Rough MABAC (Jia, Liu & Wang, 2019)

Four things change; the border-region idea does not.

Cells. This is the card's most distinctive point, and needs stating up front. In crisp MABAC every cell is a single number. In the form described on the rough data-type card, every cell is a single lower and upper bound derived from a group of experts' crisp scores. In this extension the cell is richer still. A group of experts scores every alternative on every criterion with an intuitionistic fuzzy judgement; each judgement is given as a membership degree and a non-membership degree (μ, ν).

The group scores these two degrees separately. Each (μ, ν) pair is then opened out into its own lower and upper approximation. The same logic set out on the rough-number card applies: the average of the scores less than or equal to a given score gives the lower bound, and the average of those greater than or equal to it gives the upper bound. As a result, four numbers sit in every cell: the membership and non-membership degree of the lower approximation, and the membership and non-membership degree of the upper approximation.

DecisionMind takes these four numbers as data; the step from group scores to these four numbers is carried out before the analysis, and the engine does not run this step itself. Weights are given from outside, as in crisp MABAC, as numbers summing to 1; this extension does not generate weights.

Scale equalisation. Crisp MABAC reverses the column for a "lower is better" criterion. Here the same job is done by swapping the membership and non-membership degrees. On a cost criterion, a cell's membership degree swaps places with its non-membership degree, in both the lower and the upper approximation. There is no numerical division here; the reversal is carried out by the intuitionistic fuzzy algebra's own rule.

Weighting and the border approximation area. Because the geometric mean in crisp MABAC cannot work with zero or negative numbers, a constant 1 is added to the values. Intuitionistic fuzzy numbers are already defined between 0 and 1, so this "+1" shift is not needed here. Weighting and the border approximation area are built directly with intuitionistic fuzzy operations. The border carries the same idea as in crisp MABAC: the criterion's typical magnitude within this alternative set. Only here this typical magnitude is built as a four-number interval, from the lower and upper approximations' own geometric means separately.

Distance and total score. Crisp MABAC subtracts the border value from an alternative's value and obtains a positive or negative number. Here subtraction is not performed directly. The two intuitionistic fuzzy values are first summed, and then a score function is computed for this sum. Whether the alternative's value is greater than, less than or equal to the border is determined by looking at this score.

The result is again a signed number: positive if the alternative sits above the border, negative if below. These signed numbers are summed across criteria to give the final score; the summing logic, that is, the compensation, is the same as in crisp MABAC.

Result and defuzzification. The final score is already a single real number; the four-number intuitionistic fuzzy rough structure has already been consumed at the score-function step. The ranking is read from this single number, in the same way as in crisp MABAC.

DecisionMind fixes, in this extension, that group scores are entered as a pre-computed intuitionistic fuzzy table with lower and upper approximations. Comparison, too, is done with an intuitionistic fuzzy sum and score function, not with plain subtraction.

The score says the same thing as in crisp MABAC: an alternative's net position relative to the average-performance border within this set. A positive score does not mean "good" and a negative score does not mean "bad"; it only shows the relative position within this alternative set.

The difference is here. Beneath the score now lies a two-layered uncertainty. The first is each expert's own membership and non-membership degree in their own judgement; the second is the opening of disagreement between experts into a lower and upper approximation. Both layers collapse into a single number at the score-function step and are invisible in the report. If two alternatives' score gap is small, this gap can be sensitive at once to a single expert's judgement and to how much the group agreed on that criterion.

Thus instead of writing:

"Rough MABAC accounts for group uncertainty, so the result is more reliable"

the report should read:

"Disagreement in the group's judgements and each judgement's own membership/non-membership degree have been carried through to the final step; nearby alternatives in the ranking are sensitive to this two-layered uncertainty"

This extension is suitable when a group of experts scores every alternative on every criterion separately, and these scores take the form of intuitionistic fuzzy judgements (a membership and a non-membership degree). Its suitability increases further when the disagreement between experts is itself meant to feed into the decision. Reducing the scores to an average and running crisp MABAC erases this disagreement.

The exit condition on the rough data-type card applies here too. With a single expert, or where experts give very similar scores, the lower and upper approximation narrow to a tight interval. In that case the extension says nothing different from the crisp method. Where criteria are measured, that is, already a crisp number, loading an intuitionistic fuzzy judgement and group disagreement onto that number is not modelling uncertainty but manufacturing it.

Writing the lower approximation greater than the upper approximation. In no cell may the lower approximation's membership and non-membership degree exceed the upper approximation's; if this relation breaks, the cell becomes undefined.

Comparing against the border with plain subtraction. Which of two intuitionistic fuzzy numbers is larger is determined not by the plain difference of their components but by the score function of their sum. Subtracting only the membership degrees and comparing gives a wrongly signed distance.

Averaging group scores first and only then opening them into a rough number. As explained on the rough data-type card, the moment an average is taken, disagreement between experts is erased. The lower and upper approximation must be calculated directly from the individual scores.

Skipping the membership/non-membership swap on a cost criterion and reversing only the number. In crisp MABAC, a "lower is better" criterion is handled by reversing the number. Here the membership and non-membership degrees must be swapped; a purely numerical operation is not sufficient.

The governing principle is this:

A rough MABAC score carries both each expert's own intuitionistic fuzzy judgement and the group's disagreement over these judgements. Both layers collapse into a single number at the score-function step; the report must separately explain this layered uncertainty for alternatives that come out close together.

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