q-Rung Orthopair MABAC (Wang, Wei, Wei & Wei, 2020)

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 two degrees: support (μ) and rejection (ν). The constraint is that the sum of these two degrees' q-th powers must not exceed 1. q comes bundled with the method, following the same rule as the q-Rung orthopair data-type card; the user does not choose it. DecisionMind works with q = 3 by default in this family, and checks on upload whether the cells' pairs are valid under this q. Weights are supplied from outside as crisp numbers.

Scale equalisation. Crisp MABAC places every column between 0 and 1 against its own minimum and maximum. There is no separate min-max step in q-Rung orthopair fuzzy MABAC. For cost criteria, μ and ν simply swap places; support becomes rejection, rejection becomes support. Because the cells already sit on the same 0-1 scale, the columns are comparable from the start.

Weighting and the border approximation area. Every cell is scaled by its criterion's weight, following the q-Rung orthopair algebra's own exponential rule: support grows by an amplifying rule, rejection by a diminishing one. The border approximation area is built, for every criterion, with a geometric mean specific to this algebra, applied to these scaled values; it is the q-Rung orthopair counterpart of the geometric mean in crisp MABAC.

Distance and total score. An alternative's distance to the border is a signed Hamming distance. Its magnitude is built from the differences in support, rejection and a hesitancy margin derived from them; its sign comes from a score comparison showing whether the alternative sits above or below the border on that criterion. In crisp MABAC the sign comes directly from subtraction; here, since subtraction alone does not give a direction for two-degree cells, 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 holds the q-Rung orthopair scaling, the geometric-mean border and the signed Hamming distance fixed in this classical form. q comes bundled with the method; weights are taken from outside as crisp numbers.

The total score is read as in crisp MABAC: a positive score means above the border, a negative score below it, and this holds only for this alternative set. The difference is here: beneath this score lies a two-degree support-rejection pair, and a q that determines this pair's acceptance domain. The same pairs can give a different score, and a different ranking, under a different q.

Thus instead of writing:

"According to q-Rung orthopair MABAC, this project is clearly first"

the report should read:

"With the support-rejection pairs deemed valid under q=3 and the weights given, this project sits highest above the border approximation area; which q was used must be stated in the report"

Use this extension in situations where the expert pairs' support and rejection degrees exceed both the intuitionistic fuzzy boundary (sum exceeding 1) and the Pythagorean fuzzy boundary (sum of squares exceeding 1), but their cubes do not sum to more than 1; this is the Fermatean fuzzy special case, with q = 3. If the pairs' cubes are also exceeded, q is raised to the smallest exponent that covers these pairs. If the expert pairs already fit the intuitionistic or Pythagorean boundary, there is no need to raise q; a larger q only reduces discrimination.

The crisp MABAC exit condition applies here too: if no compromise is accepted on one criterion, this extension is also fully compensatory and does not eliminate anything below a threshold. A measured criterion is not expanded directly into a support-rejection pair.

Constraint violation. A pair whose μ^q + ν^q sum exceeds 1 is invalid for the chosen q. This check must be performed separately for every cell.

Choosing a different q cell by cell. The entire matrix works with a single q; if q=2 suffices on one criterion but q=4 is needed on another, the whole matrix is assessed at the largest q required to cover the whole of the pairs.

Computing ν as 1 − μ. In that case the sum is always exactly 1, q=1 is already sufficient, and raising q serves no purpose.

Assuming a large q is "a stronger model." As q grows, the structure accepts more pairs; this does not make it more accurate. As the acceptance domain widens, different judgements move closer together, and the information reflected in the ranking diminishes.

The governing principle is this:

In q-Rung orthopair MABAC, q determines the support-rejection pair's acceptance domain; q comes bundled with the method, does not vary cell by cell, and which q was used must be clearly stated in the report.