Fuzzy MULTIMOORA

Four things change; the dominance logic does not.

Cells. In crisp MULTIMOORA every cell is a single number. Here every cell is three numbers: lowest, most likely, highest. Criterion weights can also be entered as triangles, but in this extension DecisionMind never carries the weight's uncertainty into the calculation: the weight triangle is reduced to its centre value ((lowest + most likely + highest) / 3) at the very first step and used as a crisp number throughout the remaining five steps. Uncertainty stays only in the decision matrix's cells, not in the weights. This extension does not support a group decision; if more than one expert's triangles exist, they must be aggregated outside the card, at the data-preparation stage.

Scale equalisation. Crisp MULTIMOORA divides a column by the square root of the sum of its squares. Here too a vector normalisation is performed, but simultaneously across the three components (lowest, most likely, highest). The denominator is built by summing, across the alternatives, the mean of the squares of each cell's three components and taking the square root; the triangle's three components are all divided by this SINGLE denominator. The logic of the crisp method is preserved exactly; it is simply applied jointly across the three components.

Distance/score/aggregation. The three sub-methods defuzzify the uncertainty at DIFFERENT points; this is the extension's most important design decision. The ratio system sums and subtracts the weighted triangles (using triangle arithmetic) and only AT THE END arrives at a single triangle, whose centre value it then takes (centroid defuzzification, "COA"). The reference point and the full multiplicative form, by contrast, reduce every cell to its centre value IMMEDIATELY AFTER normalisation. They run the remaining calculations, that is, the deviation from the best value and the product across criteria, on these crisp numbers. So the ratio system carries the fuzziness step by step, while the reference point and the full multiplicative form consume it right after normalisation.

Result and defuzzification. The output is a single number (the negated sum of the three sub-rankings); it does not remain a triangular number. DecisionMind holds the centroid defuzzification ((lowest + most likely + highest) / 3) fixed in this extension; no other defuzzification rule (taking the most-likely value, taking the widest interval) is used.

DecisionMind holds fixed, in this extension: the immediate reduction of weights to their centre value, component-based vector normalisation, late (end-stage) defuzzification in the ratio system versus early (immediately after normalisation) defuzzification in the reference point and the full multiplicative form, and the Borda-sum form of dominance theory.

The final ranking is, as in crisp MULTIMOORA, a combined summary of three perspectives (average performance, distance from the worst case, multiplicative balance); it is read the same way: an alternative coming first in the final ranking does not mean it came first in ALL THREE sub-methods.

The difference is this. Of the three sub-methods, only one, the ratio system, carries the fuzziness all the way through. The other two, the reference point and the full multiplicative form, defuzzify immediately after normalisation and effectively work with crisp data thereafter. So it is wrong to say "the uncertainty was carried equally in all three sub-methods"; the report should state which sub-method defuzzifies, and when.

So instead of:

"Because Fuzzy MULTIMOORA carries uncertainty through the entire calculation, the result is more reliable"

the report should read:

"Uncertainty is carried only up to the final step in the ratio system; the reference point and the full multiplicative form work with crisp values that have been defuzzified immediately after normalisation"

Use this when criteria come from expert judgement or estimation, and reducing these values to a single number would create an artificial precision. Opening a measured criterion into a triangle produces uncertainty rather than modelling it; the matrix must be a single type (a DecisionMind rule), so if a measured criterion is also to be written into a triangle, its three components are entered as the same number. Crisp MULTIMOORA's exit condition applies here in exactly the same way: if there is a zero or negative value, or if no compromise can ever be made on a criterion, this method should not be used. Because the full multiplicative form operates on the triangles' centre value, a cell whose centre value is zero or negative also renders this component undefined.

Not declaring the value range. Every triangle must satisfy lowest ≤ most likely ≤ highest, and all components must be non-negative; if this ordering breaks, normalisation and the reference point become meaningless.

Assuming weight uncertainty is carried through. Even when a weight is entered as a triangle, it collapses to its centre value at the first step; saying "the uncertainty in the weights was also reflected in the result" is wrong in this extension.

Assuming the reference point and the full multiplicative form also work with triangle arithmetic. Only the ratio system uses triangle arithmetic; the other two work on crisp values that have been defuzzified immediately after normalisation. In the illustrative example below, these two sub-methods' numbers turn out nearly equal to the numbers produced by taking the same matrix, reducing it directly to its centre value right after normalisation, and running crisp MULTIMOORA. This is not a coincidence; it is a consequence of the extension's own design.

Changing the defuzzification method, or not stating it in the report. DecisionMind uses the centroid ((lowest + most likely + highest) / 3); a different defuzzification rule (the most-likely value, the widest interval) can produce a different ranking, and the choice must be written into the report.

The governing principle is this:

In Fuzzy MULTIMOORA, uncertainty is carried only to the end in the ratio system; the reference point and the full multiplicative form defuzzify immediately after normalisation, and it must be reported that these three sub-methods do not carry the fuzziness to the same degree.