Neutrosophic MULTIMOORA (Stanujkic et al., 2017)

Four things change; the overall flow is preserved, but the dominance rule changes as well.

Cells. In crisp MULTIMOORA every cell is a single number. Here every cell consists of three independent degrees: truth (T), indeterminacy (I) and falsity (F); their sum is not forced to equal 1. Weights are crisp numbers; they are not entered in neutrosophic form.

Scale equalisation. Crisp MULTIMOORA divides the column by the root of the sum of its squares. Here the method first complements the triple for cost criteria: ⟨T, I, F⟩ is rewritten as ⟨F, 1−I, T⟩. Truth and falsity swap places, and indeterminacy is complemented. It then reduces every cell to a single number with a score function, (1 + T − 2I − F) / 2. Only after this does it equalise this score by the column-wise root of the sum of squares (vector normalisation). In other words, the neutrosophic structure collapses to a single number immediately after the cost complementing but before normalisation.

Distance, score and merging. The ratio system is the weighted sum of this normalised score. The logic is the same as in crisp MULTIMOORA, except that here the T-I-F structure has already been reduced to a score. The reference point takes, for every criterion, the largest normalised score as the ideal, and finds each alternative's largest weighted deviation from it. The full multiplicative form is DIFFERENT. Here the method does not use the normalised score but the complemented T-I-F triple itself: it builds a new ⟨T, I, F⟩ triple by taking a weighted geometric mean over this triple (a neutrosophic weighted geometric aggregation), then converts this triple into a single number with the same score function. So the full multiplicative form carries the T-I-F structure further, into its own aggregation step, than the ratio system and the reference point do.

Result and defuzzification. The most important difference lies here. Crisp MULTIMOORA, and most other extensions, use the SUM of the three sub-rankings (a Borda count). N-MULTIMOORA instead uses a "two-out-of-three dominance" (2-of-3 dominance) rule. An alternative counts as "dominant" over another if it outranks it in at least TWO of the three sub-rankings. Only when the dominance counts are tied does the sum of the three sub-rankings come into play, as a tie-breaker. This is a different merging logic from crisp MULTIMOORA's direct summation rule.

DecisionMind holds fixed, in this extension: the cost complementing ⟨T,I,F⟩→⟨F,1−I,T⟩, the score function (1+T−2I−F)/2, and the fact that the ratio system and the reference point operate on this score. It also holds fixed the neutrosophic weighted geometric aggregation used in the full multiplicative form, and the two-out-of-three dominance rule (broken by the sum in case of a tie) used in the final ranking.

The final ranking is a combined summary of three viewpoints, and it is read essentially the same way. But the merging rule differs from crisp MULTIMOORA: crisp MULTIMOORA applies "the alternative with the smallest sum of the three rankings wins", whereas here the rule is "the alternative that outranks the others in at least two of the three sub-methods wins". Because of this, the chance that the two methods give a different final ranking on the same data is somewhat higher than for crisp MULTIMOORA.

The T-I-F structure is also not carried to the same depth by the three sub-methods: the ratio system and the reference point reduce it to a single score immediately after the cost complementing, while only the full multiplicative form preserves the triple structure through to its own aggregation step.

Thus instead of writing:

"N-MULTIMOORA is more informative because it carries neutrosophic uncertainty through the entire calculation"

the report should read:

"The truth-indeterminacy-falsity triple is preserved only up to the aggregation step of the full multiplicative form; the ratio system and the reference point operate on a crisp number already reduced to a single score right after complementing, and the final ranking is built with a two-out-of-three dominance rule rather than a sum"

The rule from the neutrosophic data-type card applies here as well: use this extension when information about a criterion is missing, inconsistent or contradictory, and this matters for the decision. A measured criterion should not be written directly as T with I=0 and F=1−T; doing so zeroes out the indeterminacy share and removes the sole contribution of the neutrosophic structure. Crisp MULTIMOORA's exit condition also applies here: all values must stay positive, since the full multiplicative form becomes undefined otherwise. If no compromise can ever be accepted on one criterion, this method should not be used.

Forcing the sum of the three components to equal 1. This reduces the neutrosophic structure to an intuitionistic fuzzy pair; T, I and F must remain independent of one another.

Skipping the cost complementing. Feeding a cost criterion's ⟨T,I,F⟩ triple into the score without complementing it first (without swapping F and T) confuses "high truth" with "low cost" on that criterion and reverses the ranking.

Changing the score function, or not stating it in the report. DecisionMind fixes the function (1+T−2I−F)/2; a different score function can produce a different ranking.

Confusing it with the summation rule. It is wrong to assume that crisp MULTIMOORA's rule, "the alternative with the smallest sum of the three rankings wins", also applies here; N-MULTIMOORA looks first at two-out-of-three dominance, and the sum only comes into play in case of a tie.

The governing principle is this:

In N-MULTIMOORA, the truth-indeterminacy-falsity triple is preserved only up to the aggregation step of the full multiplicative form; the ratio system and the reference point operate on a crisp number already reduced to a single score right after complementing, and the final ranking is built with a two-out-of-three dominance rule rather than a sum.