Plithogenic TOPSIS

Four things change; the decision logic does not.

Cells. In crisp TOPSIS every cell is a single number. Here every cell is three numbers: truth (T), indeterminacy (I) and falsity (F); all three lie between 0 and 1. The sub-alternative idea from the plithogenic data-type card is carried, in this family, to the criterion level. DecisionMind treats the criteria themselves as parts of a whole. One criterion is taken as dominant, with a contradiction degree of zero. How far the other criteria oppose this dominant one is given as a contradiction degree between 0 and 1. This degree belongs to the criterion; the same number is used for every alternative under that criterion and does not vary from one alternative to another. Criterion weights (numbers summing to 1) are a separate input from the contradiction degree, and are supplied from outside as in neutrosophic TOPSIS.

Scale equalisation. Crisp TOPSIS divides every column by the square root of the sum of its squares. There is no such division in this family. First, for a cost criterion the triple is reversed: (T, I, F) is rewritten as (F, I, T), so that a high T on a "less is better" criterion still points to good. Each cell is then adjusted by its own criterion's contradiction degree: truth is enlarged towards the dominant criterion, while indeterminacy and falsity shrink by the same proportion. This contradiction adjustment replaces crisp TOPSIS's sum-of-squares scale equalisation.

Distance. The adjusted cells are weighted exponentially by criterion weight, the same power-weighting formula used in neutrosophic TOPSIS. The ideal and anti-ideal are built, component by component, from the best and worst of the weighted cells on each criterion. The value with the largest truth and the smallest indeterminacy and falsity is taken as ideal. Distance is measured as the square root of a third of the sum of the squared differences of the three components, then multiplied by the weight; this is the same vertex distance used in fuzzy and neutrosophic TOPSIS.

Closeness score. The definition is as in crisp TOPSIS: the ratio of the distance to the anti-ideal over the sum of the two distances. Indeterminacy (I) and falsity (F) are not defuzzified in a separate step; they dissolve directly into the distance calculation.

DecisionMind fixes, in this family, the cost complement, the contradiction adjustment, the power weighting, and the component-based construction of the ideal and anti-ideal. Weights and contradiction degrees are supplied from outside and separately; the method generates neither.

The output is a closeness score and a ranking, as in crisp TOPSIS, and it is read the same way: it is not a percentage, it is not compared with a different analysis, and the ideal and anti-ideal change when the alternative set changes.

The difference is here: beneath the score lie both the T-I-F uncertainty and an assumption about the contradiction between criteria. In the illustrative example below, A2 is clearly ahead at 0.7019; the gap between A3 (0.3916) and A1 (0.3861) is only 0.0055, and this pair's order reverses once C2's contradiction degree is raised from 0.33 to 0.40. The report should show which weight and which contradiction degree a narrow ranking rests on.

Thus instead of writing:

"Plithogenic TOPSIS also accounts for contradiction, so the result is more reliable"

the report should read:

"The contradiction degree between criteria has been given under this assumption; A2 is clearly ahead, and the order between A3 and A1 is sensitive to this assumption"

Taking contradiction into account does not automatically make the result more accurate; it only makes visible which assumption is bound up in the ranking.

Use this extension when your criterion scores are given as a truth-indeterminacy-falsity triple and some criteria are a more "independent" or more "contradictory" source of information than others. If the T-I-F triple alone is sufficient and no such dominance-contradiction relationship exists between the criteria, neutrosophic TOPSIS (n-topsis) is already enough; the plithogenic form adds one further assumption, and if this assumption is given without justification it creates a false distinction.

Turning a measured criterion into a T-I-F triple is not modelling uncertainty but producing it; this principle applies here too. If the table is mixed, DecisionMind requires a single data type. If no compromise is acceptable on one criterion, this extension is also compensatory and will not screen out anything below a threshold.

Violating the value space. Every cell must be a valid T, I, F triple, and a contradiction degree must be declared for every criterion; running the method with a missing or invented contradiction degree invalidates it.

Assigning the contradiction degree without justification. In the illustrative example, raising C2's contradiction degree from 0.33 to 0.40 reverses the order of A1 and A3. If this degree is given "by feel," the ranking becomes a matter of feeling too; it must rest on a measurable basis for how independent or overlapping a criterion's information is relative to the dominant one.

Choosing the dominant criterion at random. Which criterion is taken to have zero contradiction is a decision and must be justified in the report; if the dominant criterion changes, the other criteria's contradiction degrees must be reconsidered too.

Changing the defuzzification method. The score function (truth plus, indeterminacy and falsity minus, weighted) is fixed; comparing against a different defuzzification formula is a difference of definition, not of method.

The governing principle is this:

The contradiction degree is a criterion's share of independence or contradiction, drawn from a measurable basis; an unjustified number also makes the ranking TOPSIS produces unjustified.