Fuzzy Data

Rather than a single number, a fuzzy data structure expresses an alternative's standing on a criterion through a set of numbers, each carrying its own membership grade for how plausible it is. Its commonest form, the triangular fuzzy number, is defined by three values: the lowest plausible figure, the most likely figure, and the highest plausible figure. The statement "delivery time falls between 10 and 20 days, most likely 14" is the triangular fuzzy number (10, 14, 20). Membership at the midpoint is 1, it falls away steadily towards the two ends, and beyond the ends it is 0.

A membership grade is not a probability. Saying "membership 1 for 14 days" does not claim that 14 days will certainly occur; it claims that 14 days is the value that best fits this assessment. A fuzzy structure is the numerical counterpart of words such as "approximately," "around," "good" or "fair": expressions with no sharp edge.

Use a fuzzy structure when the assessment comes not from measurement but from human judgement or a forecast. It is useful wherever experts score criteria such as quality, risk, fit or reputation as "approximate" or in words ("low / medium / high"), wherever a forward-looking quantity is at stake (expected cost, expected demand), and wherever a concept has no sharp boundary by its very nature even though it could in principle be measured ("sufficient experience," "a reasonable price").

Where a value has instead been measured, counted or invoiced, moving it into a fuzzy structure is unnecessary; adding width to a measured number is not modelling uncertainty but manufacturing it.

Yes, but the width added must have a genuine source. Two steps are required.

The first is deciding what the value actually represents: a measured quantity, or a judgement or forecast? A measured quantity stays crisp. A judgement or a forecast can be fuzzified, because some "approximation" already sits in its nature.

The second is justifying the lowest, most likely and highest values individually. For an expert forecast, this triple is taken directly from the expert ("at least, most likely, at most"). For a score given in words, a pre-announced scale is used: on a seven-point scale, for instance, the term "good" might correspond to the triangular number (7, 9, 10). The scale is defined before the analysis begins, stated in the report, and applied identically to every expert.

What must not be done is building a triangle by adding a symmetric margin to either side of a crisp number. Writing 2,450 TL as (2,400, 2,450, 2,500) has no source for any of the added values.

A survey's margin of error of ±3 points, a measuring instrument's deviation of ±0.5 units, or an investment's return distribution are all probabilistic uncertainty: the value is really a single number whose exact size simply is not known. Fuzziness, by contrast, is conceptual uncertainty: the phrase "a reasonable price" has no sharp boundary of its own.

For this reason:

"Fuzzy is used because there is measurement error"

is better replaced by:

"Membership grades are kept because the underlying concept has no sharp boundary"

A stochastic structure suits measurement error and distributions; an interval structure suits cases where only a lower and an upper bound are known.

Fuzzy number forms are distinguished by the shape of the answer they give to "which values are most plausible."

Triangular (TFN, three values: l, m, u). There is a single most likely value, and plausibility falls off linearly on both sides of it. If an expert says "at least 10, most likely 14, at most 20 days," a triangular number fits. It is the commonest form; the great majority of fuzzy methods in DM3 expect it.

Trapezoidal (TrFN, four values: a, b, c, d). Here the most likely value is not one point but a band: every value between b and c is equally plausible, while a and d mark the outer limits. If an expert says "12 to 16 days is normal, it is never below 10 or above 20," the trapezoidal number (10, 12, 16, 20) fits. The triangular number is simply the special case of the trapezoidal one where b equals c.

Interval (a, b). Nothing at all is known about a most likely value; every value between the lower and upper bound is equally plausible. This is the special case of the trapezoidal number where a equals b and c equals d. When only a boundary is known, an interval structure should be used rather than building a triangle and declaring its midpoint "most likely."

Gaussian (centre, spread). Plausibility declines smoothly, bell-curve fashion, as distance from the centre grows, with no sharp outer edge. It suits an expert who says "around 14, but I cannot fix a firm limit." Few methods support this form.

LR form. The shape of the left and right flank is defined separately; it suits asymmetric uncertainty, as in "finishing before 14 days is unlikely, running late is quite likely." Triangular and trapezoidal numbers are the special cases of the LR form with straight-line flanks.

Type-2 fuzzy. Here the membership grade itself is uncertain: if an expert cannot decide whether 14 days carries membership 1 or 0.8, or if several experts' triangles fail to line up, membership is given as an interval instead of a single figure. The assessment burden rises sharply, so this form should be reserved for cases where the uncertainty in membership genuinely matters to the decision.

Some structures whose name includes "fuzzy" are not forms within this family but separate data types in their own right: intuitionistic fuzzy (a membership and a non-membership grade), Pythagorean and q-rung orthopair, picture fuzzy, spherical fuzzy, hesitant fuzzy. These answer not "how plausible is the value" but "how far do I support this judgement, how far do I reject it," and each is described on its own card.

A fuzzy structure's principal strength is that, in converting human judgement into numbers, it does not erase the approximation already built into that judgement. Where the gap between "good" and "very good" is not sharp, forcing that gap into two crisp numbers manufactures a false precision; a triangular or trapezoidal number carries the approximation through the whole calculation instead.

A pre-announced scale for word-based assessments also lets different experts' judgements be brought onto the same numerical footing and combined in a group decision.

Operations on fuzzy numbers widen them further: the width of the sum of two triangular numbers is the sum of their two widths. Across a calculation with many steps, the result can turn out to be a far wider fuzzy number than any of the inputs, and discrimination between alternatives suffers.

Ranking requires the fuzzy result to be reduced to a single number, a step called defuzzification, and more than one method exists for it: centroid, mean, most-likely value. The method chosen can affect the ranking. Comparing fuzzy numbers directly is likewise not tied to one single rule; different ranking rules can give different results.

Finally, the very scale that converts words into numbers is itself an assumption. A five-point and a seven-point scale turn the same judgement into different triangles; the choice of scale must be justified in the report.

The commonest mistake is fuzzifying a measured value so that it "looks more thorough." Adding width to a measured 2,450 TL adds no information; it simply makes the analyst the author of that width.

A second common mistake is representing measurement error or a probability distribution with a fuzzy number. A survey's margin of error is not fuzziness; it calls for a stochastic structure.

A third is failing to announce the scale when converting words to numbers, or using a different scale from expert to expert. It is likewise a formal error to declare a midpoint "most likely" and build a triangle when only a boundary is known, or to choose a triangle instead of a trapezoid when the most likely region is really a band.

The governing principle is this:

Every corner of a fuzzy number (lowest, most likely, highest) must rest on its own justification; width should never be added to a measured number after the fact, but must come from the judgement or forecast itself.