Plithogenic CoCoSo

Three things change; the five-step skeleton and the decision logic do not.

Cells. In crisp CoCoSo every cell is a single number. Here every cell is a truth (T), indeterminacy (I), falsity (F) triple. As on the plithogenic data-type card, DecisionMind treats the criteria themselves as parts of one whole: one criterion is taken as dominant and its contradiction degree is zero, and how far each other criterion opposes the dominant one is given by a contradiction degree between 0 and 1. This degree belongs to the criterion, not to the alternative; it does not vary from alternative to alternative. Criterion weights are a separate, externally supplied input, distinct from the contradiction degree.

Contradiction adjustment and score. For a cost criterion the triple is first complemented: (T, I, F) is rewritten as (F, I, T). Every cell is then adjusted by its own criterion's contradiction degree: truth is enlarged towards the dominant criterion (T + c·(1−T)), while indeterminacy and falsity are shrunk by the same proportion (I·(1−c), F·(1−c)). This adjusted triple is then reduced to a single number with the same score function as neutrosophic CoCoSo ((1+T−2I−F)/2); this score is shifted by 1 and rescaled to 0–1 on a column-by-column basis.

An important limit: the contradiction degree can be shown not to change the result. Because of the algebraic structure of the score function, a cell's contradiction-adjusted score is a fixed linear transformation of its original score (the c = 0 case): s(c) = c + (1−c)·s(0). The coefficients of this transformation (c and 1−c) depend only on c, NOT ON THE ALTERNATIVE; they are the same for every alternative in the same column. Column-wise min-max scaling absorbs exactly this kind of linear transformation by definition. The result: in preparing this card, the kernel was run directly and checked, and it was found that when the contradiction degree is set to any value between 0 and 0.95, the scaled scores, and hence the final K scores, do NOT change to 10 decimal places (see the verification notes). In other words, in DecisionMind's current P-COCOSO engine the contradiction-degree parameter has no effect whatsoever on the result after the min-max scaling step, whatever value is entered.

DecisionMind holds the defuzzification score and the balance coefficient in the third compromise strategy (λ = 0.5) fixed in this extension. Weights come from outside; the method does not generate weights.

As in crisp CoCoSo, the K score is only a ranking within this particular alternative set. Whether the three compromise strategies agree with one another indicates robustness.

The real difference shows up here: even though the report may present the contradiction-degree input as "the independence/contradiction between criteria has been taken into account," the algebraic property above means this information is never reflected in the K score in any way. Whatever contradiction degree is entered (0 or 0.95, it makes no difference), the result is identical.

Thus instead of writing:

"Because the degree of contradiction between criteria has also been taken into account, this result rests on a more detailed model than P-TOPSIS"

the report should read:

"The contradiction-degree input becomes inert after min-max scaling in this engine, because of the structure of the score function; the K score is identical to what it would be had no contradiction degrees been entered at all"

Consider this extension when your criterion scores are given as truth-indeterminacy-falsity triples and some criteria are a more independent, or more contradictory, source of information than others. But because of the finding above, this extension's contradiction-degree input does NOT provide any additional discriminating power in today's engine; the result is effectively indistinguishable from neutrosophic CoCoSo (n-cocoso), which uses only the T-I-F triple. If it genuinely matters that contradiction between criteria be reflected in the result, this need is not met by P-COCOSO for the time being; the detail is in the verification notes.

Turning a measured criterion into a T-I-F triple is producing uncertainty, not modelling it; this principle holds here too. The exit condition of the crisp method holds here as well: the matrix must be of a single type, and if no compromise is acceptable on one criterion, this extension is compensatory too.

Taking the contradiction degree to be a meaningful input. As proven above, in today's engine the contradiction degree does not change the result after min-max scaling. Writing in the report "the contradiction degrees were set as follows and affected the result in this way" describes an effect that does not occur.

Choosing the dominant criterion arbitrarily. Even though the contradiction degree has no effect on the result, which criterion is treated as dominant (zero contradiction) must still be stated openly and justified in the report; this is required for the consistency of the data entry.

Values that spike when a column's spread narrows. If an alternative's score on a criterion draws very close to the other values in that column (near-equal), the min-max denominator shrinks, and the normalised values, and hence the K scores, can jump to unrealistic magnitudes. This is also a known weakness of crisp CoCoSo (the constant-column case); it can be more pronounced in the plithogenic extension because of the exponential weighting.

Skipping the cost complementation. If the calculation proceeds without rewriting (T, I, F) as (F, I, T) for a cost criterion, the highest-cost alternative appears to have been drawn towards the ideal.

The governing principle is this:

P-COCOSO's contradiction-degree input does not change the result after min-max scaling in today's engine, because of the algebraic structure of the score function; this is a proven property, independent of the dataset, and must not be concealed in the report.