Neutrosophic Spherical Set CODAS (Bhuvaneshwari & Sweety, 2024)

Five things change. The only thing shared with crisp CODAS is the idea of two distances; here, though, the distances are not Euclidean and taxicab but a single NSS distance type to the ideal and the anti-ideal.

Cells. In crisp CODAS every cell is a single number. Here every cell has three components: truth (T), indeterminacy (I) and falsity (F). The difference from the single-valued structure (SVN) on the neutrosophic data-type card is this: in NSS the three components not only lie between 0 and 1, the sum of their squares is also required not to exceed √3 (T² + I² + F² ≤ √3). This constraint sits somewhere between two extremes. At one extreme is the single-valued neutrosophic structure, which sets no bound at all; the sum can take any value between 0 and 3. At the other extreme is the spherical fuzzy structure, which bounds the sum of squares by 1. NSS leaves a wider region of acceptance than the spherical structure, but it is not entirely unbounded either.

Aggregation. If the decision is made by more than one expert, every expert's T-I-F triple is first combined with their own weight. This aggregation is carried out with the weighted neutrosophic arithmetic mean (SWAM) or the weighted neutrosophic geometric mean (SWGM) rule, producing a single NS matrix. This step is absent from crisp CODAS, which already starts from a single numerical matrix.

Scale equalisation and score. Crisp CODAS divides a benefit criterion's value by the column's largest value, and for a cost criterion takes the ratio of the column's smallest value to the value itself. NSS-CODAS has no such division; the T-I-F components already lie within 0-1, and the criterion weight is applied to the cell through NSS multiplication. For ranking, every cell is reduced to a single number by a score function: S = (T − F)² − (I − F)². This score differs from the simple (T − F + 1)/2 score used by N-CODAS; it takes the I component directly into account too.

Ideal and anti-ideal. Crisp CODAS builds only a single "worst" reference, the negative-ideal. NSS-CODAS turns to TOPSIS here: an ideal (NS-PIS) is built from the best score-wise value on every criterion, and an anti-ideal (NS-NIS) from the worst. Both references exist.

Distance and the closeness ratio. A weighted NSS distance to the ideal and to the anti-ideal is computed for every alternative. Then, from the set, the value at which an alternative is furthest from the anti-ideal (Dmax) and the value at which it is closest to the ideal (Dmin) are taken; the closeness ratio is computed as ξ = (distance to the ideal / Dmin) − (distance to the anti-ideal / Dmax). The ranking direction is reversed: a lower ξ marks the better alternative. This is the exact opposite of crisp CODAS's "higher score is better" rule.

For classical NSS-CODAS, DecisionMind fixes the norm constraint (√3), the score function and the definition of the closeness ratio. Criterion weights are taken from outside; the method does not generate weights.

The closeness ratio ξ shows how close an alternative is to the ideal relative to the others in this set. The best alternative takes ξ=0; this does not come from that alternative scoring zero against itself, but from the fact that the alternative closest to the ideal and furthest from the anti-ideal settles at zero under this definition. The value cannot be compared with a ξ value from a different analysis; the ideal and anti-ideal are built afresh, in every analysis, from that analysis's own data.

Thus instead of writing:

"The ξ value came out high, so this alternative is the best"

the report should read:

"The alternative with the lower ξ is closest to the ideal; ξ=0 marks the alternative in this set that is both closest to the ideal and furthest from the anti-ideal, it is not an absolute measure of quality"

This extension is used when criterion assessment rests on incomplete, contradictory or inconsistent evidence, and the judgements of more than one expert need to be combined. The same rationale applies where the sum of squares of the T-I-F triples sits in a region narrower than the single-valued neutrosophic structure's unbounded range, but wider than the spherical fuzzy structure's constraint (≤1). If the triples already fit within the spherical fuzzy constraint, NSS's extra region of acceptance is unnecessary. If no constraint at all is sought between the triples, single-valued neutrosophic CODAS (n-codas) is sufficient.

Measured criteria should not be carried into this extension. Where the matrix must be of a single type, and a measured value has a satisfaction degree of t, the honest embedding (t, 0, 1−t) is used; the sum of squares of this triple always satisfies the NSS constraint. The base CODAS's exit condition applies here exactly as it does there: where no compromise is acceptable on one criterion, a compensatory method is unsuitable.

Reading the ranking direction as in crisp CODAS. This is the most common error. In crisp CODAS a higher score is better; in NSS-CODAS a lower ξ is better. Reading this direction the wrong way round reverses the entire ranking.

Applying the norm constraint incorrectly. The NSS constraint is T² + I² + F² ≤ √3. Mistaking it for the spherical fuzzy structure's constraint (≤1), or for the single-valued neutrosophic structure's linear constraint (T+I+F≤3), is wrong. Testing the data against the wrong constraint can reject valid cells or accept invalid ones.

Multiplying the criterion weight as an ordinary number. The weight is applied to the cell through the NSS multiplication rule; doing this with ordinary real-number multiplication breaks the definition of neutrosophic weighting.

Deriving indeterminacy (I) from truth and falsity. Writing I = 1 − T − F reduces the neutrosophic structure to crisp data. Each component must come from its own body of evidence.

The governing principle is this:

NSS-CODAS's one contribution is to open up a region between the unboundedness of the single-valued neutrosophic structure and the narrow constraint of the spherical fuzzy structure, and to read this through a lower-is-better closeness ratio. Any application that reads the ranking direction the wrong way round, or confuses the norm constraint, invalidates this contribution.