Fuzzy ROV

Three things change. ROV's idea of averaging an optimistic and a pessimistic score stays exactly as it is.

Cells. In crisp ROV every cell is a single number. Here every cell is made of three numbers: lowest, most likely, highest. Criterion weights, too, can come from outside as triangular numbers, or equally as a single crisp number. Fuzzy ROV does not generate its own weights.

Scale equalisation. Crisp ROV places every criterion between 0 and 1 relative to its own minimum and maximum. Fuzzy ROV carries out this same operation on all three components of the triangle at once. For cost criteria the components are reversed, so that the lowest component takes the place of the highest. This preserves the triangle's own ordering, namely that the lowest component stays below the most likely one.

Score and combination. The method first defuzzifies the normalised triangles and the weight triangles, that is, it takes the average of the three components. It then uses these crisp numbers to compute the best-case score from the benefit criteria and the worst-case score from the cost criteria. DecisionMind defuzzifies immediately after normalisation here; it does not carry the triangles any further.

Result. The average of the two scores is a single number, as in crisp ROV.

DecisionMind fixes, for this extension, the order of normalisation and the point at which defuzzification occurs. Weights come from outside and may be either a single number or a triangle.

The score is read the same way as in crisp ROV. It is meaningful only for this alternative set and these weights. See the ROV card.

The difference is here: the normalisation step works separately on the triangle's three corners, but defuzzification follows immediately after this step. The final score therefore carries the uncertainty in the input only through the normalisation stage. If two alternatives' scores sit close together, it should be checked separately how sensitive this closeness is to the width of the input triangles.

Thus instead of writing:

"The Fuzzy ROV analysis found alternative K to be best"

the report should read:

"Alternative K obtained the highest score with these weights and this alternative set. The gap to the next-ranked alternatives can close with a change in the criterion weights"

Fuzzy ROV is preferred when criteria are scored verbally or approximately, and compensation is accepted. Opening up a measured criterion into a triangle afterwards adds no information. The Fuzzy data-type card explains this distinction in detail.

DecisionMind requires a single data type. Crisp ROV's exit condition applies here too: if a criterion column is constant, normalisation stays undefined, and this criterion must either be dropped or measured differently. If no compromise is acceptable on one criterion, this extension is also unsuitable because it is compensatory.

Marking criterion direction wrongly. If a cost criterion is marked as a benefit, the triangle's components are reversed in the wrong direction and the ranking breaks down.

Defuzzifying before normalisation. Reducing the triangles to a single number before normalisation and then running crisp ROV is not Fuzzy ROV. Uncertainty is lost at the very first step.

Feeding in a constant criterion column without noticing. If a criterion carries the same triangle across all three alternatives, the normalisation denominator becomes zero and the calculation stays undefined.

The manifest's shared warning also applies: cells must follow the order l ≤ m ≤ u. The choice of defuzzification method affects the result; DecisionMind uses the average of the three components as a fixed rule here.

The governing principle is this:

Fuzzy ROV is normalisation carried out at the triangle's three corners, followed immediately by defuzzification. An application that does not show how much of the input uncertainty is reflected in the score conceals the method's true behaviour.