Plithogenic GRA

Four things change. The grey relational formula in the fifth step does not.

Cells. In crisp GRA every cell is a single number. Here every cell is a truth (T), indeterminacy (I) and falsity (F) triple. As on the plithogenic data-type card, one criterion is taken as dominant, with a contradiction degree of zero. How far each other criterion opposes this 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.

Cost complementation and contradiction adjustment. 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 grows towards the dominant criterion (T plus the contradiction degree times one minus T), while indeterminacy and falsity shrink by the same proportion.

The reference is built from the positive ideal alone. In crisp GRA the reference sequence is a fixed, hypothetical alternative. Here the reference is a plithogenic ideal built from each column's own highest truth value and lowest indeterminacy-falsity values. Only the positive ideal is considered; no second, anti-ideal is built as it would be in hesitant GRA. Distance is computed as the square root of one-third of the sum of the squared differences of the three components.

Grey relational coefficient and degree. These distances are converted to a grey relational coefficient with the same formula as in crisp GRA, using a discrimination coefficient of ρ = 0.5, and then to a grey relational degree by weighted summation.

An important difference from its sibling extensions (P-COCOSO, P-VIKOR) needs stating here. In those two methods the contradiction degree never affects the result after column scaling, because of the score function's linear structure; this has been proved with Python in the earlier extension cards. In P-GRA the situation is different. The grey relational coefficient is a min-max ratio and is not linear; an independent Python test has shown that changing the contradiction degree also changes P-GRA's ranking (see the detail in Case 1 and in the verification notes).

DecisionMind holds the single-positive-ideal reference, the SVN Euclidean distance and the discrimination coefficient (ρ = 0.5) fixed in this extension. Weights are taken from outside as crisp numbers.

As in crisp GRA, the grey relational degree shows an alternative's relative closeness, within this analysis, to the plithogenic reference; it cannot be compared with a different analysis.

The difference lies here. The P-COCOSO and P-VIKOR cards proved that the contradiction degree never affects the result. In P-GRA the opposite holds: the contradiction degree can change both the grey relational coefficients and the final ranking. The report must therefore describe how the contradiction degree was set not as "an indicative input" in the manner of P-COCOSO, but as a decision that genuinely affects the result.

Thus instead of writing:

"As in P-COCOSO, the contradiction degree here too is a nominal parameter that does not change the result"

the report should read:

"In P-GRA the contradiction degree genuinely changes the result; A1 and A3 swap places when C2's and C3's contradiction degrees are raised to 0.95, so the contradiction degrees must be chosen with justification and stated in the report"

Consider this extension when your criterion scores are given as truth-indeterminacy-falsity triples and some criteria naturally split into sub-options whose mutual opposition matters to the decision. Detail is on the Plithogenic data-type card.

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 crisp GRA 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 and does not screen out anything below a threshold.

Assuming the contradiction degree is inert, as in P-COCOSO. Assuming this degree will not change the result here just because it does not in other members of the same family is wrong; in P-GRA the contradiction degree can genuinely change the ranking.

Choosing the dominant criterion without justification. Because it affects the result here, which criterion is treated as dominant (zero contradiction) must be chosen even more carefully, openly and with justification than in P-COCOSO.

Skipping the cost complementation. If (T, I, F) is not rewritten as (F, I, T) for a "less is better" criterion, the reference sequence is built relative to the most expensive, or worst, alternative.

Confusing it with the two-ideal family. Hesitant GRA looks at both a positive and a negative ideal and gives a closure ratio. P-GRA looks only at the positive ideal and gives a grey relational degree directly; the two calculations can produce different rankings.

Never questioning the discrimination coefficient. ρ = 0.5 is DecisionMind's fixed value; in tables where the distances are close to one another, this choice can affect the result.

The governing principle is this:

In P-GRA, unlike its sibling extensions, the contradiction degree genuinely changes the result; if this degree is assigned without justification, or assumed to be merely nominal, the ranking is read incorrectly.

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