Neutrosophic DNMA

Three things change. One of them affects crisp DNMA's defining structure.

Cells. In crisp DNMA every cell is a single number. Here every cell has three independent components: truth (T), indeterminacy (I), falsity (F). Each lies between 0 and 1, and their sum is not expected to equal 1. Criterion weights remain crisp, that is, single numbers.

Scale equalisation. The method first reduces the neutrosophic triple to a single score number. This score counts truth positively, twice the indeterminacy negatively, and falsity negatively. Crisp DNMA's three separate aggregation models (weighted sum, weighted product, closeness to the ideal) here collapse into a single model. Two normalisations are computed on this score: linear min–max normalisation and vector normalisation. Both use direction information here too; for a benefit criterion a large value is good, for a cost criterion a small value is good.

Score and combination. The two normalisations are averaged and collapse into a single table. This table is multiplied by the weights and summed, giving a single intermediate score. This intermediate score is then rescaled so that the smallest value becomes zero and the largest becomes one. The step that defines crisp DNMA is absent here: the combined score formula that jointly rewards the score and rank consistency of the three separate aggregation models. Structurally, N-DNMA resembles a doubly-normalised weighted sum method more than it resembles crisp DNMA.

Result. Defuzzification happens at the very start, through the score function. The output is a single number rescaled between 0 and 1. A larger value is better.

DecisionMind fixes, for this extension, the score function and the equally weighted average of the two normalisations. One further point should be noted: crisp DNMA's philosophical core is that three different aggregation logics corroborate one another. This core is absent in N-DNMA; the manifest records this extension as having no published literature source, an internal adaptation of DecisionMind's own. This difference is flagged as an open point for scientific review.

The output is a single number between 0 and 1, and a rank. It is not a percentage or a probability. It only shows a ranking relative to the other alternatives computed from the same triple inputs.

The difference is this. The score is built from the truth-indeterminacy-falsity triple after it has already been dissolved by a single score function at the very first step. If two alternatives' scores are very close, the difference may come from a small indeterminacy gap in the input data; the score itself does not distinguish this.

Thus instead of writing:

"By the N-DNMA score, A3 is the best alternative"

the report should read:

"A3 has received the highest score (1.0) once the truth-indeterminacy-falsity triples were combined by a single score function. The gap between A1 and A2 (0.036) is very small and is sensitive to a small change in Criterion 1's weight"

This extension is suitable when information about a criterion is incomplete, inconsistent or contradictory, and this matters for the decision itself. It is also used when assessing alternatives with no track record, or in expert opinions based on conflicting sources. Which situation calls for a neutrosophic structure is explained on the Neutrosophic data-type card.

Crisp DNMA's exit condition applies here too. If no compromise is acceptable on one criterion, this extension is not suitable.

Violating the value-domain constraint. T, I and F must each lie between 0 and 1, and their sum must not exceed 3. If this is violated, the score function and normalisation can produce invalid numbers.

Changing the score function without stating it. The canonical choice is truth, plus minus twice the indeterminacy, minus falsity, divided by two. A different score function can produce a different ranking.

Looking for three aggregation models in this extension. The idea from the crisp DNMA card, that "three models corroborate one another," is absent here. N-DNMA is a single, doubly-normalised model. Confusing the two presents the output as a stronger guarantee than it is.

A criterion column giving the same score for every alternative. In this case linear min–max normalisation becomes undefined, since its denominator approaches division by zero. Such a criterion should be reviewed.

The governing principle is this:

In N-DNMA, defuzzification happens at the very start, through the score function, and the remaining steps form a single, doubly-normalised model. Crisp DNMA's three-model guarantee does not carry over to this extension; this must be stated plainly in the report.