Spherical Fuzzy Z-Number MARCOS (Niu, 2024)

A school's English department has assessed four teachers (T1–T4) on four criteria: lesson-planning skill, classroom management, student feedback and professional-development engagement. All four are "higher is better." In this example the rejection degree is zero and the reliability degrees are set to one; only the support and hesitancy degrees vary. Weights are chained in from Spherical Fuzzy Z-Number CRITIC.

The method builds a PIS and NIS for each criterion, computes every cell's distance to these two points over value×reliability products, and reduces this to a closeness coefficient. The extended table is built on these coefficients, normalised and weighted, and the final utility degree (U) is calculated.

The result reads as follows. T3 sits relatively close to the PIS on all four criteria and is strong on Y3, the most heavily weighted criterion; T4 follows close behind. T1 and T2 sit in the rearward ranks.

An important disclosure is needed here: the final utility degrees above (0.8100; 0.7821; 0.4681; 0.2760) were obtained by independently recomputing this table with today's DecisionMind engine, and the engine's ranking matches the order reported in the paper (T3>T4>T1>T2). But the engine's own internal verification gate reports that these values do not exactly match the numbers recorded in the manifest (T1=0.700; T2=0.550; T3=0.696; T4=0.507); the manifest's numbers are themselves derived from the paper's own Table XXIV/XXV values, not from the engine recomputing this table from scratch (from the Table IV/V input). This has been recorded as a separate finding (see the verification notes); a similar difference has also been reported on the SFZN-CRADIS card.

The board's hesitation is this: what would happen if T3's assessment on all four criteria rested only on the teacher's own self-report, without an external observer's confirmation? That is, if reliability were low on all four criteria. This scenario has been recomputed in Python using the engine's own algorithm. When T3's reliability on all four criteria is lowered from 1.0 to 0.4, the ranking genuinely changes: T4 (0.9063) moves to first place, T1 (0.7192) second, T2 (0.6522) third, and T3 (0.3232) drops to last. By contrast, something different happens if only T3's reliability on a single criterion (Y3) is lowered by the same proportion. T3 drops from first to second place (T4 becomes first at 0.7762, T3 second at 0.7251), not to last. This shows that whether reliability is lowered generally (consistently low across all criteria) or locally (low on a single criterion) changes the outcome.

In the report: "T3 obtained the highest final utility degree in the current assessment (0.8100); but this advantage rests on the reliability of T3's assessment across all four criteria. If reliability is lowered on all four criteria together, T3 drops to last; if it is lowered on only one criterion, T3 falls to second place, so the source of T3's assessment should be stated separately in the report."

Source: Niu (2024), IJACSA 15(3), Tables IV–V (input); a ranking matching the paper's order (T3>T4>T1>T2) has been produced by today's independent engine recomputation, but the magnitude of the final utility degrees does not match the numbers recorded in the manifest (detail in the verification notes). The reliability sensitivity scenario was independently computed by this card's author.

A nursery chain will carry out the annual performance assessment of four care-giving staff. Criteria: child-safety practice, parent communication, daily activity planning and emergency-response knowledge; all four are "higher is better." For each staff member, both the nursery director and a parent representative give support, rejection and hesitancy degrees separately; the chain merges these two views into a single collective assessment. Reliability is derived from how many months each assessor has been observing that staff member: high for long-standing observers, low for those who have only just started.

The method computes each staff member's distance from their spherical fuzzy Z-numbers to the PIS and NIS, reduces this to a closeness coefficient, normalises and weights it in the extended table, and finds the final utility degree. Suppose the result places a staff member who is very strong on child safety but weak on parent communication first, and a staff member who is moderate but consistent across all four criteria second.

The chain's hesitation is this: the first-ranked staff member's child-safety score was recorded with low reliability, because this staff member started recently. If reliability were raised, the first rank might not change, but if it were lowered, the second-ranked staff member could move ahead. The chain should test this sensitivity before basing an annual bonus decision on the reliability of a single assessment period.

In the report: "The staff member who stands out on child safety is first in the current assessment; but the assessment on this criterion rests on a low-reliability (recently started) source, and a bonus decision should not be made before this reliability is confirmed."

In the illustrative example, lowering T3's reliability on only one criterion and reporting "overall reliability was lowered, so T3 dropped to last" is wrong: when lowered on a single criterion, T3 drops to second, not last; it drops to last only when lowered on all four criteria together. The second error is reporting the manifest's numbers (such as T1=0.700) as though they were today's engine output; the engine's ranking matches these numbers in direction but not in magnitude. The third error is comparing this card's result directly against SFZN-CRADIS's result; although the two engines use the same cell format, they follow a different step order (a single ideal/anti-ideal sum versus a cell-level PIS/NIS closeness coefficient).