Bipolar neutrosophic ELECTRE I (Akram, Shumaiza & Smarandache, 2018)

Four things change; the core kernel-extraction logic does not.

Cells. In crisp ELECTRE I every cell is a single number. Here every cell is six numbers: on the positive side of a judgement, a degree of truth, indeterminacy and falsity (T⁺, I⁺, F⁺ ∈ [0,1]), and on the negative side, the same triple (T⁻, I⁻, F⁻ ∈ [−1,0]). For example, both how strongly and how weakly a judgement such as "this supplier is reliable" is supported are kept separately, in the positive and negative pole. Weights, by contrast, are a single crisp number (supplied from outside in this manifest, not generated) and are applied to cells through a bipolar neutrosophic weighted-product operation.

No scale equalisation. Crisp ELECTRE I divides columns by their own magnitude. BN-ELECTRE-I has no such step: because all six components are already defined on a fixed range ([0,1] and [−1,0]), they are not separately normalised; weighting is applied directly to these six components with a bipolar neutrosophic product operation.

Concordance / discordance. Two different operations come into play here. Concordance rests on a score that is the sum of every cell's six components (the BNS score, ρ = T⁺+I⁺+F⁺+T⁻+I⁻+F⁻). This score can also be negative, because the negative components lie in [−1,0]. As soon as an alternative is judged "at least as good" as another on a criterion (by comparing ρ), that criterion enters the concordance set and the criterion's weight is added to the concordance index. Discordance, by contrast, is not a score but a normalised Euclidean distance computed over the six components themselves (the neutrosophic counterpart of the Hamming distance in fuzzy TrFNs, but square-root based here).

The form of the thresholds. In crisp ELECTRE I the concordance threshold (c̄) and discordance threshold (d̄) are chosen by the analyst. Here, just as in Fuzzy ELECTRE I, the thresholds are not set by the user: ê is the average of all pairwise concordance indices, and f̂ is the average of all pairwise discordance indices. One alternative outranks another when its concordance stays above this average and its discordance stays below it.

Result and defuzzification. Reducing the six components to a single score (ρ) happens very early, while the concordance set is being built. But this is not a defuzzification that collapses the entire input to a single number, as in Fuzzy ELECTRE II/III; the discordance calculation still uses all six components, as a distance. The final output, as in crisp ELECTRE I, is an outranking graph (the π matrix) and the kernel set that follows from it; what DecisionMind fixes here is that the thresholds (ê, f̂) are automatically derived as the data's own average.

What stays the same as the base method: the output is not a ranking score but a breakdown of which alternatives are outranked by none of the others (the "kernel set") and of the pairwise relations (outranking, tie, incomparability).

What differs is this: a high total BNS score (ρ) for an alternative does not, by itself, mean that alternative will outrank the others. Although concordance rests on a score comparison, discordance rests on the full distance across all six components, and this distance is sensitive to which component (T, I or F; positive pole or negative pole) the difference between two alternatives is concentrated in. A high total score does not always offset a large distance (high discordance) on one criterion.

Thus instead of writing:

"S4 has the highest total score, so S4 must outrank the other three as well"

the report should read:

"Even though S4's weighted BNS score is the highest, the outranking relation depends not only on the score but also on the discordance distance; the π matrix should not be read without examining, criterion by criterion, which alternative actually outranks which"

Use this method when the assessment of a criterion carries evidence both for and against, and these two directions (positive/negative) need to be recorded separately, independently of one another. This applies, for example, when a supplier has both positive references and a negative piece of industry news. Carrying a measured value into a bipolar neutrosophic cell here, too, first requires converting it into a judgement (the steps on the neutrosophic data-type card); skipping this step and intuitively inventing six components around a crisp number adds no information.

Crisp ELECTRE I's exit condition applies here too: if a kernel set, rather than a ranking, is sufficient, this method is suitable. If there is no need to keep positive and negative evidence separate (if the judgement is adequately represented by a single T-I-F triple), the simpler single-valued neutrosophic ELECTRE I form should be preferred.

Using the BNS score (ρ) on its own as a ranking score. ρ is used only to build the concordance sets; the final outranking decision is determined together with the discordance distance. Saying "the highest score wins" by looking at ρ alone reduces the method to a compensatory scoring scheme.

Assuming the thresholds (ê, f̂) are fixed numbers chosen by the analyst. These are the data's own average; when one alternative's score changes, not only the relevant pair but the threshold itself, being the average of all pairs, shifts.

Leaving the discordance distance unnormalised. The distance is brought into [0,1] by ratioing it against the largest distance across all criterion pairs; skipping this normalisation leaves the discordance indices on different scales and makes them incomparable with f̂.

Defuzzifying first and running single-valued neutrosophic or crisp ELECTRE I instead. Reducing the six components first to a single T-I-F triple, or to a single number, and then running the single-valued or crisp method removes the requirement that positive and negative evidence be carried independently of one another; that is this method's sole contribution.

The governing principle is this:

In BN-ELECTRE-I, outranking depends not on a single total score but on concordance (a score comparison) and discordance (a distance over all six components) being satisfied together; the thresholds derive from the data, and a high total score alone does not guarantee outranking.