Fuzzy MOORA

Cells. In crisp MOORA 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 DecisionMind reduces this weight to its centroid (the average of the three components) before it enters the calculation; the calculation itself proceeds with a single number. So the uncertainty is carried only in the decision matrix's cells, not in the weights. Writing a weight as a TFN adds no width to the result whatsoever; only the centre value is used.

Scale equalisation. Crisp MOORA divides every column by the square root of the sum of squares of its own values. Fuzzy MOORA applies the same operation to the three components (lowest, most likely, highest) separately: each component is divided by the magnitude of the same component in its own column. The reason is this: when this method is applied to a degenerate triangle whose three components are all the same number, that is, one that actually corresponds to a crisp value, the calculation automatically reduces to crisp MOORA. No separate special-case code is required.

Ratio system. The same logic runs in parallel across the three components: for each weighted component, the difference between the sum on the "more is better" criteria and the sum on the "less is better" criteria is computed separately (the difference of the lowest components, the difference of the most-likely components, the difference of the highest components). Where crisp MOORA has this difference as a single number, here it is three numbers.

Result and defuzzification. In crisp MOORA the net score is already a single number. Here the three-component difference is reduced to a single number by taking the components' average (the centroid); this is the last step, not the first. The output is not a fuzzy number but a net score and rank directly, as in crisp MOORA's output structure; only the three components have been carried in parallel throughout the calculation.

DecisionMind holds this three-component normalisation and the final centroid defuzzification fixed for classical Fuzzy MOORA; weights are reduced to their centre even when entered as TFNs.

The output is a net score and rank, as in crisp MOORA, and it is read the same way: it is not a percentage or a probability, and it is not compared with a different analysis. The difference is this. This net score is the average, taken at the final step, of the three-component uncertainty in the cells; the uncertainty affects the calculation, not the outcome, and becomes invisible by the end. Because of this, the score gap between two alternatives is sensitive not only to the weights but also to how wide, that is, how uncertain, the inputs were given.

So instead of:

"Fuzzy MOORA accounts for uncertainty, so the result is more reliable"

the report should read:

"Because expert scores were given approximately, the uncertainty has been carried through the calculation and reduced to a single score at the final step; whether the ranking is robust to a one-notch score change or a weight swap should be shown separately"

Use this when criteria come from expert judgement or approximate estimation and reducing that judgement to a single number would create an artificial precision. Opening a measured criterion (such as price or duration) into a triangle is not modelling uncertainty but manufacturing it; DecisionMind requires a single data type in the matrix, so a measured criterion in the same matrix is written with all three components equal. The exit condition is the same as for crisp MOORA: if no compromise is acceptable on a criterion, this extension is also compensatory and will not eliminate anything below a threshold; the ratio system's reference-point-free character also applies here.

Violating the value space. Every cell must satisfy lowest ≤ most likely ≤ highest and non-negativity; an out-of-scale triangle (for example, a most-likely value greater than the highest) invalidates the calculation from the start.

Ignoring the choice of defuzzification method. The centroid (the average of the three components) is DecisionMind's fixed choice; a different defuzzification rule (such as taking only the most-likely value) can give a different ranking, and this choice should be stated in the report.

Writing a weight as fuzzy and assuming "now the weight also carries uncertainty". DecisionMind reduces the weight to its centre before it enters the calculation; the TFN weight's width has no effect on the outcome at all. The only place uncertainty is genuinely carried is the decision matrix's cells.

Reducing every cell to its centre first and running crisp MOORA. In the illustrative example below, because the triangles are roughly symmetric, this shortcut happens to give an apparently close result on this particular data set; but this is not a guarantee. With asymmetric or wide triangles the two routes can diverge, and in every case the information about which criterion was judged more uncertain is erased from the outset.

The governing principle is this:

In Fuzzy MOORA, uncertainty is carried only in the decision matrix's cells and collapses to a single score at the final step; writing the weight as fuzzy, or defuzzifying the input from the outset, removes this carrying altogether.