Spherical Fuzzy

A spherical fuzzy data structure expresses an alternative's standing on a criterion through three degrees: support (membership, μ), rejection (non-membership, ν) and hesitancy (π). Its key difference from intuitionistic, Pythagorean and q-rung structures is that the hesitancy share is not derived from the other two but given separately by the expert. The constraint is that the sum of the squares of the three degrees must not exceed 1. For the judgement "this supplier is reliable", (0.5, 0.4, 0.3) is admissible: the sum of squares is 0.50.

This structure is a relative of the picture fuzzy structure, where the sum of the three degrees themselves, not their squares, must not exceed 1; bounding the squares instead admits a wider range of triples. Its difference from the neutrosophic structure is that the three degrees still constrain one another, whereas in a neutrosophic structure they are independent and may sum as high as 3.

This structure suits cases where an expert's hesitancy is not merely what is left over from support and rejection but information measured and stated in its own right: surveys where "yes / no / undecided" are asked separately, experts giving three distinct degrees such as "agree / disagree / no opinion", and triples too high to fit the picture fuzzy constraint (sum at most 1).

Conversely, where hesitancy is not separately measured, an intuitionistic or Pythagorean structure carries the same information with two degrees, and where triples already fit the picture fuzzy constraint, moving to a spherical structure is unnecessary.

Yes, but two steps are needed.

The first is converting the criterion into a judgement. This structure carries the degrees of support, rejection and hesitancy attached to a proposition, not a triple-valued quantity. "Gold is at 2,450 TL" is a measurement; "gold is a suitable investment for the next six months" is a judgement.

The second is deriving the three degrees from three separate sources: μ from evidence in favour, ν from evidence against, and π from evidence that remains undecided or undirected. If there is no third source, so that hesitancy can only be calculated as what is left over, a spherical structure is not needed.

What must not be done is calculating π as 1 − μ − ν and writing it into a spherical structure. Here π carries no new information, and the structure becomes an intuitionistic fuzzy one whose constraint has been needlessly widened.

In an intuitionistic fuzzy structure, the hesitancy share is calculated: 1 − μ − ν. In a spherical structure, the expert gives all three degrees directly, and hesitancy is itself a measurement. The two figures share a name but not a nature: a derived share is the mathematical remainder of something never said; a given share is a judgement explicitly stated.

For this reason:

"The expert gave μ = 0.5 and ν = 0.4; π = 0.1 was calculated and written into a spherical structure"

should give way to:

"The expert was separately asked for a degree of hesitancy and gave 0.3; the triple (0.5, 0.4, 0.3) was recorded"

If hesitancy was not asked for, a spherical structure should not be used.

The chief advantage of this structure is that it records an "I don't know" answer as a genuine measurement. In two-degree structures, hesitancy is derived from a figure the expert never gave; here it is a figure the expert did give, and the difference in hesitancy between experts can be compared directly in group decisions.

The constraint, defined over squares, also admits triples that a picture fuzzy structure could not: an expert giving three high degrees such as (0.7, 0.5, 0.4), summing to 1.6 but with a sum of squares of only 0.90, has the judgement recorded without being clipped.

Asking for three separate degrees increases the burden on experts; in practice, they may not consistently distinguish "degree of hesitancy", and the third degree can end up filled in arbitrarily.

Ranking requires reducing the three degrees to a single figure; more than one score and distance function has been proposed, and the choice affects the result. Because the degrees still constrain one another, cases where information is genuinely incomplete or contradictory, with high support, rejection and unknown all at once, may not fit here; that is the domain of the neutrosophic structure.

Where an expert gives the three degrees as intervals rather than single figures ("support between 0.5 and 0.7"), the interval-valued spherical fuzzy structure applies, with the constraint holding for the upper bounds. This carries, beyond the judgement itself, the hesitation involved in reducing it to a number; the assessment burden rises markedly, and it should be chosen only where this second layer matters to the decision.

The most frequent mistake is calculating the hesitancy share and writing it into a spherical structure; if the expert did not give a third degree, the structure is not needed.

The second is choosing this structure for triples that already sum to at most 1, where the wider margin adds nothing. Splitting three degrees out of a single source, using a measured quantity directly as μ, and moving from a two-degree structure here on the reasoning that "three numbers are richer" all cause similar problems.

The governing principle is this:

The degree of hesitancy must be obtained from the expert separately, the three degrees must rest on three distinct sources, and any move to the spherical constraint must be justified by triples that genuinely exceed the picture fuzzy constraint.