Fuzzy TODIM (Krohling & de Souza, 2012)

Three things change; the pairwise gain-loss logic and the role of the loss-aversion coefficient θ do not.

Cells. In crisp TODIM every cell is a single number. Here every cell is three numbers: lowest, most likely, highest. Criterion weights, too, are triangular in the same way. An expert may give a score directly as a triple ("at least 0.65, most likely 0.70, at most 0.75") or give it as a word from a pre-declared scale and have it converted into a triangle; DecisionMind does not support a group decision in this extension, and every cell belongs to a single assessment.

Scale equalisation. In crisp TODIM, scale equalisation takes the form of dividing by the column sum (a share of the total). Here the same logic is applied separately to the triangle's three components: each component is divided by the sum of the components in its own column. This is the fuzzy, component-wise extension of crisp TODIM's normalisation; the direction (benefit/cost) distinction is used before this step, as in crisp TODIM.

Defuzzification and distance. In crisp TODIM, the gain-loss comparison is made directly on the normalised numbers. Here, the normalised triangle is first reduced to a single number by its centroid (the average of the three components); the gain-loss comparison, and hence the loss term magnified by θ, is calculated on this defuzzified number. In other words, an intermediate step that reduces the fuzzy input to a single number is added to crisp TODIM's single-stage flow. The gain-loss logic itself, that is, θ magnifying losses and the reference criterion determining relative weight, continues exactly as before.

Result. The global value is again a single number normalised between 0 and 1; the fuzzy input does not stay fuzzy in the end, because defuzzification is carried out before the fourth step, before entering the gain-loss comparison.

DecisionMind fixes the centroid defuzzification (the average of the three components) for this extension; θ defaults to 1 and, as in crisp TODIM, can be changed by the user.

The global value is read the same way as in crisp TODIM: the lowest total dominance takes 0, the highest takes 1. This is not an absolute "good/bad" measure, and the choice of θ also affects the result.

The difference is this. This value is the defuzzified, embedded form of the fuzziness in the input. The width of the triangle, that is, the uncertainty, is not carried through the calculation leading to the ranking; only its average is carried. Because of this, when reading the gap between two alternatives, attention should be paid not only to θ but also to the width of the triangles in the input. A wide triangle's average may equal a narrow triangle's average, but the two do not carry the same reliability.

Thus instead of writing:

"The Fuzzy TODIM result with θ=1 puts A2 first"

the report should read:

"Expert scores were entered as triangular fuzzy numbers, defuzzified by centroid, and ranked with a loss-aversion coefficient of θ=1; A2 has the highest global value, and this ranking does not change between θ=0.5 and θ=5"

This extension is suitable when criterion values come from expert judgement or estimation, and reducing this assessment to a single number would create an artificial precision. It is equally suitable when the decision-maker is more sensitive to losses than to gains, that is, when TODIM's underlying assumption, fits the nature of the decision.

If criteria are measured, crisp TODIM should be kept; opening a measured time or price into a triangle is not modelling uncertainty but manufacturing it. If the table is mixed, meaning some criteria are measured and some judgemental, DecisionMind requires a single data type; the measured criterion is written as a triangle with all three components equal, and this embedding adds no information. TODIM's exit condition applies unchanged: if no compromise is acceptable on one criterion, screening should be applied first rather than this compensatory method. If the loss-aversion assumption does not fit the nature of the decision, a symmetrically compensatory method such as crisp or fuzzy TOPSIS is simpler.

Defuzzifying and then forgetting θ. Defuzzification is only for averaging after scale equalisation; the choice and reporting of θ, which enters the gain-loss comparison, remains as obligatory as in crisp TODIM. The defuzzification step does not remove the need for θ.

Not checking the triangle's l ≤ m ≤ u order. A value-space violation (as the manifest warns) silently corrupts the calculation; every cell's lowest, most likely, highest order must be verified on entry.

Using a defuzzification method other than the centroid and comparing the result. DecisionMind fixes the centroid (the average of the three components); a different defuzzification (such as taking only the most-likely value) can produce a different normalised matrix and a different ranking, which is why two pieces of software can diverge on this point.

Opening a measured value into a triangle "to look more holistic." A measured price or time stays crisp; if there is no genuine source for the width, no triangle should be built.

The governing principle is this:

Fuzzy TODIM carries the judgemental approximation in the input as far as scale equalisation; defuzzification takes place before entering the gain-loss comparison, and the choice of θ on these defuzzified numbers is exactly as decisive as in crisp TODIM.