DecisionMind combines them and then converts them into a single number for use.

Reducing the difference to a single number. In crisp PROMETHEE the difference between two alternatives on a criterion is a straight subtraction. Here every shared judgement is first reduced to a single number by a score function (in the logic of the support side's cube minus the rejection side's cube); the difference is then taken between these reduced numbers. This does the same job as the centroid reduction in fuzzy PROMETHEE: it produces a real number that can be compared against the preference function's thresholds. Everything after this step, the preference function, the entering and leaving flows and the net flow, is exactly the same as crisp PROMETHEE's own steps.

Preference function and flows. The default preference function is the thresholdless Gaussian form (spread 0.5); crisp PROMETHEE's simplest, absolute, function can also be chosen. The function type and spread are settings exposed to the user. The reduced difference is passed through this function, multiplied by the criterion weight and summed; the entering and leaving flows are computed with the same average definition as crisp PROMETHEE.

Result and defuzzification. The output, as in crisp PROMETHEE, is directly a single net flow number. The uncertainty, that is, the term-shift and support-rejection structure, has already been consumed by the score function before the pairwise comparison; it is not carried through to the result in layered form.

DecisionMind holds fixed, for this entry point, the combination of multiple decision-makers' judgements and weights, the early defuzzification through the score function that follows, and crisp PROMETHEE's own preference-function/flow mechanics.

The net flow is read as in crisp PROMETHEE: it is assessed together with the entering and leaving flows, and it cannot be compared with a score from a different method. See the PROMETHEE card.

The difference is here: this net flow has already reduced both the views of multiple decision-makers and every judgement's term-shift and support-rejection structure to a single number at the very first step. In the literature example below, the gap between the highest net-flow candidate and the second is large, and this gap is preserved even when the preference function is changed from Gaussian to the simplest type; the position of the second- and third-ranked candidates, however, is sensitive to this choice.

Thus instead of writing:

"According to FF-PROMETHEE, T5 is the best candidate"

the report should read:

"Four decision-makers' term-shift and support-rejection judgements have been combined and processed with the Gaussian preference function; T5's net flow of 4.51 is clearly first, and this first place is preserved even when the preference function type is changed; the position of the second- and third-ranked candidates, however, is sensitive to this choice"

Use this when multiple decision-makers' judgements need to be combined, and these judgements are expressed both with a linguistic term and with a separate support-rejection split. This is Akram and Bibi's (2023) own field of application: a corporate personnel selection in which several assessors report, separately, how strongly they support and how much reservation they hold on criteria such as "leadership" and "technical skill".

If the criteria are measured, or if there is only one decision-maker, this layered structure adds no information; a measured criterion could be pinned to the highest term with zero shift and its rejection side set to the lowest term, but this only repeats crisp PROMETHEE in a more complicated form. Crisp PROMETHEE's own exit conditions apply here too: if the number of alternatives exceeds fifty, the pairwise-comparison burden grows; if a full ranking is not needed but an honest partial relation is, PROMETHEE I's reading should be considered.

Violating the value space. The sum of the cubes of the numerical counterparts of the support and rejection terms must not exceed 1; otherwise the score function is pushed into an invalid range.

Taking decision-maker weights as equal without question. If multiple decision-makers' judgements are combined with equal weight, this is an assumption; if one decision-maker is less familiar with the subject, equal weighting can give that person's view undue prominence.

Leaving the preference function type and spread at their defaults without question. The Gaussian spread (default 0.5) determines how much a small difference is treated as a preference; this value should be calibrated to the criterion's own scale, and the choice should be stated in the report.

Passing over the direction of the difference on a cost criterion without confirming it. According to the founding paper, the sign of the difference must be reversed on cost criteria. In the example below, this step does not come into play because all the criteria are "higher is better"; if a "lower is better" criterion is to be added, whether the direction reversal has been applied correctly must be separately tested before the calculation.

The governing principle is this:

FF-PROMETHEE's contribution is to reduce multiple decision-makers' linguistic, support-rejection judgements honestly to a single number and carry it into PROMETHEE's own preference mechanics; if the assumptions behind this reduction (decision-maker weight, preference function, direction) are not stated, the result is left unjustified.

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