Neutrosophic AROMAN

Three things change. Two of them also affect crisp AROMAN's defining structure.

Cells. In crisp AROMAN every cell is a single number. Here every cell is made up of 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 stay crisp, that is, single numbers. The method does not generate weights; it takes weight from outside.

Scale equalisation. The method first reduces the neutrosophic triple to a single score number. This score counts truth as positive, twice indeterminacy as negative, and falsity as negative. Crisp AROMAN performs defuzzification at the very last step; here this order reverses, and defuzzification comes first. A ratio-type normalisation is then applied to this score: for a benefit criterion it is divided by the column's largest score, for a cost criterion by the column's smallest score. Crisp AROMAN's normalisation, by contrast, divides the gap between the value and the minimum by the gap between the maximum and the minimum. The two are different calculations.

Score and aggregation. In crisp and fuzzy AROMAN, linear and vector normalisation are computed independently from the raw data and then averaged. Here, vector normalisation is computed not from the raw score but from the previous step's ratio-normalised value. That is, the steps are chained one after another rather than running in parallel. The two are averaged, multiplied by the criterion weight, and summed. The step that defines crisp and fuzzy AROMAN is absent here: keeping the benefit total and the cost total separate and combining them with a visible balance parameter (λ). The direction information has already been dissolved into the ratio normalisation at the second step. The final score is simply a plain weighted sum.

Result. Because defuzzification happens at the start, through the score function, there is no separate defuzzification step at the end. The output is a single crisp number. Larger is better.

DecisionMind fixes, for this extension, the score function, the ratio-type normalisation, and the linear-vector average. One further point should be noted: crisp and fuzzy AROMAN's philosophical core is combining a separate benefit total and cost total with a visible λ parameter. This core is absent in N-AROMAN. Structurally, N-AROMAN resembles a ratio-normalised weighted sum method more than it resembles crisp AROMAN. This discrepancy is an open finding for scientific review.

The output is a single number and a ranking, as in crisp AROMAN. It is not a percentage or a probability. It shows only a ranking relative to the other alternatives computed with the same triple inputs.

The difference is this. The score is built from the truth-indeterminacy-falsity triple, already dissolved into a single number by a single score function at the very first step. Where two alternatives' scores are close, checking which one has the higher indeterminacy component adds a separate layer of interpretation; the score function has already penalised this component.

Thus instead of writing:

"According to the N-AROMAN score, A3 is the best option"

the report should read:

"A3 has the highest score (0.705), obtained by combining the truth-indeterminacy-falsity triples with a weighted score function. Its gap over A1 (0.607) is clear. The order between A1 and A2 (0.594), however, is sensitive to the weight of Criterion 1"

Where the information about a criterion is incomplete, inconsistent or contradictory, and this matters for the decision itself, this extension is suitable. It is also used when assessing alternatives with no track record, or with expert opinions resting on contradictory sources. Which situations call for the neutrosophic structure, and the principle of not converting measured data directly into a triple, are covered on the Neutrosophic data-type card.

Crisp AROMAN's exit condition applies here too. This extension is also compensatory. It is not suitable where no compromise is acceptable on one criterion.

Violating the value-range constraint. T, I and F must each lie between 0 and 1, and their sum must not exceed 3. Violating this can make the score function and normalisation produce an invalid number.

Changing the score function without stating it. The canonical choice is truth plus one minus twice indeterminacy minus falsity, all divided by two. Indeterminacy is penalised here twice as heavily as falsity. A different score function, for instance one that counts indeterminacy only once, can produce a different order.

Looking for λ in this extension. The λ balance parameter found on the crisp and fuzzy AROMAN cards is absent here. The direction information has already been folded into the normalisation at the second step. Carrying crisp AROMAN's habit of "I computed it with λ=0.5" over here is a mistake.

A near-zero score on a cost criterion. If a cost criterion's smallest score is very close to zero, the ratio normalisation becomes undefined; this denominator approaches division by zero. Such a criterion should be reviewed.

The governing principle is this:

In N-AROMAN, defuzzification happens at the start, through the score function. Which function this is, and that no λ-based balancing step exists here, must be stated explicitly in the report.