Probabilistic linguistic TOPSIS (Lu et al., 2019)

Four things change; the decision logic does not.

Cells. In crisp TOPSIS every cell is a single number. Here every cell is a list of terms together with their probabilities. Rather than a single word, the expert gives several words with probabilities; a store branch's service quality, for instance, might be judged "60 per cent medium, 40 per cent high." Every probabilistic linguistic value on the same criterion is brought to equal length, and each one's probabilities are made to sum to one. Criterion weights remain 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 within the set is multiplied by that term's probability and the products are summed within the criterion. This step, which summarises a distribution into a single number, comes before crisp TOPSIS's vector normalisation. Once the expected values are obtained, DecisionMind applies crisp TOPSIS's own normalisation: each column is divided by the square root of the sum of its squared values, then multiplied by the weight. This step does not exist in 2-tuple linguistic TOPSIS; there, the weight enters the distance formula directly, whereas here it is applied to the column first.

Distance. The ideal and anti-ideal are built from the weighted, normalised expected values, according to the direction of each criterion. The distance is the same straight-line Euclidean distance as in crisp TOPSIS; the weight has already been worked into the values at the previous step.

Result and defuzzification. The closeness score is again a single number between 0 and 1. The difference lies here: reducing to an expected value discards the shape of the term distribution, that is, which terms were how probable; only the distribution's average remains. Two different distributions can give the same expected value; one might say "100 per cent medium" while another says "50 per cent low, 50 per cent high," and both come down to the same number. This is a fundamental difference from 2-tuple linguistic TOPSIS: there, the passage from a single term plus a small shift to a number loses no information, whereas here the passage from a probability distribution to a single summary number does lose information.

DecisionMind fixes, for classical probabilistic linguistic TOPSIS, the reduction to expected value followed by crisp TOPSIS's vector normalisation. Weights are taken from outside as crisp numbers.

The closeness score is read as in crisp TOPSIS: it is not a percentage, and it is not compared with a different analysis. The difference is this: beneath the score now lies a probability distribution, and this distribution disappears once it is reduced to an expected value. The score gap between two alternatives depends not only on which term was chosen but also on the expert's probability distribution across the terms. The report should therefore note not only the closeness scores but also how spread out or concentrated the distribution is; an expected value coming from a concentrated distribution is more reliable than the same expected value coming from a spread-out one.

Thus instead of writing:

"Because probabilistic linguistic TOPSIS uses the expert's full opinion, the result is the most accurate"

the report should read:

"The closeness score, computed over expected values, has produced this order; whether the probability distribution beneath the score is concentrated or spread out should be checked separately"

Use this extension when an expert, or a panel of experts, judges a criterion not with a single term but with an opinion spread across several terms. The typical situation is a group assessment in which several reviewers or auditors score the same criterion with different terms, and this spread is to be preserved rather than collapsed into a single average.

It must not be confused with the other member of the same family. If the expert selects a single term and the fusion result is to be kept as a shift rather than rounded to a term, 2-tuple linguistic TOPSIS is used; there is no probability there, only a single term and its shift. If the expert gives a range such as "at least medium" and no probability, the hesitant linguistic structure is used.

The condition for staying with the base method is the same: if a criterion is measured, it remains measured. If the table is mixed, DecisionMind requires a single data type. The exit point is the same as for crisp TOPSIS: if no compromise is acceptable on a criterion, this extension too is compensatory and will not eliminate anything below a threshold.

Leaving the term list's probabilities unreconciled. The probabilistic linguistic values on the same criterion must have their probabilities sum to one and be brought to equal-length term lists; otherwise the expected value stops being comparable.

Interpreting the expected value on its own. 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 identical across every expert and alternative; two sets with a different number of steps cannot be compared directly.

Giving weights as terms. In this kernel, criterion weights are crisp numbers; if a linguistic weight is to be given, how it was reduced to a crisp number must first be stated in the report.

The governing principle is this:

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