Fuzzy ELECTRE I (Hatami-Marbini & Tavana, 2011)

Four things change; the concordance-discordance logic and the veto principle do not.

Cells. In crisp ELECTRE I every cell is a single number. Here every cell is a trapezoidal fuzzy number made of four numbers (lowest, two middle values, highest); decision-makers give performance and criterion weights in verbal terms such as "very good" or "medium", and these terms are converted into a trapezoid using a scale declared in advance. The method is designed directly for group decisions: the trapezoids of K decision-makers are merged into a single group matrix by taking the minimum of the most pessimistic lower bound, the average of the two middle values, and the maximum of the most optimistic upper bound (the min-average-average-max rule). Weights are likewise trapezoidal and need not sum to 1. Here a weight is a "voting weight", not a probability distribution as in TOPSIS.

Scale equalisation. In crisp ELECTRE every column is divided by its own magnitude. Here, too, the division follows the criterion's direction: for a benefit criterion the column is divided by its highest upper value, for a cost criterion by its lowest lower value; but the operation runs on the four-component trapezoid, with the order of the components reversed. The result remains a comparable, four-component trapezoid.

Concordance and discordance. This is where the method departs most from crisp ELECTRE I: the question of which of two trapezoids is larger cannot be answered by direct comparison as if they were single numbers, so the method instead applies an outranking rule based on the vertex (Hamming) distance between the two trapezoids. Whether one alternative counts as "at least as good as" another on a given criterion is decided by comparing the distances of the two trapezoids to their union's upper bound. The concordance set is built with this rule, and the concordance index is summed to the extent of the criterion's own weight, that is, still as a trapezoidal number; it stays undefuzzified. Discordance, by contrast, is calculated as a ratio of the two trapezoids' vertex distances and collapses at this point into a single number between 0 and 1.

Shape of the thresholds and defuzzification. In crisp ELECTRE I the concordance threshold (c̄) and the discordance threshold (d̄) are chosen directly by the analyst (for example c̄ = 0.75, d̄ = 0.25 in the base card's own example). Here matters differ: C̄ and D̄ are not chosen by the user but derived from the data itself, being the average of the concordance and discordance values across all pairwise comparisons. Defuzzification also happens at exactly this point, at the moment of comparison against the threshold: the still-trapezoidal concordance index is reduced to 0/1 by comparing its vertex distance against C̄'s. What DecisionMind fixes for this classical Fuzzy ELECTRE I is precisely this: the fuzziness is carried through to the final comparison, with no early defuzzification at any intermediate step; the output is a Z matrix (outranking graph), as in crisp ELECTRE I, and the core set derived from it.

What stays the same as the base method, briefly, is this: the output is not a ranking but an inventory of those outranked by no other alternative (the "core set") and of the pairwise outranking relations (outranking, equality, incomparability).

What differs is this: the thresholds (C̄, D̄) here are not fixed numbers chosen by the analyst but the data's own average. This recasts crisp ELECTRE I's question of what happens if the thresholds are relaxed into a different shape: the thresholds are not in the analyst's hands, but when one alternative's score changes, not only the pairs involving that alternative move, C̄ and D̄ themselves shift too, and with them the pass line for every pair. In the report this should be expressed not as "the threshold was this, had it been drawn there", but as "had this alternative's score changed by this much, the average threshold would also have shifted, and the decision could have changed with it."

Thus instead of writing:

"Fuzzy ELECTRE I placed A2 and A3 clearly first"

the report should read:

"With thresholds derived from the data's own average, A2 and A3 emerge as a pair, neither outranking the other, counted as equal; this equality is consistent with their raw scores also being close to one another"

Use it when performance and weight judgements come verbally from more than one decision-maker, and these judgements need to be combined by pooling rather than reduced to a single number. If a measured performance figure exists, for example adherence to a delivery date as a percentage, opening it into a trapezoid models nothing, it manufactures uncertainty; DecisionMind requires a single data type, so a measured cell is embedded by writing all four components of the trapezoid equal (a = b = c = d).

The exit condition of crisp ELECTRE I also applies here: if a core set is enough and a full ranking is not required, this method is suitable; if a complete order is needed, other members of the family should be considered (Fuzzy ELECTRE II, III). But unlike what is described here, those two defuzzify performance immediately and run the crisp ELECTRE II/III logic; only this Fuzzy ELECTRE I form carries the fuzziness through to the end.

Defuzzifying the trapezoids before the Hamming distance is calculated. Which criterion the concordance set falls into is determined over the two trapezoids' full membership function; comparing "which is larger" on single numbers defuzzified by the centre of gravity erases the asymmetry and can produce a different concordance set.

Using ELECTRE I as a ranking method rather than a selection method. Incomparability or equality is not a computational fault; it is a signal that the data cannot reliably separate the two.

Treating fuzzy weights as if they were a probability simplex summing to 1. In this method the concordance index is the fuzzy sum of the weights; the weights in the seminal paper's own example do not sum to 1 either. Forcibly normalising the weights changes the concordance index's scale.

Assuming the thresholds (C̄, D̄) are fixed numbers chosen by the analyst. In this method they are the data's own average; a sentence in the report such as "the threshold was chosen as 0.75", which belongs to crisp ELECTRE I, is a mistake here.

The governing principle is this:

Fuzzy ELECTRE I exists to carry more than one decision-maker's verbal judgement, as fuzzy quantities, through to the final comparison; the thresholds are not chosen here, they are derived from the data itself. This is why a change in one alternative's score moves not only that alternative but the pass line itself.