Neutrosophic TOPSIS (Biswas, Pramanik & Giri, 2016)

Three things change; the decision logic does not.

Cells. In crisp TOPSIS every cell is a single number. Here every cell is three numbers: truth (T), indeterminacy (I) and falsity (F). Each lies between 0 and 1, and the three are independent of one another; their sum is not expected to equal 1. For cost criteria, truth and falsity swap places, that is, their complement is taken. When a criterion is "lower is better," its T and F values are reversed and converted to the benefit direction, while I stays as it is. Criterion weights remain crisp, single numbers; the method does not generate weights, it takes them from outside.

Weighting. In crisp TOPSIS, weighting is multiplication. Here the neutrosophic triple is weighted by a crisp weight through a scalar-multiplication operation; this operation preserves the triple's T-I-F structure (all three stay within 0-1) but is defined by a formula different from crisp-number multiplication.

Ideal, anti-ideal and distance. In crisp TOPSIS, the ideal and anti-ideal are each column's single best/worst value. Here the ideal for each column is built component-wise from the weighted triples: in the benefit direction, T takes its highest value while I and F take their lowest. The anti-ideal is built the opposite way. Where crisp TOPSIS's distance comes from a single difference, here the method takes the T difference, the I difference and the F difference together on every criterion and measures a normalised distance across them. All three components enter the calculation; because none is derived from another, all three are carried separately.

Result and defuzzification. The definition is the same: the ratio of the distance to the anti-ideal over the sum of the two distances. The result is again a single number between 0 and 1. For this extension, DecisionMind fixes the complementation rule (the T-F swap on a cost criterion), the weighted scalar multiplication and the three-component normalised distance. Indeterminacy (I) enters the result but is not reported separately; it dissolves, together with T and F, into a single distance figure.

The output is a closeness coefficient and a ranking, as in crisp TOPSIS, and it is read the same way: it is not a percentage, and it cannot be compared with a different analysis.

Beneath the score lie three independent components, truth, indeterminacy and falsity, and this is where the difference emerges. One of them, indeterminacy, matters particularly, because it is indeterminacy that carries how incomplete or contradictory the evidence is. When two alternatives' scores are close, looking at which one carries the higher indeterminacy shows which rests on the sounder evidence; the score itself does not distinguish this.

Thus instead of writing:

"N-TOPSIS takes indeterminacy into account, so the result is more reliable"

the report should read:

"How incomplete or contradictory the evidence behind the assessment is has been carried in a separate component; A3, at 0.623, sits slightly ahead of A1 (0.597), and this gap closes if A3's truth-indeterminacy-falsity values on the third criterion worsen somewhat"

This extension is used when the information about a criterion is incomplete, inconsistent or contradictory, and this matters for the decision itself. Typical situations include evaluating alternatives with no track record, expert opinions drawn from conflicting sources, and analyses where the unknown share must be reported explicitly. Where the opposing view needs to be held under a separate sign, that is, a negative pole, BN-TOPSIS is used. Where support and rejection alone are sufficient and their sum does not exceed 1, IF-TOPSIS suffices; carrying a third, independent component (I) is an unnecessary burden in that case.

Where criteria are measured, the base method should be kept; this condition holds here too. Converting a measured value directly into a triple, that is, taking T from the score, F from 1-T and I from zero, violates the core principle of the Neutrosophic data-type card: each component must come from its own body of evidence. When the table is mixed, DecisionMind requires a single data type; a measured criterion can be written as (t, 0, 1-t), but this states "no indeterminacy" and adds no information. If no compromise is acceptable on one criterion, this extension too is compensatory and will not eliminate anything below a threshold.

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 weighting and distance calculations produce invalid numbers.

Applying the score function too early. The canonical choice for ranking is the score function S=(T-F+1)/2, but this function is used only at the RESULT stage. Reducing every cell to a single number with this function at the input stage and then running crisp TOPSIS drops the indeterminacy component (I) from the calculation entirely. The illustrative example below shows this component having no effect on the score at all.

Forcing the sum to equal 1. Expecting the three components to sum to 1, or computing I as 1-T-F, collapses the neutrosophic structure into an intuitionistic or picture fuzzy structure; N-TOPSIS's three components are independent of one another.

Choosing a different score function without stating it. More than one score function has been proposed in the literature, each weighting indeterminacy differently; if which one was used is not stated in the report, a comparison against different software becomes misleading.

The governing principle is this:

Truth, indeterminacy and falsity must come from separate sources, and reduction to a single number for ranking must happen only at the final step, not at the input stage.