Neutrosophic VIKOR (Tooranloo & Ayatollah, 2024)

Three things change; the idea behind the two conditions (acceptable advantage and acceptable stability) does not change, but DecisionMind does not compute either of them in this extension.

Cells. In crisp VIKOR every cell is a single number; here every cell is a degree of truth (T), a degree of indeterminacy (I) and a degree of falsity (F), the three independent of one another, and their sum may exceed 1. Criterion weights are crisp. DecisionMind does not combine several decision-makers' scores in the presentation layer for this extension; it works with a single matrix that has already been aggregated beforehand.

Scale equalisation (direction reversal). In crisp VIKOR, the best and worst value on a cost criterion simply swap places. Here the same idea is applied at the level of the triple. On a cost criterion, the degrees of truth and falsity swap places, while the degree of indeterminacy is complemented, that is, one minus itself is taken. The truth of the judgement "this alternative is good on this criterion" turns into its falsity on a cost criterion.

Distance, score and combination. The method builds the best triple by bringing together the highest truth, the lowest indeterminacy and the lowest falsity on every criterion; the worst triple is built the opposite way. It computes each alternative's distance to this best triple by taking the mean of the squared truth, indeterminacy and falsity differences and then its square root; this is a normalised distance that reduces the three degrees to a single number. The ratio of this distance to the distance between the best and worst triple is the counterpart of crisp VIKOR's normalised difference. The method weights and sums this ratio criterion by criterion (S) and separates out its largest value (R).

Result and defuzzification. Defuzzification sits in the middle, as in Intuitionistic Fuzzy VIKOR. The T-I-F triple is reduced to a single number inside the distance calculation; from that point on, S, R and Q are entirely crisp numbers. Indeterminacy is not carried through to the end as a separate component within S or R; it is taken into account only once, together with the other two degrees, in the distance formula.

For this extension, DecisionMind fixes distance in this form and offers the compromise coefficient v as a parameter the user can change (default 0.5). Unlike Fuzzy VIKOR, it does not compute the two conditions; it produces only the Q ranking.

A lower Q is better. However, the two conditions described on the base VIKOR card are not tested by DecisionMind in this extension; the user must check this by inspecting the S and R columns directly. As before, the alternative that ranks first on S may differ from the one that ranks first on R. The neutrosophic input's separation of truth, indeterminacy and falsity sharpens, further, which alternative is "good overall" and which is "not badly off on any single criterion."

Thus instead of writing:

"According to Neutrosophic VIKOR, the best alternative is A1"

the report should read:

"A3's compromise index is the smallest, and it stands well both overall and on the single worst criterion; because DecisionMind does not test the compromise conditions in this extension, whether its gap to the second-ranked alternative is large enough must be checked separately"

Use this extension in problems where the information about a criterion is incomplete, inconsistent or contradictory, and this matters for the decision itself: evaluating alternatives with no track record, expert opinions drawn from conflicting sources, and analyses where the unknown share must be reported explicitly.

If the assessment rests on a reliable measurement, or experts express no share of uncertainty, moving to the neutrosophic structure merely to use a more elaborate model is not warranted. When the table is mixed, DecisionMind requires a single data type. The base VIKOR's exit condition applies here exactly as it does there: if no compromise is acceptable on one criterion, dominance-based methods should be used instead.

Value-domain violation. T, I and F must each lie between 0 and 1; it is enough for their sum to stay between 0 and 3, and it should not be forced to equal 1.

Computing indeterminacy as 1 − T − F. This reduces the three components to a dependent triple and removes the very justification for using a neutrosophic structure; I must come from its own source, the incompleteness or inconsistency of the evidence.

Applying the cost-criterion swap only partly. Swapping only T and F while forgetting to complement I (or the reverse) corrupts the indeterminacy information of the cost criterion and misselects the best/worst triple.

Defuzzifying with a score function first and running crisp VIKOR. Reducing every triple at the outset to a single score (truth minus falsity, plus one, divided by two) and then applying crisp VIKOR is not Neutrosophic VIKOR; in the table for Case 1, this shortcut reverses the ranking completely.

The governing principle is this:

In Neutrosophic VIKOR, truth, indeterminacy and falsity must come from three independent sources and must be brought together only within the distance formula; deriving one from another, or reducing all three to a single number at the outset, erases the method's own contribution.

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