Fuzzy DNMA

Four things change. The decision logic stays the same.

Cells. In crisp DNMA every cell is a single number. Here every cell carries three numbers: the lowest, the most likely and the highest plausible value. Criterion weights can also be entered as triangles. But the method reduces the weights to a single number in the very first step, taking the average of each weight's three components. The triangular structure is not preserved in the weights, only in the decision matrix.

Scale equalisation. As in crisp DNMA, two different normalisations are computed here too. Vector normalisation divides by the column magnitude; for a cost criterion the result is subtracted from one to turn its direction into the benefit side. Linear-max normalisation is built against the column's largest value for a benefit criterion and its smallest value for a cost criterion. Both are applied separately to the triangle's three components; there is no defuzzification yet at this step.

Score and aggregation. The two normalised tables now descend to their centre value; the centre is the average of the three components. From these centre values, three sub-utilities are computed: a weighted sum over vector normalisation, a weighted sum over linear normalisation, and a weighted product over linear normalisation. As in crisp DNMA, the internal ranking of these three sub-utilities is then extracted. Next the normalised square of the score magnitude is summed, the normalised square of the rank inconsistency is subtracted from it, and the combined score is built. This last step uses exactly the same formula as crisp DNMA.

Result. Defuzzification happens in two stages. First the criterion weights descend to a single number at the very start. Then the normalised matrices descend to their centre value immediately before the three sub-utilities are computed. The output is a single crisp number and, as in crisp DNMA, it can come out negative; a negative score is not an error, it only shows a relatively weak position.

DecisionMind keeps the weight given to the score and rank components of the three models (α and β) equal in this fuzzy extension, just as in crisp DNMA; all three are one-third. These weights are not user input here either.

The output is a combined score, as in crisp DNMA. A higher score is better, but this number arises not from a distance measure but from how much the three separate models confirm one another.

The difference is this. These three models here operate over the triangles' centre value. So there is an uncertainty allowance per model, and this allowance is lost the moment it descends to the centre value. If an alternative's score is high, this means that alternative is consistently ranked well by all three models at their centre values; the width of the input triangles does not show how robust this consistency is.

Thus instead of writing:

"According to the Fuzzy DNMA score, A2's performance is excellent"

the report should read:

"A2 is the alternative that all three aggregation models consistently rank best at their centre values. This score shows only a ranking among these three alternatives, not an absolute percentage of success"

This extension is suitable when criteria come from expert judgement or estimation and you do not want to rely on a single aggregation logic alone. Which situations can be fuzzified, and the principle of not converting measured data, are explained on the Fuzzy data-type card. DecisionMind requires a single data type. If the table is mixed, a measured criterion is also written as a triangle, with all three components the same number.

Crisp DNMA's exit condition applies here too. If a criterion column carries the same triangle for every alternative, linear normalisation becomes undefined. Where no compromise is acceptable on one criterion, this extension is not suitable.

Violating the value-range constraint. In every triangle the lowest value must be less than or equal to the most likely value, and the most likely less than or equal to the highest; none may be negative.

Changing the defuzzification method without stating it. The canonical choice is the centre value. A different defuzzification, such as taking only the most likely value, can change the three sub-utilities and hence the ranking.

Never checking the three models separately. Reporting only the combined score without ever looking at whether the three sub-utilities favour the same alternative; if they disagree, the combined score conceals this disagreement.

Entering the weights as triangles and expecting a triangular result. Criterion weights are reduced to a single number by the method immediately. The triangular width of the weights never carries through to the result; this can surprise the user and should be stated in the report.

The governing principle is this:

Fuzzy DNMA's combined score is a summary of how consistent the three aggregation models' score and rank are over the centre values. If an alternative looks good in only one model, the report must show this.