Plithogenic CODAS

Four things change. The two-distance comparison logic does not.

Cells. In crisp CODAS every cell is a single number. Here every cell is three numbers: truth (T), indeterminacy (I), falsity (F); all three lie between 0 and 1. One criterion is taken as dominant, with a contradiction degree of zero; how far each other criterion opposes this dominant criterion is given separately 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 input from the contradiction degree and are supplied from outside.

Scale equalisation. Crisp CODAS divides the value by the column maximum for a benefit criterion, and divides the column minimum by the value for a cost criterion. There is no such division here. First, for a cost criterion, the triple is reversed: (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, while indeterminacy and falsity are shrunk by the same proportion. This contradiction adjustment takes the place of crisp CODAS's division step.

Weighting and the negative-ideal. The adjusted cells are combined with the criterion weight using the power weighted-average rule (PNWA). The negative-ideal is built, component by component, from the worst value on every criterion (the smallest T, the largest I, the largest F). As in crisp CODAS, a single "worst" reference is used here too; no separate ideal point is constructed as it would be in TOPSIS.

Distance and the assessment score. Euclidean and Taxicab distance are computed from the T-I-F differences between the weighted cells and the negative-ideal. The pairwise comparison rule (threshold ψ = 0.02, Euclidean first, Taxicab if the difference does not exceed it) and the assessment score are built in exactly the same form as in crisp CODAS. Uncertainty is not defuzzified separately at any step; it dissolves directly inside the distance calculation.

DecisionMind holds the contradiction adjustment, the power weighting and the threshold value fixed in classical P-CODAS. Weights and contradiction degrees are taken from outside, separately; the method generates neither.

The assessment score is read exactly as in crisp CODAS: it is a relative positional measure, not a percentage, and the negative-ideal is rebuilt whenever the alternative set changes. The difference lies here: beneath the score sits both T-I-F uncertainty and an assumption about the contradiction degree between criteria. In the illustrative example below, A2 leads clearly at 0.0878; the gap between A3 (−0.0321) and A1 (−0.0471) is only 0.0150, and the ranking of these two reverses once C2's contradiction degree is raised from 0.33 to 0.50.

Thus instead of writing:

"The result is more reliable because plithogenic CODAS also takes contradiction into account"

the report should read:

"The contradiction degree between criteria has been assigned under the following assumption; A2 leads clearly, and the ranking of A3 against A1 is sensitive to this assumption"

Taking contradiction into account does not automatically make the result more correct; it merely makes visible which assumption is bearing on the ranking.

Use 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. The same reasoning applies where close rivals among the alternatives need to be separated by a second distance measure. Where the T-I-F triple alone suffices and no such dominance-contradiction relationship exists between criteria, neutrosophic CODAS (n-codas) is already sufficient; the plithogenic form adds one further assumption, and if that assumption is supplied without justification it creates a spurious distinction.

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

Assigning the contradiction degree without justification. In the illustrative example, raising C2's contradiction degree from 0.33 to 0.50 reverses the ranking of A1 and A3. If this degree is set "by feel," the ranking becomes a matter of feeling too; the degree must rest on a measurable justification for how independent, or how overlapping, a criterion's information is relative to the dominant criterion.

Choosing the dominant criterion arbitrarily. Which criterion is treated as having zero contradiction is a decision, and it must be justified in the report.

Skipping the cost complementation. Moving to the contradiction adjustment without rewriting (T, I, F) as (F, I, T) for a cost criterion builds the negative-ideal from the wrong end of the criterion, and the ranking becomes meaningless.

Leaving the threshold value (ψ) unquestioned. The same mistake as in crisp CODAS applies here: a report should not be written without testing whether the ranking between two close alternatives is sensitive to the threshold.

The governing principle is this:

The contradiction degree is a criterion's share of independence or contradiction, grounded in a measurable basis; an unjustified number makes the ranking CODAS produces unjustified too.