Plithogenic TODIM

Three things change; the reference-criterion mechanism and loss aversion's magnifying of losses do not.

Cells. In crisp TODIM 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, DecisionMind treats the criteria themselves as parts of a whole: one criterion is taken as dominant, with a contradiction degree of zero, and the others' opposition to the dominant one is given as a contradiction degree between 0 and 1. This degree belongs to the criterion, not the alternative, and does not vary from one alternative to another. Criterion weights are supplied from outside as crisp numbers.

Contradiction adjustment and the reference criterion. For a cost criterion, the triple is complemented first: (T, I, F) is rewritten as (F, I, T). 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. After these two operations, crisp TODIM's reference-criterion step runs unchanged: the heaviest criterion is chosen as the reference, and the other criteria's weights are scaled relative to it.

Distance and gain-loss. In crisp TODIM, the gain-loss comparison is made over a single number. Here a score is first computed for each cell, and this score decides which alternative "wins." The magnitude, however, is not the score difference but a distance between the contradiction-adjusted T-I-F triples, based on the square root of the sum of the squared differences of the three components. On the winning side a positive contribution is summed, and on the losing side a negative contribution magnified by the loss-aversion coefficient θ; this follows the same logic as crisp TODIM's third step.

The contradiction degree's effect here differs from that in P-CoCoSo. In DecisionMind's Plithogenic CoCoSo extension it has been proven that, after a column-wise scaling, the contradiction degree never reaches the result at all. P-TODIM has no such column scaling; the contradiction degree directly changes the size of the pairwise distance, and this change carries through into the comparison between alternatives. In preparing this card the kernel was run directly, and it was confirmed that varying the contradiction degree changes the global values, and in some cases even the ranking between alternatives (see the verification notes).

Result. The global value is again a number normalised between 0 and 1; the lowest total dominance takes 0, the highest takes 1.

DecisionMind fixes, in this family, that the cost complement is applied before the contradiction adjustment. Weights are supplied from outside; the method does not generate weights.

As in crisp TODIM, the global value is a ranking only within this alternative set; the choice of θ also affects the result.

The difference is here: this value carries the combined effect of both θ and every criterion's contradiction degree. The contradiction degree here is not an ineffective ornament, as in P-CoCoSo; if one criterion's contradiction degree is raised, the contribution that criterion's differences make to the global value also changes.

Thus instead of writing:

"The contradiction degree was also entered, but the weights and θ remain the real determinants"

the report should read:

"The contradiction degree has a measurable effect here; which criterion's contradiction degree is set to what should be stated in the report as clearly as the weights and θ"

This extension is used 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. It is equally suitable when the assumption that the decision-maker is more sensitive to losses than to equivalent gains fits the nature of the decision.

Opening a measured criterion into a T-I-F triple "to look more comprehensive" is not modelling uncertainty but producing it. The table must be of a single data type. The exit condition on crisp TODIM 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. If the loss-aversion assumption does not fit the nature of the decision, a symmetrically compensatory method such as Plithogenic TOPSIS is simpler.

Assuming the contradiction degree is unimportant. Unlike the known finding in P-CoCoSo, in this extension the contradiction degree genuinely changes the result. Entering it at random on the assumption that "it has no effect anyway" is a mistake.

Choosing the dominant criterion at random. Because the contradiction degree has a real effect on the result, which criterion is taken as dominant (zero contradiction) and how the others' contradiction degrees are set must be justified in the report.

Skipping the cost complement. If (T, I, F) is not rewritten as (F, I, T) for a cost criterion, the highest-cost alternative appears to have been pulled towards the ideal.

Leaving θ at its default without ever reporting it. As in crisp TODIM, the value used for θ, and the result's sensitivity to it, must be visible in the report.

The governing principle is this:

In Plithogenic TODIM, unlike in P-CoCoSo, the contradiction degree is a genuine input that measurably changes the result; the contradiction degrees, θ and the weights must therefore all be reported together.