Interval-valued intuitionistic fuzzy MABAC (Xue, You, Lai & Liu, 2016)

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

Cells. In crisp MABAC every cell is a single number. Here every cell consists of four numbers: the lower and upper bound of the degree of support, and the lower and upper bound of the degree of rejection. The sum of the upper support and the upper rejection cannot exceed 1. Criterion weights are supplied from outside as crisp numbers; DecisionMind does not derive weights from incomplete weight information in this family, the weights must already be to hand.

Scale equalisation. Crisp MABAC places every column between 0 and 1 relative to its own minimum and maximum. Here there is no separate min–max step. On cost criteria the four numbers simply change places: the support interval takes the position of the rejection interval, and the rejection interval takes the position of the support interval. This is a different form of equalisation from the ideal–anti-ideal points TOPSIS builds; MABAC already assesses a column against the set's own average rather than against a single extreme.

Weighting and the border approximation area. Every cell is weighted by its criterion's weight using interval-valued intuitionistic fuzzy algebra's own exponential rule: the support bounds change according to a reinforcing rule, the rejection bounds according to a diminishing one. The border approximation area is, as in crisp MABAC, a geometric mean; here it is taken separately for each of the four numbers, using the same interval-valued intuitionistic fuzzy geometric-mean rule. The border itself is thus also a four-number interval pair.

Distance and total score. An alternative's distance to the border is a signed Euclidean distance. Its magnitude is built from the differences of the four numbers; its sign comes from a score comparison that shows whether the alternative is above or below the border on this criterion (when scores are equal, a second accuracy comparison decides the sign). In crisp MABAC the sign already comes from the subtraction itself; here, because subtracting the four-number cells alone gives no direction, the sign is established separately. These signed distances are summed across criteria, and alternatives are ranked from the highest total score to the lowest.

DecisionMind fixes, for this classical form, interval-valued intuitionistic fuzzy weighting, the geometric-mean border and the signed Euclidean distance. Weights are taken from outside as crisp numbers; the optimisation step the founding paper uses to derive weights from incomplete weight information does not run in this engine.

The total score is read as in crisp MABAC: a positive score means above the border, a negative one below, and this holds only for this particular alternative set. The difference lies beneath this score: a support–rejection interval has been carried through the steps as four numbers, and the score does not show the width of that interval.

Thus instead of writing:

"Interval-valued intuitionistic fuzzy MABAC finds this alternative to be the clear winner"

the report should read:

"With the given weights, this alternative sits highest above the border approximation area; the size of the gap is sensitive to the width of the support–rejection intervals and to the weight distribution"

This extension suits situations where only an interval is known about a judgement's degree of support and rejection, and that interval's width needs to be carried through to the end of the calculation. A typical case is a decision table where the lower and upper bounds of several assessors' support–rejection scores are known. If support and rejection are given as a single point each, that is, if the interval width is zero, the extension reduces to intuitionistic fuzzy MABAC.

Crisp MABAC's exit condition applies here too: if no compromise is acceptable on one criterion, this extension is equally fully compensatory and will not screen out anything below a threshold. Where criteria are measured, they remain measured; the interval-valued intuitionistic fuzzy cell is only for a judgement's support–rejection interval.

Constraint violation. In every cell the lower bound must be less than or equal to the upper bound, and the sum of the upper support and the upper rejection must not exceed 1. This constraint must be checked separately in every cell.

Assuming the signed distance comes from the subtraction itself. Plain subtraction of four-number cells gives no direction; the sign is established separately, through a score comparison. Skipping this step and summing only the magnitudes loses track of which alternative sits above the border and which below.

Assuming this engine will complete missing weight information. The founding paper's optimisation step is for situations where weights are only partially known. DecisionMind's engine in this family requires weights to be supplied from outside; running it with incomplete weights does not produce a faulty result, it produces an input the engine cannot run at all.

Mistaking the support–rejection swap on a cost criterion for normalisation. This extension has no separate min–max normalisation step; the benefit–cost distinction is made solely through the support–rejection swap. Attempting to apply crisp MABAC's min–max step on top of this adds an incorrect double conversion.

The governing principle is this:

In interval-valued intuitionistic fuzzy MABAC, the support–rejection interval is carried as four numbers until the border approximation area is built; the direction of the distance comes not from subtraction but from a separate score comparison.