Probabilistic linguistic MULTIMOORA (Wu et al., 2018)

Three things change; the dominance logic stays the same, only the scoring direction is flipped.

Cells. In crisp MULTIMOORA every cell is a single number. Here every cell is several terms chosen from a previously declared term set, each term given a probability: "term s3 at 40 per cent, term s4 at 60 per cent," for example. This cell form is used when an expert cannot settle on a single term, or when several experts' terms are combined with probabilities. Criterion weights are crisp numbers; they are not entered as probabilistic linguistic values.

Scale equalisation. Crisp MULTIMOORA divides a column directly by the square root of the sum of its squares. Here every cell is first reduced to a single number with the probabilistic linguistic expectation function (PLEF): each term's position in the set is multiplied by its own probability and summed. This is the term's "expected position," and it gives a single real number. Only after this does vector normalisation, exactly as in crisp MULTIMOORA, apply column by column. In other words, the term-probability structure dissolves into a single number before normalisation, at the very first step.

Distance / score / combination. The ratio system, the reference point and the full multiplicative form all operate on this normalised expected value, and there is no difference between them in this respect. The ratio system subtracts the weighted sum of the cost criteria from the weighted sum of the benefit criteria. The reference point finds the best expected value on each criterion, and takes the largest of each alternative's weighted distances to it. The full multiplicative form divides the weighted product of the benefit criteria by the weighted product of the cost criteria.

Result and combination. Crisp MULTIMOORA sums the ranks across the three sub-rankings and the smallest total wins. Here an "improved Borda rule" is used: for each sub-method, an alternative's position is checked; coming last earns 0 points, one place up earns 1 point, and first place earns the highest score. The scores from the three sub-methods are summed, and the alternative with the highest total comes first. This carries the same logic as crisp MULTIMOORA's "smallest rank-sum wins" rule, only with the scoring direction reversed; the two rules make the same alternative come first.

DecisionMind fixes, in this extension, the PLEF (expected value) and the subsequent vector normalisation, the fact that all three sub-methods operate on this one normalised value, and the improved Borda scoring.

The final ranking is, as in crisp MULTIMOORA, a combined summary of the three viewpoints; it is read the same way.

The difference is this. A term's probability distribution, that is, how much the expert hesitated between two terms, is used only at the first step and collapses into a single expected value there. All three sub-methods then operate on this same crisp number from that point on; there is no difference here of the kind found in intuitionistic fuzzy or neutrosophic MULTIMOORA, where "one sub-method defuzzifies later than another."

Thus instead of writing:

"PL-MULTIMOORA is more informative because it carries the probability distribution through the entire calculation"

the report should read:

"Every term's probability distribution has been reduced to an expected value at the first step only; all three sub-methods then operate on this single number, and the shape of the distribution is not used separately in the later steps"

Use this extension when experts decide on a criterion not with a single term but with a probability spread across several terms; for example, when an expert says "40 per cent good, 60 per cent very good." The rule on the linguistic data-type card applies here too: if a criterion is measured, it should not be converted into a term. If an expert can settle on a single term, classical linguistic MULTIMOORA is sufficient; a probability distribution adds an unnecessary layer.

Crisp MULTIMOORA's exit condition also applies here: the full multiplicative form requires every normalised expected value to be strictly greater than zero; a zero expected value makes this component undefined. If no compromise is ever acceptable on one criterion, this extension too is compensatory and will not eliminate anything below a threshold.

Ignoring an expected value that comes out zero or negative. The full multiplicative form becomes undefined in this case; the lowest term in the term set should either never be used in a probabilistic linguistic assessment, or it should be confirmed that it does not come out as zero after normalisation.

Not bringing the probabilities to sum to 1. The probabilities assigned to the terms in a cell must be declared so that they sum to 1; otherwise the expected-value calculation shifts to a different scale in different cells and the comparison breaks down.

Confusing the expected value with the most likely term. Reading "40 per cent good, 60 per cent very good" directly as "very good" forgets that the expected value is a weighted average; when the probabilities of two terms are close, the expected value falls between them.

Changing the term set during the analysis. As the linguistic data-type card warns, the number and order of terms must stay the same for all experts and all alternatives; the PLEF calculation depends on a term's position within the set.

The governing principle is this:

The probability distribution is reduced to an expected value only at the first step; from that point on, PL-MULTIMOORA is a calculation that repeats crisp MULTIMOORA's three sub-methods on this same number and combines the result with a Borda score.