Spherical fuzzy Z-number CRADIS (Niu, 2024)

Four things change; CRADIS's own logic of keeping closeness to the ideal and distance from the anti-ideal as separate ratios and then averaging them does not.

Cells. In crisp CRADIS every cell is a single number. Here every cell consists of six numbers: a degree of support (ε), rejection (ν) and hesitancy (∂) for a judgement, whose squares sum to no more than 1 as in the spherical fuzzy structure, and, in addition, a separate reliability (τ_ε, τ_ν, τ_∂) accompanying each of these three degrees. Reliability is the B component of the Z-number, and here it is not a single number but three reliabilities distributed separately across the three degrees. Weights come from outside; in DecisionMind they usually come chained from the Spherical fuzzy Z-number CRITIC card, but any weight source is accepted. The method supports group decisions: several experts' spherical fuzzy Z-number matrices can be merged into a single collective matrix using the experts' weights.

Scale equalisation. Crisp CRADIS equalises direction by dividing the benefit column by its largest value and the cost column by its smallest. Here there is no division; on a cost criterion the support-and-hesitancy pair swaps places (ε,τ_ε is exchanged with ∂,τ_∂). This swap ensures that the "good" and "bad" ends work the same way in both directions within the ideal-anti-ideal logic.

Weighting. Crisp CRADIS multiplies the column directly by the weight. Here the weight is applied as a power (exponent); the scalar power operation of spherical fuzzy algebra is used. This is not a linear multiplication, and the way it shrinks a low-weighted criterion's three degrees in the calculations that follow differs from crisp CRADIS.

Ideal, anti-ideal and distance. CRADIS's distinguishing feature relative to TOPSIS is preserved here too: for every criterion, a SINGLE spherical ideal and a SINGLE spherical anti-ideal point are built, not an ideal that varies by alternative as in TOPSIS. The ideal point takes the highest of the weighted support×reliability product and the lowest of the rejection and hesitancy×reliability products; the anti-ideal is the reverse. Distance is the root of the sum of the squared differences of these three products (value×reliability). From there, the total distances to the ideal and anti-ideal (S+, S−), the best ratios (K+, K−) and the compromise score (Q) are calculated exactly as in crisp CRADIS.

DecisionMind holds fixed, in this extension, the three-degree spherical fuzzy structure, the distribution of reliability separately across each degree, and the single spherical ideal/anti-ideal logic.

The compromise score Q is, as in crisp CRADIS, the average of an alternative's closeness to the ideal and its distance from the anti-ideal; it is read the same way (see the CRADIS card).

The difference is here. In some Z-number extensions the reliability component appears to enter the calculation but algebraically cancels out at the last step and never reaches the result; this is a situation seen, and separately reported, on DecisionMind's Z-TOPSIS, Z-VIKOR, Z-EDAS, Z-COPRAS and Z-WASPAS cards. In this extension the situation is different: reliability genuinely changes the result here, because the weighting is a power operation and is not linear, and the reliability ratio does not cancel between numerator and denominator. This has been verified by running DecisionMind's SFZN-CRADIS engine: in the illustrative example below, when T3's reliability is lowered on every criterion, the ranking actually changes.

Thus instead of writing:

"In spherical fuzzy Z-number CRADIS too, the reliability component does not feed into the result, as in some Z-number extensions, and is indicative only"

the report should read:

"Reliability genuinely enters the calculation here; two alternatives with the same support-rejection-hesitancy triple receive different compromise scores if their reliability differs, and the ranking can be affected by this difference"

Use this when an expert's judgement is given both as a support-rejection-hesitancy triple and with a separate reliability attached to each degree of that triple. The typical situation is one where several evaluators (a student survey, peer review, a manager's observation, say) contribute to the same judgement from sources of differing reliability, and this difference needs to feed into the calculation. The data-collection burden is heavy, six numbers per cell; this burden should only be taken on when the reliability difference will genuinely affect the decision.

If hesitancy is not measured separately, or if all sources carry the same reliability, the Spherical fuzzy data-type card and the Z-number data-type card may point to a simpler extension (spherical fuzzy only, or Z-number only). Crisp CRADIS's exit condition applies here too: if no compromise is acceptable on one criterion, sub-threshold alternatives should be screened out first and only the remainder ranked with this extension.

Confusing Niu's (2024) K+ formula with Puška et al.'s (2022). Niu's definition of K+ is S°+/S+; the Puška formula on the Z-fuzzy CRADIS card is in the form S/S+, where S here is the average of S°+ and S°−. The two papers use a different normalisation reference; confusing the two produces the wrong compromise score.

Confusing CRADIS's single spherical ideal/anti-ideal logic with TOPSIS's criterion-by-criterion ideal. CRADIS builds a SINGLE ideal and a SINGLE anti-ideal point for the whole decision matrix and then sums the distances; it is not, as in TOPSIS, a separate ideal for every alternative.

Skipping the (ε,τ_ε)↔(∂,τ_∂) swap on a cost criterion. If this swap is not made, the ideal-anti-ideal logic works inconsistently across benefit and cost criteria, and the "good" and "bad" ends on the cost criterion are read in reverse.

Carrying reliability as a separate term. The distance formula uses the products (ε×τ_ε), (ν×τ_ν) and (∂×τ_∂). Putting ε and τ_ε separately into the squared difference gives the wrong distance. Reliability reduces each degree to a single number before the squared difference is taken.

Writing the same reliability into every cell and calling this "using a Z-number." If reliability is identical everywhere, and constant for each of the three degrees (ε, ν, ∂), it carries no discriminating information; as the illustrative example below shows, reliability's contribution becomes visible only once it differs between alternatives.

The governing principle is this:

Here, unlike in some of DecisionMind's other Z-number extensions, reliability genuinely changes the result; the report must therefore state separately which alternative rests, on which criterion, on a lower-reliability source.