Probabilistic linguistic MABAC (Wei, Wei, Wu and Wang, 2019)

Four things change; the border-approximation-area logic does not.

Cells. In crisp MABAC every cell is a single number. Here every cell is a list of terms together with their probabilities. The expert gives several words with a probability rather than a single word; for example, a supplier might be assessed as "50 per cent moderate, 50 per cent high." Criterion weights are crisp numbers; the method does not generate weights, it takes them from outside.

Scale equalisation. The method first reduces every term list to an expected value: each term's rank in the set is multiplied by its probability and summed within the criterion. This step summarises the distribution into a single number, and it comes before crisp MABAC's own min-max normalisation. Once the expected values are obtained, DecisionMind applies crisp MABAC's own min-max normalisation: every column is placed into the 0-to-1 range according to its own smallest and largest expected value, and this scale is inverted for a cost criterion.

Weighting and the border approximation area. The equalised value first has 1 added to it, then is multiplied by the criterion's weight. This "+1" shift comes from the same technical necessity as in crisp MABAC: the geometric mean in the next step cannot work with a zero or negative value. The border approximation area is, for each criterion, the geometric mean of these weighted values; this is exactly the same rule as in crisp MABAC, only the input is now an expected value derived from a term distribution.

Result and defuzzification. Each alternative's distance to the border is a plain subtraction, exactly as in crisp MABAC, and is summed across the criteria. Defuzzification actually happens at the first step, when the term distribution is reduced to the expected value; from that point on, the calculation follows the same five steps as crisp MABAC. Two different distributions can give the same expected value; one might say "100 per cent moderate" while another says "50 per cent low, 50 per cent high," and both reduce to the same number.

DecisionMind fixes, in the classical probabilistic linguistic MABAC form, the reduction to expected value and, following it, crisp MABAC's min-max normalisation, the "+1" shift and the geometric-mean border. Weights are taken from outside as crisp numbers.

The total score is read as in crisp MABAC: a positive score means above the border, a negative score means below it, and this only holds for this particular set of alternatives. The difference is this: beneath this score there is now a probability distribution, and this distribution becomes invisible once it is reduced to the expected value. The score gap between two alternatives depends not only on which term was chosen but also on the probability distribution the expert placed across the terms.

Thus instead of writing:

"Probabilistic linguistic MABAC uses the expert's entire opinion, so the result is the most accurate"

the report should read:

"This ranking arises from a border approximation area built on expected values; whether the probability distribution beneath the score is concentrated or scattered should be checked separately"

Use this extension when an expert, or a group of experts, evaluates a criterion not with a single term but with an opinion spread across several terms. The typical situation is a group evaluation in which several assessors score the same criterion with different terms and this distribution is to be preserved rather than collapsed into a single average. If an expert selects a single term and gives no probability, the classical linguistic structure is sufficient; if the expert gives a range such as "at least moderate," the hesitant linguistic structure is used instead.

Crisp MABAC's exit condition also applies here: if no compromise is acceptable on one criterion, this extension too is fully compensatory and will not eliminate anything below a threshold. If a criterion is measured, it stays measured; the probabilistic linguistic cell is used only when the criterion's natural measure is a word, and that word is spread across several terms.

Leaving the term list's probabilities unreconciled. The probabilities of the probabilistic linguistic values within the same criterion must sum to one; otherwise the expected value stops being comparable.

Interpreting the expected value in isolation. Two alternatives reaching the same expected value through different distributions must remain visible to the decision; reporting only the number hides the information the distribution carries.

Keeping the term set different from expert to expert. The number and order of terms must be the same for all experts and alternatives; two sets with a different number of steps cannot be compared directly.

The "more sophisticated" fallacy. Probabilistic linguistic MABAC does not produce a "more accurate" ranking than classical linguistic MABAC; it only preserves the spread of expert opinion, without losing it, when that opinion is distributed across several terms.

The governing principle is this:

Probabilistic linguistic MABAC computes by reducing the expert's opinion, spread across terms, to an expected value; any report that forgets that this reduction discards the shape of the distribution is incomplete.