Stochastic

A stochastic data structure expresses an alternative's standing on a criterion not as a single figure but as a probability distribution: the value has such-and-such a mean and such-and-such a standard deviation; or it is spread uniformly across a given range; or it takes the empirical distribution built from past observations. "Six-month return, mean 8 per cent, standard deviation 5 per cent" is a stochastic assessment.

Uncertainty in this structure rests on the assumption that the value is in fact single but simply not known precisely: a measurement error, a sampling error, a random outcome yet to occur. The decision method draws many samples from the distribution and works out how often each alternative comes first, second, or falls within an acceptable region: the result is not a single ranking but an acceptability measure for each alternative.

Use it wherever a value can be described by a distribution with a known source: a mean and deviation from repeated measurements, a survey's sampling error, a historical return series, simulation output, or a device's calibration error bounds.

It also suits problems with no firm information about criterion weights: where decision-makers avoid or cannot agree on weights, stochastic methods treat the weights as a distribution and show which alternative leads under which weight conditions.

By contrast, where the uncertainty is not of the "the value is really single, merely unknown" kind but reflects the concept's own lack of a sharp boundary, a fuzzy structure suits better; where several plausible states exist at once, a hesitant structure fits better.

Yes, but the distribution must have its own source. Two steps are required.

First, decide what the value represents: a single exact measurement, or an estimate with a margin of error, a sample statistic, or a quantity yet to occur. A single exact measurement is not moved into a stochastic structure; a margin of error, a sample, or a future occurrence lets a distribution be built.

Second, take the distribution's shape and parameters from a source: the mean and deviation of repeated measurements, the margin of error and confidence level a survey reports, a historical series' empirical distribution, or a device's documented error bound. The distribution is not guessed; it is calculated or read from a record.

What must not be done is attaching a standard deviation with no source to an exact figure. Saying that 2,450 Turkish lira "deviates by roughly 2 per cent" builds a distribution out of nothing.

Under stochastic uncertainty the value is single but unknown: blood pressure is a definite number, and the device measures it with an error of ±5 mmHg. Under fuzzy uncertainty the value itself is not sharp: where "high risk" begins and ends is, by definition, undetermined.

For that reason:

"There is uncertainty, so I will use a fuzzy structure"

is better replaced by:

"Where uncertainty comes from measurement, sampling or chance, use a stochastic structure; where it comes from the concept's own lack of a sharp boundary, use a fuzzy one"

A survey's ±3-point margin of error, a device's deviation, and a return distribution all belong to the stochastic structure; representing them with a fuzzy number misidentifies the type of uncertainty involved.

The chief advantage of the stochastic structure is that uncertainty is carried through to the result in the language of probability. The outcome is not "A comes first" but "A comes first with 72 per cent probability, B with 21 per cent"; the decision-maker sees directly how robust the ranking is.

It also works where weight information is missing. In problems where decision-makers avoid or cannot agree on weights, stochastic methods show under which weight conditions each alternative is acceptable, letting the weight argument be settled concretely, after the fact.

The result is governed by the choice of distribution. Whether it is normal, uniform or empirical, and how the parameters are estimated, changes the acceptability figures. A distribution built from very few observations is unreliable.

The calculation relies on drawing many samples from the distribution (simulation); the result can shift slightly between runs, and the sample count must be stated in the report. Presenting the result in the language of probability can also make it harder to read for someone expecting a single ranking.

One family of stochastic methods treats not only the criterion values but also the weights as a distribution. Where nothing is known about the weights, every possible combination is scanned with equal probability; where partial information exists (say, "cost matters at least as much as delivery time"), the scan is narrowed accordingly.

The result answers, for each alternative, "in what percentage of the possible weight combinations does it come first?" This suits group decisions where members cannot agree on weights, and settings where assigning weights is politically sensitive.

The most frequent mistake is constructing a distribution with no source: adding an assumed deviation to an exact figure, or choosing the distribution shape simply because "let it be normal."

The second is conflating stochastic and fuzzy uncertainty: representing an expert's approximation with a standard deviation, or a measurement error with a triangular fuzzy number. A third is reading an acceptability figure as a firm ranking: "A comes first with 55 per cent probability" says the ranking is fragile, not that A comes first.

The governing principle is this:

Every distribution in a stochastic structure must rest on a source derived from measurement, sampling, a historical series, or a document; the shape of the distribution and the number of samples must be stated in the report.

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