Plithogenic SAW

Four things change; the decision logic does not.

Cells. In crisp SAW every cell is a single number. Here every cell consists of three degrees: truth, indeterminacy, falsity. In addition, every criterion is given a contradiction degree relative to whichever criterion is taken as dominant within the criterion set. The dominant criterion's contradiction is zero; the other criteria take a number according to how far they oppose it. This applies the sub-alternative idea from the plithogenic data-type card at the criterion level: here the criteria themselves are treated as sub-alternatives of a single overall suitability question.

Scale equalisation. Crisp SAW divides every column by its own best value. Here equalisation works differently. For a cost criterion, the truth and falsity degrees are swapped first; this is the plithogenic counterpart of reversing the three numbers. Each criterion's contradiction degree is then worked into the cell: the cell is pulled, by an amount equal to its contradiction, towards the dominant criterion's reference value, that is, fully true, zero indeterminacy, zero false. Because the dominant criterion's contradiction is zero, this step never changes it; the most contradictory criterion's cell changes the most.

Combination. Crisp SAW multiplies the equalised columns by their weights and sums them directly. Here a plithogenic weighted average is used instead of a sum. Each criterion's adjusted three degrees combine with a power equal to the criterion's weight; on the truth dimension the weighted product of the probability of "not being true at all" is taken and subtracted from one, while on the indeterminacy and falsity dimensions the weighted product is taken directly. This is a multiplicative combination, unlike crisp SAW's additive average.

Result and defuzzification. A single score is drawn from the three combined degrees: truth plus, twice indeterminacy and falsity minus, divided by two. Crisp SAW has no defuzzification step because the calculation already starts from a single number; here defuzzification happens at the last step, through this score function that reduces the three degrees to a single score.

DecisionMind fixes, for classical Plithogenic SAW, the way the contradiction degree pulls towards the dominant value, the weighted combination, and this score function. Weights are taken from outside and must sum to 1; the method does not generate weights.

The score is read like crisp SAW's total score: it produces a ranking only within this alternative set, and it is not a percentage or a probability. The difference is here. Beneath the score now lies a three-degree structure and an assumption about contradiction between criteria; if that assumption changes, so does the score. Which criterion is dominant, and how the contradiction degrees are assigned, determine the result as much as the weights do.

Thus instead of writing:

"Plithogenic SAW has found the best alternative"

the report should read:

"With these weights, these truth-indeterminacy-falsity degrees and this contradiction assumption, the alternative with the highest score is this one; the ranking may change if the dominant criterion or the contradiction degrees change"

If criteria are measured, crisp SAW is enough. Plithogenic SAW is used when criteria are given by expert judgement as degrees of truth, indeterminacy and falsity, one criterion is clearly dominant over the others, and the contradiction between them rests on a measurable basis. If the contradiction degree does not come from a measurable basis, that is, if it has been given by feel, this structure should not be built, as the plithogenic data-type card warns; the result would then rest on an invented number. The matrix must be of a single kind: if some criteria are crisp, these too are written as a cell in which all three components are equal, fully true, zero indeterminacy, zero false. SAW's exit condition applies here too: if no compromise is acceptable on one criterion, this extension is also compensatory and will not screen out anything below a threshold.

Confusing the contradiction degree with the criterion's importance. As the plithogenic data-type card also warns, the contradiction degree measures how far a criterion opposes the dominant criterion, not how important it is. Importance is already given by the weight; raising the contradiction as if it were importance too counts the same information twice.

Assigning the contradiction degree without justification. The calculation uses this number to decide how far to pull each criterion's cell towards the dominant criterion's reference value. If the number is given by feel, the defuzzification is also done by feel.

Changing the dominant criterion after the analysis is finished. Which criterion is taken as dominant, and the others' contradiction relative to it, are declared before the analysis. Changing the dominant criterion afterwards invalidates every contradiction degree and, with it, the result.

Inventing a contradiction degree for a criterion that has no sub-alternative. If a criterion genuinely has no contradiction positioned relative to a dominant reference, a measured price, say, this structure becomes indistinguishable from a neutrosophic one. Adding a contradiction degree then only adds apparent complexity.

The governing principle is this:

The contradiction degree is a number that must come from a measurable basis measuring how far a criterion opposes the dominant reference; if it is confused with importance or invented, Plithogenic SAW's one distinctive contribution, this distinction, is lost.