Fuzzy MABAC

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

Cells. In crisp MABAC every cell is a single number. Here every cell is three numbers: lowest, most likely, highest. Criterion weights are likewise triangular; where a crisp weight is used, its three components are written identical. DecisionMind's Fuzzy MABAC performs no group-decision pooling; every cell is entered as a single triangle.

Scale equalisation. Crisp MABAC places every column between 0 and 1 relative to its own smallest and largest value. Fuzzy MABAC does this separately for each of the three components, but uses one shared denominator: the gap between the column's highest upper value and its lowest lower value. For a benefit criterion, each component is scaled up by dividing by this gap; for a cost criterion, the order is reversed. This is the same linear logic as the crisp method, applied to all three components at once.

Weighting and shifting. A value of 1 is first added to the equalised triangle, and the result is then multiplied, component by component, by the criterion's triangular weight. This "+1" shift comes from the same technical necessity as in crisp MABAC, because the geometric mean in the next step cannot work with a triangle that is zero or negative.

Border approximation area and distance. The border, for each component, is the geometric mean taken separately; the border itself is therefore also a triangle. An alternative's distance to the border is found by subtraction, as in crisp MABAC, but the subtraction between triangles is carried out using cross-matched components. One triangle's lowest end is subtracted from the border's highest end, and its highest end from the border's lowest end. This way, the genuine share of uncertainty in both triangles is reflected in the distance; one triangle may be wide, the other narrow.

Result and defuzzification. Each distance triangle is first reduced to a single number by its centre. The centre is the average of the three components, abbreviated COA. These single numbers are then summed across the criteria. DecisionMind keeps this centre defuzzification fixed. Using a different defuzzification rule, for instance taking only the most likely value, can produce a different score order.

The score is read the same way as in crisp MABAC. A positive score means the alternative sits above the border, a negative score means it sits below, and this reading holds only for this particular set of alternatives. The difference is this: although the score appears as a single number, it carries three layers of uncertainty beneath it. This uncertainty becomes invisible because it is reduced to a single number at the defuzzification step. If the gap between two alternatives' scores is small, that gap is sensitive to the width of the estimates in the input; this width shows how open the triangle is.

Thus instead of writing:

"Because Fuzzy MABAC takes uncertainty into account, A2's superiority is certain"

the report should read:

"The expert estimates were entered as triangles, and the uncertainty has been carried through to the border-distance calculation; A2 sits highest above the border, A1 lowest, and the order between A1 and A3 is sensitive to a shift in the estimate for a single criterion"

Use this extension when the criteria are not measured, but rest on expert estimation or judgement, and both a range and a most-likely point for that estimate are known separately. Typically, a performance ratio that has not yet materialised is estimated as "at least, most likely, at most"; this ratio could be a criterion such as yield, compliance, or satisfaction.

Where criteria are measured, counted or invoiced, stay with the base method. Adding width to a measured ratio does not model uncertainty, it manufactures it. DecisionMind requires a single data type; if the table is mixed, a measured criterion is also written as a triangle, with all three components equal. This adds no information; it only conforms to the format.

The exit point is the same as for crisp MABAC. This extension is compensatory too; if no compromise is acceptable on a criterion, the ELECTRE family should be considered instead.

Breaking the l ≤ m ≤ u order of the value range. If the triangle's components do not keep this order, the whole calculation becomes invalid. This happens, for example, if the most likely value is written smaller than the lowest. Every cell must be checked before entry.

Comparing the result against a different source without changing the defuzzification method. Using a defuzzification other than the centre (COA), for instance taking only the most likely value, can produce a different order. This is the rule DecisionMind keeps fixed; this should be stated when comparing against another piece of software.

Defuzzifying first and then running crisp MABAC. Reducing the triangles to a single number at the outset and applying the crisp method is not Fuzzy MABAC. Doing so strips the border approximation area and distance calculation of any uncertainty, and exaggerates the gaps between scores.

The "more advanced" fallacy. Fuzzy MABAC does not produce a more "correct" ranking than crisp MABAC; it is only a more honest representation, and only when the nature of the input, an estimate, calls for it.

The governing principle is this:

Fuzzy MABAC exists to carry the approximation in expert estimation through to the border approximation area; any application that hardens the input at the outset, or breaks the component order, destroys the extension's only contribution.