2-tuple linguistic TOPSIS (Wei et al., 2010)

Four things change; the decision logic does not.

Cells. In crisp TOPSIS every cell is a single number. Here every cell consists of a term and a shift. An expert score is chosen from a nine-level term set: very low, low, quite low, slightly low, medium, slightly high, quite high, high, very high. When more than one expert's scores are aggregated, the result often falls between two terms. The method does not round this result to the nearest term; it keeps it as a term with a small shift away from it. For instance, (quite high, −0.4) means "between quite high and slightly high, closer to quite high." Criterion weights here are crisp numbers, not terms; the method does not generate weights, it takes them from outside.

Scale equalisation. Crisp TOPSIS divides every column by the square root of the sum of its squares. 2-tuple linguistic TOPSIS does not run this step. Every term converts directly into a number by adding the shift to its position in the set. No further division by magnitude is applied to this number; the ideal and anti-ideal alternatives are built directly from these numbers, according to each criterion's direction.

Distance. Every alternative's distance to the ideal and the anti-ideal is a weighted Euclidean distance. The difference is taken, multiplied by the weight, squared, summed across criteria, and the square root taken. The weight sits directly inside the distance formula here, not inside a normalisation step; this differs from crisp TOPSIS's order, where the weight is applied to the column first.

Closeness score and defuzzification. The definition is the same as in crisp TOPSIS: the score is the ratio of the distance to the anti-ideal over the sum of the two distances. The transformation that converts a term into a number is one-to-one and reversible; a result that falls between two terms is kept together with its shift, and no information is discarded. This differs from extensions that work with triangular fuzzy numbers, where uncertainty is a three-cornered interval defuzzified at the last step; here it is a fine shift around a single term.

DecisionMind fixes, for classical 2-tuple linguistic TOPSIS, a nine-level term set and the term-to-number conversion. Weights are taken from outside as crisp numbers; the method does not generate weights.

The closeness score is read as in crisp TOPSIS: it is not a percentage, it is not compared with a different analysis, and the ideal point shifts when the alternative set changes. The difference is here: the uncertainty beneath the score is not a wide triangle but a single term and a fine shift around it. This is why the robustness of the gap between two alternatives' scores is tested by asking "how many levels of term change does it rest on"; a single expert shifting one score by one level may be enough.

Thus instead of writing:

"2-tuple linguistic TOPSIS does not lose information, so the result is exact"

the report should read:

"Expert scores were chosen from the term set, and the aggregation result was preserved with its shift; the ranking is sensitive to a term change of this many levels"

Preserving the shift does not make the result exact; it merely restores the information that rounding to a term would have discarded.

Use this method when experts assess a criterion not with a number but with a word chosen from a pre-declared term set, and when the aggregation result needs to be preserved without rounding to a term. Typical situations: criteria whose natural measure is a word, such as service quality, corporate reputation or degree of compliance, and board decisions where several experts' scores are combined without collapsing to an average.

It should not be confused with the other member of the same family. If an expert gives not a single term but several terms with probabilities ("60 per cent high, 40 per cent very high"), probabilistic linguistic TOPSIS is used; here there is a single term and its shift. If the expert has an "approximately this number" in mind and wants it converted into a triangle, fuzzy TOPSIS is appropriate.

The case for staying with the base method is unchanged: if a criterion is measured, it stays measured and is not converted into a word. If the table is mixed, DecisionMind asks for a single data type; a measured criterion is written with a fixed term at one end of the term set. The exit point is the same as for crisp TOPSIS: if no compromise is acceptable on one criterion, this extension is compensatory too and will not eliminate anything below a threshold.

Dropping the shift and rounding to the nearest term. Pulling the aggregation result directly to the nearest term defeats the method's one contribution: preventing information loss. In the illustrative example below, this does not change the ranking of the two alternatives but hides how robust the gap between them is.

Moving the shift value outside the term set. The shift should stay between −0.5 and 0.5; a value such as 0.6 actually signals a move to the next term up and needs recalculating.

Changing the term set during the analysis. The number of levels and the order of the terms must remain the same for every expert and every alternative; two sets with different numbers of levels cannot be compared directly.

Giving weights as terms. In this kernel, criterion weights are crisp numbers. If weights are also to be given as terms, linguistic weighting methods should be consulted first, and how they were reduced to a crisp number must be stated in the report.

The governing principle is this:

2-tuple linguistic TOPSIS exists to carry the shift produced when terms are aggregated without losing information; any application that drops the shift or moves it outside the term set destroys the method's one contribution.