Intuitionistic Fuzzy

An intuitionistic fuzzy data structure expresses an alternative's assessment on a criterion not as a single figure but through two degrees: the degree of supporting the judgement (membership, μ) and the degree of rejecting it (non-membership, ν). Both lie between 0 and 1, and their sum cannot exceed 1. For the judgement "this supplier is reliable," one might write μ = 0.6, ν = 0.3; the remaining 0.1 is the hesitation margin.

What sets this structure apart from fuzzy data is the question it asks. Fuzzy data answers "how plausible is the value" and expresses a quantity (a duration, a price) with approximation. Intuitionistic fuzzy data instead deals with a judgement, asking separately how far you stand for it and how far against it. The hesitation margin is not asked separately; it is what remains once the two degrees are set.

Use it where the assessment comes from a judgement and the expert's opposing view also carries information worth measuring. It is useful in recruitment, supplier selection, and project appraisal, where the 0.6 support given to a judgement such as "this candidate is suitable" and the 0.3 reservation held about the same candidate need to be recorded separately; in short, wherever evidence naturally splits into for and against.

By contrast, if the expert only answers "how suitable" and there is no separate source for the opposing view, there is no need to write ν as 1 − μ and move to the intuitionistic fuzzy structure; that carries the same information as fuzzy or classical data.

Yes, but two steps are required.

First, turn the criterion into a judgement. An intuitionistic fuzzy pair carries the support and rejection degrees of a proposition, not of a number. "Gold is 2,450 Turkish lira" is a measurement; it is neither supported nor rejected. "Gold is a suitable investment for the next six months" is a judgement, and evidence for and against it can be gathered separately.

Second, derive μ and ν from separate sources: μ from favourable evidence (the share of supporting indicators, the share of experts saying "suitable"), ν from unfavourable evidence (the share of opposing indicators, the share saying "not suitable"). If the two do not sum to more than 1, the pair is valid, and the remaining share gives the hesitation margin.

What must not be done is computing ν as 1 − μ. In that case the hesitation margin is zeroed out, ν carries no new information, and the structure's one contribution disappears.

The hesitation margin in the intuitionistic fuzzy structure (π = 1 − μ − ν) is not a component the expert assesses separately; it is what is left over from the support and rejection degrees. It should not, therefore, be read as an independent measure of "the unknown." Where how incomplete or conflicting the underlying information is gets measured from a separate source, independent of support and rejection, that information does not fit this structure; it belongs to the neutrosophic structure.

For that reason:

"The expert stated their hesitation as 0.4, so π = 0.4 was written"

is better replaced by:

"The expert gave support as 0.5 and rejection as 0.1; the hesitation margin of 0.4 was derived from these"

Where hesitation is given directly, a spherical fuzzy or neutrosophic structure should be considered instead.

The chief advantage of the intuitionistic fuzzy structure is that it preserves the difference between "I do not support this" and "I am against this." A single figure of 0.6 support does not say whether the remaining 0.4 is opposition or ignorance; the two-degree structure does.

It also offers a natural question form for experts: "How far do you agree, and how far do you disagree?" The answers to these two questions are usually gathered more reliably than a single score, and they capture differences between experts better in group decisions.

The constraint that the sum cannot exceed 1 invalidates the pair wherever the expert reports both strong support and marked reservation. An assessment such as μ = 0.8, ν = 0.5 does not fit this structure; the expert is forced either to shrink their values or a different structure is required.

For ranking, the two degrees must be reduced to a single value, and several score and accuracy functions exist for this; the choice can affect the ranking. Deriving the two degrees from separate sources is also not always possible for every problem; where no such source exists, the structure gets filled in artificially.

Where an expert can give support and rejection not as a single figure but as a range ("support is between 0.5 and 0.7, rejection between 0.1 and 0.2"), the interval-valued intuitionistic fuzzy structure is used. The constraint stays the same: the sum of the upper bounds cannot exceed 1.

This structure carries a second layer of uncertainty, arising not from the expert's judgement itself but from the hesitation felt in converting that judgement into a number. It raises the assessment burden and should be chosen only where this second layer matters to the decision.

The most frequent mistake is writing ν as 1 − μ. This reduces the intuitionistic fuzzy structure to classical data and zeroes out the hesitation margin in every cell.

The second frequent mistake is deriving μ and ν from a single source: writing the proportion of experts saying "suitable" as μ and the remainder as ν treats undecided experts as opposed. Likewise, using a measured quantity (price, duration) directly as μ, reading the hesitation margin as though it were a separate figure supplied by the expert, and "shrinking a little" to fit a pair whose sum exceeds 1 into the structure all lead to methodological problems.

The governing principle is this:

Support and rejection degrees must come from separate sources, their sum must not exceed 1, and the hesitation margin must be derived from these two degrees, not from the expert.