Fuzzy ELECTRE II (Govindan, Grigore & Kannan, 2010)

One thing changes in the input; everything else, unlike Fuzzy ELECTRE I, stays identical to crisp ELECTRE II.

Cells. Decision-makers score performance verbally or approximately; these scores are converted into triangular fuzzy numbers (lowest, most likely, highest). In crisp ELECTRE II every cell is a single number; here it is three. But these three numbers are not carried through the whole calculation.

Where defuzzification happens: here, at the very start. This is where the method departs most from the family's other fuzzy members, particularly Fuzzy ELECTRE I. Fuzzy ELECTRE I carries the fuzziness through to the final comparison. What DecisionMind fixes in this classical Fuzzy ELECTRE II is exactly the opposite: the triangular number is reduced to a single crisp number immediately after the input, in one step, by the centre of gravity ((l+m+u)/3); if the criterion weights are also fuzzy, they are defuzzified the same way. After this step there is a single crisp decision matrix. None of the remaining seven steps, that is vector normalisation, weighting, direction-sensitive concordance and discordance, Roy-Bertier's five-threshold strong/weak outranking, cycle removal, and two-way distillation, sees a fuzzy number again. All of them are identical to the steps described on the crisp ELECTRE II card.

Scale equalisation and concordance/discordance. Crisp ELECTRE II's vector normalisation and weighting are applied on the defuzzified matrix. Direction (benefit/cost) enters only once, when the concordance and discordance sets are built, exactly as on the crisp ELECTRE II card. The concordance and discordance indices are crisp numbers from start to finish; there is no fuzzy intermediate step.

Result and defuzzification. The result is not a single closeness score but a partial order built, as in crisp ELECTRE II, by two-way (descending and ascending) distillation from the strong/weak outranking graphs; incomparable pairs remain possible here too. Because defuzzification already finished in a single step at the outset, the result itself requires no further defuzzification.

What stays the same as the base method: the output is a position, not a score; strict preference, equality and incomparability are three separate relations.

What differs: this ranking is built after the verbal scores have been reduced early to a single number. The fuzziness is never carried into the ranking itself at any point. So how robust a position is in the final ranking should be read not from which term the verbal score was given in, but from ELECTRE II's own threshold sensitivity, that is, the concordance thresholds and the discordance ceiling. Two routes, a verbal score converted to a triangle and then defuzzified early before crisp ELECTRE II runs, versus a crisp score fed directly into crisp ELECTRE II, give bit-for-bit identical results once they reach the same input value. The only difference lies in where the input came from: a verbal judgement or a direct measurement.

Thus instead of writing:

"Fuzzy ELECTRE II is more reliable because it carries the uncertainty in the verbal scores through to the final ranking"

the report should read:

"The verbal scores were converted into triangular fuzzy numbers and reduced to a single number right at the start; the robustness of the final ranking should be checked against ELECTRE II's own concordance thresholds and discordance ceiling, not against this number"

Use it when the performance judgement is verbal or approximate, and converting this judgement into a triangle with a scale declared in advance, then feeding it into a single crisp ELECTRE II run, is sufficient. Opening a measured performance into a triangle manufactures uncertainty here too; a measured cell is embedded by writing all three components of the triangle equal (l=m=u). In this case early defuzzification changes nothing at all.

If the aim is to carry the uncertainty through to the final ranking, that is, to distinguish whether an alternative's order is sensitive to the verbal score itself rather than to early defuzzification, or to the ELECTRE II thresholds, this classical Fuzzy ELECTRE II form is not sufficient; Fuzzy ELECTRE I meets this need, with its core-set output. Crisp ELECTRE II's exit condition, that a full ranking and non-compensatory logic are wanted while graded pseudo-criterion thresholds are not needed, applies here exactly as it does there.

Treating early defuzzification as a shortcoming and trying to carry the fuzziness through to the end. This changes the method itself; what DecisionMind fixes here is precisely this design. The triangular number exists only to collect the input; the calculation is crisp ELECTRE II.

Applying the criterion direction (benefit/cost) both at the normalisation step and when building the concordance set. Direction enters only once, when the concordance and discordance sets are built; applying it twice, reversing once at normalisation and again at the concordance set, cancels the direction out or reverses the result.

Choosing the concordance thresholds (s1, s2) equal to or very close to one another. This mistake in crisp ELECTRE II applies here just the same; the distinction between "strong" and "weak" outranking loses its meaning with fuzzy input too.

Defuzzifying first and then skipping normalisation. If the input has already been collected as a triangle, skipping the defuzzification step and feeding a raw (l, m, u) component, say only the most likely value, directly into crisp ELECTRE II bypasses the method's own defuzzification rule, the centre of gravity, and can produce a different result.

The governing principle is this:

In Fuzzy ELECTRE II fuzziness is only a way of collecting the input; DecisionMind reduces it to a single number right at the start and runs everything else as crisp ELECTRE II. The robustness of the ranking should be read against ELECTRE II's own thresholds, not against the verbal score.