Bipolar Neutrosophic TOPSIS (Akram, Shumaiza and Smarandache, 2018)

Four things change; the decision logic does not.

Cells. In crisp TOPSIS every cell is a single number. Here every cell is six numbers: a positive pole carrying truth, indeterminacy and falsity (T⁺, I⁺, F⁺ ∈ [0,1]) and a negative pole carrying the same triple (T⁻, I⁻, F⁻ ∈ [−1,0]). The positive pole answers "to what extent is this judgement true/indeterminate/false"; the negative pole answers, separately, "to what extent is the opposite of this judgement true/indeterminate/false"; one is never derived from the other. Criterion weights remain crisp, single numbers. If weights are not supplied, the method itself can derive them, which sets it apart from most other TOPSIS extensions in DecisionMind.

Source of the weights. Crisp TOPSIS always takes weights from outside. BN-TOPSIS can generate its own weights, when none are given, using the maximizing-deviation approach: the greater the spread between alternatives on a criterion, the higher that criterion's discriminating power is judged to be, and its weight grows accordingly. This step is skipped whenever weights are supplied externally. A criterion that carries a constant value (the same figure across every alternative) receives a weight of zero under this method and effectively drops out of the analysis.

Distance. In crisp TOPSIS, straight-line (Euclidean) distance sits between two numbers. Here the method squares and sums the difference in each of the six components (T⁺, I⁺, F⁺, T⁻, I⁻, F⁻) separately, then normalises by six times the number of criteria. This distance is broader than IF-TOPSIS's three-component distance (μ, ν, π) and broader still than the three-component distance (T, I, F) used by the single-pole neutrosophic extensions, because both the positive and the negative pole enter at once.

Outcome and defuzzification. In crisp TOPSIS the closeness coefficient is S⁻/(S⁺+S⁻), and the denominator is never zero. BN-TOPSIS instead uses a "revised closeness degree": the ratio of the distance to the ideal, taken against its own smallest value, is subtracted from the ratio of the distance to the anti-ideal, taken against its own largest value. Akram, Shumaiza and Smarandache (2018) proposed this revision to avoid the zero-denominator problem of the classical closeness coefficient; the resulting value can come out negative. In the final step this value is divided by the smallest value obtained, yielding a "deficiency ratio". The method then ranks alternatives in ascending order of this ratio: the lowest ratio is the best alternative. This is the exception to the "highest score wins" rule that governs every other TOPSIS family member in DecisionMind, and the report must state it explicitly.

Unlike crisp TOPSIS, the output is read in ascending order: the lowest deficiency ratio marks the alternative closest to the ideal, and the highest ratio (1.000) is the reference alternative. Beyond this, the reading logic is the same: the ratio is neither a percentage nor a probability, it shows only relative standing within this particular alternative set, and it cannot be compared across a different analysis.

Beneath the ratio sit two layers of evidence, positive and negative, and this is where the difference shows up. An alternative can carry both a high T⁺ and a high T⁻ (in absolute terms) at once, meaning both strong positive and strong negative evidence can coexist, and this contradiction is masked within a single ratio. When two alternatives' ratios sit close together, it is worth examining which criterion holds this contradiction.

Thus instead of writing:

"BN-TOPSIS uses both positive and negative evidence, so the result is more reliable"

the report should read:

"The ranking is read in ascending order of deficiency ratio; the alternative with the lowest ratio is closest to the ideal. But this ratio does not show on which criteria the alternative carries both strong positive and strong negative evidence at once; that must be examined separately"

Use this extension when the evaluation needs positive and negative effect expressed with separate signs. It is useful, for example, when a supplier's positive contribution and negative risk are being assessed through independent components, and the case for or against "this is a good supplier" is being weighed separately from the case in favour and the case against. If a single truth-indeterminacy-falsity triple is enough on its own, that is, if a separate negative pole is not actually needed, N-TOPSIS suffices, and carrying six components would be unnecessary weight.

Where criteria are genuinely measured, the base method should be kept; this condition applies here as well. Expanding a measured value into a six-component pair adds no information, as the Neutrosophic data-type card also warns. For a fair embedding, every component must come from its own source and cannot be filled in by guesswork. If the table is mixed, DecisionMind asks for a single data type. Where no compromise is acceptable on a criterion, this extension is compensatory too, and does not screen out anything below a threshold.

Ignoring the case where the deficiency ratio is undefined. If the smallest revised closeness degree is zero (which the structure allows, or can push very close to zero), dividing by it is undefined; a small correction should then be applied, or the ranking should be based on the raw closeness degree instead. Forcing the deficiency ratio through regardless and reporting it gives a false sense of precision.

Overlooking the limit of the maximizing-deviation method. If every alternative carries the same value on a criterion (a constant criterion), this method drives that criterion's weight to zero and it effectively drops from the analysis; this does not mean the criterion is unimportant, only that it fails to discriminate, and this should be reported separately.

Confusing the distance normalisation factor. The factor that normalises across six components (six times the number of criteria) must not be confused with the factor used in single-pole neutrosophic or intuitionistic fuzzy extensions (two or three times the number of criteria); a different factor produces numbers on a different scale.

Collapsing the bipolar sextuple into a single average too early and scoring it as an ordinary number. Reducing the six components to a single "net score" at the input stage and then running crisp TOPSIS confuses the positive and negative poles; the information that an alternative carries both strong positive and strong negative evidence is lost in that average.

The governing principle is this:

The positive and negative poles must come from separate sources, the deficiency ratio must be read in ascending order, and any case where it is undefined must be reported openly rather than concealed.