Neutrosophic PROMETHEE (Xu, Wei, Ding & Bin, 2020)

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 three degrees: truth (T), indeterminacy (I) and falsity (F); each lies between 0 and 1, all three are independent of one another, and their sum is not forced to equal 1. Weights remain crisp and are taken from outside, summing to 1; the method does not generate weights, and group decisions are not supported by a separate mechanism at this entry.

Complementing for cost criteria. In crisp PROMETHEE, the difference on a cost criterion is taken with a reversed sign. Here every (T, I, F) triple in a cost column is converted to its complement, (F, 1−I, T), so that every criterion is reduced to a single "higher is better" representation.

Score and signed distance. In crisp PROMETHEE the difference is a direct subtraction. Here two things are computed: first, every triple is reduced to a single number with Xu and colleagues' (2020) score, (2 + T − I − F)/3; whichever score is larger determines the sign of the difference. The magnitude of the difference is computed with a three-component Euclidean distance, the square root of the mean of the squared differences of the T, I and F components taken separately. The sign comes from the score, the magnitude from the distance; the two are not multiplied, the sign is applied to the magnitude.

Preference function and flows. This signed distance is passed through a thresholded preference function (default threshold 0.5): the preference is zero if the distance is negative, exactly one if it exceeds the threshold, and rises linearly in between. It is multiplied by the criterion weight and summed; the entering flow (φ⁺) and the leaving flow (φ⁻) are computed with the same averaging definition as in crisp PROMETHEE.

DecisionMind fixes, at this entry, Xu and colleagues' (2020) score, (2+T−I−F)/3, and the T-I-F distance; this should not be confused with the general neutrosophic score shortcut on the data-type card, (T−F+1)/2, which gives a different number. According to the manifest's own note, these steps are faithful to the algorithm in Xu and colleagues' (2020) article; there is no early simplification here of the kind found in the fuzzy and intuitionistic fuzzy PROMETHEE entries.

The net flow is a value as in crisp PROMETHEE, and it is read the same way: it is assessed together with the entering and leaving flows, and is not compared with another method's score.

What differs is this: this net flow has reduced the truth-indeterminacy-falsity triple to both a score and a distance. The indeterminacy (I) component enters only the magnitude of the distance, not the sign of the score directly; that is, the difference between two alternatives with high indeterminacy can produce a larger degree of preference than the same score difference with low indeterminacy. For this reason, alongside the net flow, the report should separately state on which criteria indeterminacy (I) is high and how this has enlarged the degree of preference.

Thus instead of writing:

"According to neutrosophic PROMETHEE, A3 is clear because contradictory evidence has been taken into account"

the report should read:

"The truth-indeterminacy-falsity triples have been processed with Xu and colleagues' (2020) score and the T-I-F distance; A3 has the highest net flow, and this result is also influenced by the enlarged distance on the criteria where indeterminacy is high"

Use this extension when information about a criterion is missing, inconsistent or contradictory, and this matters enough to affect the decision: evaluating an alternative with no track record, conflicting expert reports, partially observable systems. If the evaluation rests on a reliable measurement, or experts do not report a separate indeterminacy share, moving to the neutrosophic structure adds nothing.

If the sum of the three components is forced to equal 1 (T+I+F=1), the intuitionistic fuzzy structure should in fact be used; the rationale for the neutrosophic structure is that its components are independent of one another. The base method should be kept when criteria are measured. Turning a measured proportion directly into T and deriving I as 1−T−F does not produce neutrosophic data; it zeroes out the independence. If the table is mixed, DecisionMind requires a single data type. Crisp PROMETHEE's exit conditions apply here in exactly the same way: if the number of alternatives exceeds fifty, the burden of pairwise comparison grows large, and if a full ranking is not needed but an honest partial relation is, PROMETHEE I's reading, or the ELECTRE family, should be considered.

Forcing the sum of the three components to equal 1. T, I and F come from independent sources; trying to complete their sum to 1 (for instance, computing I as 1−T−F) reduces the structure to an intuitionistic fuzzy pair and removes the rationale for using neutrosophic data.

Value-range violation. T, I and F must each lie between 0 and 1; a component that falls outside this range must be corrected before it enters the preference function.

Changing the score function without stating it. DecisionMind uses the score (2+T−I−F)/3 at this entry; the general shortcut on the data-type card, (T−F+1)/2, gives a different number and can lead to a different ranking. Which score is used must be stated in the report.

The governing principle is this:

Neutrosophic PROMETHEE's contribution is to carry truth, indeterminacy and falsity without collapsing them into one another; forcing the sum of the components, exceeding their range, or changing the score function without stating it removes this contribution.