Z-Number

A Z-number data structure expresses an alternative's assessment on a criterion through two parts: A states what the value is ("delivery time is roughly 14 days"); B states how reliable that information is ("most likely," "certain," "I am not very sure"). It is written Z = (A, B). A is usually a fuzzy number, because the value itself is already given as "approximate"; B, too, is either a fuzzy number or a term on a declared reliability scale.

The structure's distinctive feature is holding two separate answers to two questions: what is the value, and how far can the source giving it be trusted? The same estimate of "roughly 14 days" carries high reliability from a supplier's ten-year record, and low reliability from the verbal statement of a firm new to the sector; A stays the same, B changes.

Use it where assessments come from sources of differing reliability and this difference ought to affect the decision: group decisions with an experienced expert and a new one; matrices where one criterion comes from measurement and another from a verbal statement; analyses resting on periods of unequal data quality. Where an expert says "roughly 14 days, but I am not very sure," the point is to keep that reservation in B, rather than discard it.

By contrast, where every assessment comes from a source of the same reliability, the B component is identical in every cell and carries no discriminating information; in that case the A component alone, that is, a fuzzy structure, is enough.

Yes, but the two components must be justified separately.

First, for the A component, consider what the value represents: a measured quantity stays classical, and its B component becomes "certain," in which case the Z-number structure adds no information. A forecast or an expert judgement, by contrast, is written as a fuzzy number for A: a lowest, most likely, and highest value.

Second, for the B component, the source of the reliability is determined: the expert's experience, how old the data is, the sample size, the source's past accuracy. B is derived from this source, and is expressed on a reliability scale declared before the analysis ("very low … very high," or a fuzzy number between 0 and 1).

What must not be done is deriving B from the width of A. A wide triangle does not mean "I am not sure"; an expert can give a wide range while being very sure ("certainly between 10 and 20 days"), or a narrow range while not sure at all ("they said 14 days, but I do not trust the source"). The width of A is the value's own uncertainty; B is the source's reliability; the two are asked separately.

The width in a fuzzy structure, the I component in a neutrosophic structure, and the B component in a Z-number are all called "uncertainty," yet the three measure three different things. Fuzzy width says how spread out the value itself is. Neutrosophic I says how incomplete or conflicting the information behind the judgement is. The B component in a Z-number says how far the source giving the value is trusted.

An example shows the difference. For the estimate "delivery is roughly 14 days": A = (12, 14, 16) is the value's spread; if the person giving this estimate is looking at ten years of records, B is high, and if they are guessing, B is low; the width of A is the same in both cases.

For that reason:

"The expert is not sure, so let us widen the range"

is better replaced by:

"The range the expert gives stays in A; their lack of confidence is recorded separately, as a component in B"

The chief advantage of the Z-number structure is that it does not treat sources of differing reliability giving the same value as equal. An experienced expert's estimate of "roughly 14 days" and a new expert's identical estimate carry different weight in the decision process thanks to the B component; this difference is not buried in an average.

Its second advantage is that it lets the decision-maker ask "what sources does this result rest on?" How much the result depends on low-reliability cells can be seen, and this shows which information is worth improving.

Asking for two components doubles the burden on experts, and giving the B component consistently is difficult: experts can be optimistic or pessimistic in assessing their own reliability. B should, as far as possible, be derived not from the expert's own statement but from an external measure (years of experience, data age, past accuracy).

For calculation, a Z-number usually has to be reduced first to a single fuzzy number, and then to a single value. This reduction applies the B component as a weight over A, and there is more than one way of doing so; the path chosen can affect the result. The literature is also relatively young, and there is less accumulated standard practice and cross-method comparison than for the fuzzy structure.

In practice, a Z-number cell consists of six figures: the three corners of the A triangle and the three corners of the B triangle. In the file, each criterion is represented by these six columns; a missing or misordered column renders the cell unreadable.

A Z-number is for the case where the assessment carries a single A value and a single reliability attached to it. Where an expert gives separate degrees of support to several assessments that do not exclude one another, and these degrees can sum to less than 1 (some support may not have been assigned to any option at all), the D-number structure is used. A D-number generalises the mass assignment of evidence theory by dropping the constraints that "options are mutually exclusive" and "the supports sum to 1."

In practice, the D-number structure is represented by relatively few methods; it may be considered for early-stage assessments where missing or overlapping expert support matters to the decision and data collection could not be completed.

The most frequent mistake is deriving the B component from the width of A: giving a low reliability to an expert who provides a wide range. Width and reliability are separate questions.

The second mistake is writing the same B value into every cell and using a Z-number regardless; if B is the same everywhere, it carries no discriminating information, and a fuzzy structure is enough.

A third mistake is leaving B to the expert's own statement, with no link to an external measure; experts cannot consistently assess their own reliability. Also, writing "medium reliability" for a B component when A is a measured value ("2,450 Turkish lira") adds doubt where none belongs to the measurement; for a measured value, B is "certain," and a Z-number is unnecessary for that cell.

The governing principle is this:

A carries the value itself, B carries the reliability of the source giving that value; the two are answered from separate questions and separate sources, and neither is derived from the other.

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