Hesitant fuzzy linguistic MABAC (Sun, Hu, Zhou and Chen, 2018)

Four things change; the logic of comparing against the border area stays the same.

Cells. In crisp MABAC every cell is a single number. Here every cell is a comparative verbal expression: it can be a single term such as "s_2," an interval such as "at least s_1" or "between s_{-1} and s_1," or an open-ended expression such as "greater than s_2." The scale is symmetrically indexed so that its middle ("medium") is zero; on a seven-term scale, for example, s_{-3} means "none," s_0 means "medium," and s_3 means "extreme." Every expression is first converted into a set built from these indices.

Scale equalisation. Crisp MABAC places every column into the 0-1 range with min-max normalisation. No such division exists here; on a benefit criterion the set is left as it is, and on a cost criterion the sign of every index in the set is reversed (thanks to the scale's symmetric structure, this corresponds to the linguistic complement). Every set is then converted into a linguistic term set scaled by the criterion's weight (a weighted HFLE).

Border approximation area and distance. In crisp MABAC, the border is built from the geometric mean, and the distance is a signed difference. Here the border is the arithmetic mean of the indices of every alternative's weighted set on that criterion (a different design choice from classical MABAC's geometric mean). The distance is not a subtraction but a projection-based signed difference: every cell and the border are projected onto an axis running through the scale's most extreme term ("s_g"), and the difference between these projections shows whether the alternative sits above the border (positive) or below it (negative).

Result and defuzzification. In crisp MABAC, every alternative's distances to the border are summed. Here, the signed projection differences per criterion are first shifted to a non-negative level, then combined with a Bonferroni mean. The special case of the Bonferroni mean at p = 1, q = 0 reduces to a simple average across criteria, and comes closest to crisp MABAC's summing logic; DecisionMind uses p = 1, q = 1 in this card, a richer combination that also accounts for interaction between criteria.

For classical hesitant fuzzy linguistic MABAC, DecisionMind fixes the projection axis (the scale's most extreme term) and the Bonferroni parameters (p = 1, q = 1).

The total score (CC) is read similarly to crisp MABAC's score: a larger CC shows a stronger position relative to the border. But the score is no longer a signed distance to a geometric-mean border; it is the Bonferroni mean of the projection differences, and it contains a shift constant. What matters, therefore, is not its absolute magnitude but the order and relative gap among alternatives.

Thus instead of writing:

"The hesitant fuzzy linguistic MABAC score is an exact measure that directly reflects the expert's verbal expression"

the report should read:

"The score is the result of converting the comparative verbal expressions into indexed sets, the projection difference to the border, and the Bonferroni mean; narrowing or widening the expression can change the score"

This extension is used when the expert gives a criterion not as a single verbal term but as a comparative expression such as "at least," "at most" or "between." Where a single verbal term is enough, a simpler linguistic extension may suffice instead. Measured criteria should not be carried into this extension. Where the matrix must hold a single data type, a measured value is written as the single nearest term on the scale. Base MABAC's exit condition applies here exactly as before.

Reducing a comparative expression to a single midpoint. Taking only the lower bound (s_1) of "at least high," or only its midpoint, erases the open-ended uncertainty the expression carries, at an early step.

Missing a cost criterion. Skipping the sign reversal on a cost criterion reverses every alternative's position relative to the border on that criterion, whether above it or below.

Changing the Bonferroni parameters (p, q) without stating them. Choosing p = 1, q = 0 comes closer to classical MABAC's simple sum, while p = 1, q = 1 also factors in interaction between criteria; this choice can produce a different CC score from the same data, and must be stated in the report.

Using the source paper's printed intermediate-table values as input without question. Some of the intermediate results in Sun et al.'s (2018) paper may not be reproduced exactly when the engine runs its own formula; the engine stays faithful to the formula, and the source of any difference must be stated in the report.

The governing principle is this:

Hesitant fuzzy linguistic MABAC converts the comparative verbal expression into an indexed set and compares it against the border through a projection difference. If the expression is reduced early, or the sign and parameter choices are not reported, it is not the calculation itself that breaks down, only its apparent precision.