Plithogenic PROMETHEE

Four things change; the pairwise-comparison and flow logic does not.

Cells. In crisp PROMETHEE every cell is a single number. Here every cell consists of three numbers: a truth (T), an indeterminacy (I) and a falsity (F) degree, all three between 0 and 1. Each criterion also carries its own contradiction degree (c), stating how much "tension" that criterion has with the decision problem, that is, how cautiously it needs to be treated. Criterion weights stay crisp.

Contradiction adjustment and cost complementing. Crisp PROMETHEE reverses the sign of the difference on a cost criterion. Here every cell in a cost criterion is first complemented (T and F swap places), and then every cell is adjusted according to its own criterion's contradiction degree: as the contradiction degree rises, truth (T) is pulled up while indeterminacy (I) and falsity (F) are pulled down. This ensures that small score differences on a highly contradictory criterion are not exaggerated; in DecisionMind this shows up as the magnitude of every pairwise difference shrinking as the contradiction degree rises.

Difference and preference function. In crisp PROMETHEE the difference is a direct subtraction. Here a score ((1+T-2I-F)/2) is first computed for every cell; the signed difference between two alternatives takes its direction from which score is higher, and its magnitude comes from the real distance in T-I-F space (the root of the mean of the three squared components). This signed difference is passed through a fixed-threshold (V-shaped) preference function with a threshold of 0.5; the threshold is not taken from the user, as in crisp PROMETHEE, but is fixed inside the engine.

Result and defuzzification. The output, as in crisp PROMETHEE, is a single net-flow number directly. Indeterminacy, that is, the T-I-F triple, together with the contradiction degree, is already consumed in the score and distance calculation before the pairwise comparison; it is not carried through in layers to the result.

DecisionMind fixes, in this entry, the contradiction adjustment, the fixed V-shaped preference function (threshold 0.5), and crisp PROMETHEE's own flow mechanics. In some members of the P-* family (P-VIKOR, P-COCOSO) the contradiction degree has been found to cancel out algebraically together with the score function, leaving the result unchanged; this is not the case in P-PROMETHEE. In the illustrative example below, distributing the contradiction degrees differently among the criteria actually changes the ranking; this has been independently tested in Python.

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

The difference is here: the magnitude of the net flow depends directly on the criteria's contradiction degree. If every criterion's contradiction degree is raised together, all the net flows shrink in the same proportion, but the ranking does not change; this should be read as "when everything is treated more cautiously, the differences also shrink." If contradiction degrees are distributed DIFFERENTLY across criteria, that is, one criterion is treated as highly contradictory and another as not contradictory at all, the ranking can genuinely change; the illustrative example below shows this.

Thus instead of writing:

"According to P-PROMETHEE, A2 is the best alternative"

the report should read:

"With the stated contradiction degrees, A2's net flow of 0.109 places it first; this ranking is sensitive to how contradictory each criterion is judged to be, and this distribution should be justified in the report"

Use this extension where criteria are expressed as a truth-indeterminacy-falsity triple, and where how "tense" a criterion is with the decision problem, its contradiction degree, is separately known. Crisp PROMETHEE's exit conditions apply equally here: if the number of alternatives exceeds fifty, the pairwise-comparison burden grows large; where a full ranking is not needed and an honest partial relation is preferred, PROMETHEE I's reading should be considered. Where criteria are already measured, or the contradiction degree is undefined, this extension adds no information; a measured criterion can be written as T=value, I=0, F=1-value, but this only reproduces crisp PROMETHEE in a more complicated form.

Violating the value space. Each of the T, I and F degrees must lie between 0 and 1; the contradiction adjustment defined by the founding source becomes meaningless with inputs that fall outside this range.

Leaving the contradiction degree at zero (or all equal) without ever questioning it. A zero contradiction degree means treating that criterion with no caution at all; this is an assumption, and the report should show how the same score difference could feed through differently into the ranking under a different contradiction degree.

Forgetting that the preference-function threshold (0.5) is fixed, and reporting it as though it were a user setting. This threshold is fixed inside the engine; defining a per-criterion p-threshold, as in crisp PROMETHEE, is not possible here.

Defuzzifying first and then running crisp PROMETHEE. Reducing the T-I-F triple to a single number from the outset and then applying the crisp method is not this extension; the contribution of both the contradiction adjustment and the T-I-F distance is erased at the first step.

The governing principle is this:

Plithogenic PROMETHEE's contribution is to take into account, before the pairwise comparison, how "contradictory" a criterion needs to be treated; if this rating is left at its default without ever being questioned, the extension's only contribution is effectively switched off.