Triangular Neutrosophic MULTIMOORA (Stanujkić et al., 2021)

Four things change; the idea behind the ratio system, the reference point and the full multiplicative form stays the same.

Cells. In ordinary neutrosophic MULTIMOORA (n-multimoora), every cell is three numbers: truth, indeterminacy, falsity. Here EACH of these three components is itself a triangle: a lower-mid-upper for truth, a lower-mid-upper for indeterminacy, a lower-mid-upper for falsity; nine numbers in total. This triangle is the result of merging several experts' single-valued neutrosophic judgements. DecisionMind splits the experts' scores in every cell around the median: the average of those below the median becomes the triangle's lower corner, the median itself becomes the middle corner, and the average of those above becomes the upper corner. In this way the triangle's width carries the board's own disagreement: a narrow triangle shows consensus, a wide triangle shows disagreement. This extension directly supports group decisions; weights, however, are taken from outside as crisp numbers.

Scale equalisation. Crisp MULTIMOORA has a separate normalisation step dividing by the square root of the sum of squares; there is no such step here. Instead, each of the three sub-methods aggregates benefit criteria and cost criteria directly on the triangles, each with its own weighted-combination rule.

Distance / score / aggregation. The ratio system combines the triangles of the benefit criteria among themselves, and the triangles of the cost criteria among themselves, each with weights; the two sides remain separate triangles. BOTH sides are then converted into a single number by a score function (Ye, 2015), and the cost side's score is subtracted from the benefit side's score. The reference point builds, for each criterion, an ideal triangle according to its direction (for a benefit criterion, the highest truth and the lowest indeterminacy-falsity; for a cost criterion, the lowest truth and the highest falsity, with indeterminacy again wanted low), and takes the LARGEST of each alternative's weighted distance to this ideal; this is the same "look at the worst criterion" logic as in crisp MULTIMOORA. The full multiplicative form likewise aggregates benefit and cost criteria separately, this time with a multiplicative rule. The two sides are again converted to a single number by the score function, and the benefit side's score is divided by the cost side's score.

Result and defuzzification. All three sub-methods carry the benefit and cost sides as triangles all the way to the END; descending to a single number happens only at the moment the two sides are compared (by subtraction or division). The final rank is built, as in crisp MULTIMOORA, by dominance theory: the alternative that sits at the top across the three sub-rankings wins.

DecisionMind fixes the median-based triangulation, the separate aggregation of benefit and cost criteria, the Ye (2015) score function and dominance theory for this extension.

The final rank is the combined summary of the three viewpoints, exactly as in crisp MULTIMOORA; it is read the same way.

The difference is this. The triangular structure is preserved until the benefit and cost criteria have each been aggregated internally; descent to a single number happens only at the moment these two sides are compared. This differs from intuitionistic fuzzy or ordinary neutrosophic MULTIMOORA: there, some sub-methods descend to a single number as early as the first step; here, all three reach this defuzzification at the same, late point.

Thus instead of writing:

"Triangular neutrosophic MULTIMOORA is the most informative form because it carries uncertainty through the whole calculation"

the report should read:

"The truth-indeterminacy-falsity triangles are preserved until the benefit side and the cost side have each been aggregated internally; descent to a single number happens only at the step where these two sides are compared, and this happens at the same point in all three sub-methods"

Use this extension when a neutrosophic judgement comes from several experts and how much the board agrees, that is, how narrow or wide the triangle is, matters for the decision. If there is only a single expert, or the degree of consensus will not be separately reported, ordinary neutrosophic MULTIMOORA (n-multimoora) is sufficient; the triangular form adds a further layer, and using this layer without justification only adds computational weight.

Turning a measured criterion into a T-I-F triangle is producing uncertainty, not modelling it; this principle applies here too. If there is full consensus, that is, all experts gave the same score, the triangle's three corners come out equal and this extension becomes indistinguishable from ordinary neutrosophic MULTIMOORA. Crisp MULTIMOORA's exit condition applies here too; if no compromise at all is acceptable on one criterion, this extension, too, is compensatory.

Fabricating or equalising the triangle's three corners. The triangle's width must come from the board's real spread of opinion; arbitrarily equalising the three corners hides a real disagreement, and arbitrarily widening them fabricates a disagreement that does not exist.

Aggregating benefit and cost criteria together without direction information. The reference point's ideal triangle is built according to direction; if a cost criterion is processed as though it were a benefit, the ideal point is built in the wrong corner and the ranking is reversed.

Changing the score function, or not stating it, in the report. DecisionMind fixes the Ye (2015) score function; a different score function can produce a different rank.

Failing to read a wide triangle from few experts as "data quality is low". A wide triangle does not mean low data quality; it means a real disagreement within the board, and the report should make this distinction.

The governing principle is this:

The triangle's width is the board's own disagreement; this width is preserved until the benefit and cost sides are each aggregated internally, and descent to a single number happens only at the moment the two sides are compared.