Hesitant Fuzzy ELECTRE I (Chen, Xu and Xia, 2015)

Four things change; the logic of building the concordance set and extracting the core set does not.

Cells. In crisp ELECTRE I every cell is a single number. Here every cell becomes a set (a hesitant fuzzy element, HFE) in which several plausible membership degrees for the same judgement are held together. Three values such as "0.4, 0.6, 0.7" all count as valid; none is collapsed into a single average. Two alternatives' HFEs on the same criterion can differ in length; when they are compared, the shorter set is padded with additional elements. This method uses the pessimistic rule here: the missing elements are filled by repeating the set's own smallest value. Weights remain crisp numbers; the method does not generate weights, it takes them from outside.

Score and consistency instead of scale equalisation. In crisp ELECTRE I, columns are equalised by dividing through by their magnitude. There is no normalisation of this kind here; instead, two numbers are extracted from every HFE: a score (the mean of the values in the set) and a deviation (the set's own standard deviation, a measure of how "consistent" the judgement is). Whether one alternative counts as superior to another on a given criterion is decided by looking at both a higher score and a lower deviation, that is, a more consistent judgement.

Concordance and discordance. In crisp ELECTRE I, the concordance set asks only "which alternative scores better." Here four separate sets are built: the criteria where one alternative is superior to another on both score and consistency, the criteria where it is superior on score alone, the criteria where it trails on score but is more consistent, and the criteria where it genuinely trails. The concordance index is the weighted sum of these sets' weights, weighted in turn by the relative importance the user assigns to "strong" and "weak" concordance. The discordance index is the ratio of the worst inconsistency among the trailing criteria to the largest inconsistency across all criteria; this uses a distance measure between HFEs, the hesitant distance defined by Chen, Xu and Xia.

The form of the thresholds and defuzzification. In crisp ELECTRE I, the concordance threshold (c̄) and the discordance threshold (d̄) are chosen by the analyst. Here, as in the family's classical Fuzzy ELECTRE I member, C̄ and D̄ are not chosen by the analyst either; they derive from the mean of the concordance and discordance values across all pairwise comparisons. What DecisionMind fixes in this extension is that score and consistency (deviation) are carried together at every step and never collapsed early into a single "mean membership" number; the core set is built on this paired structure.

What stays the same as in the base method is this: the output is not a ranking but a list of alternatives that no other alternative outranks (the "core set").

The difference is this: whether an alternative stays in the core set depends not only on its average performance but also on how consistent that performance is. Even when two alternatives share the same average score, an outranking can emerge in favour of whichever judgement is less scattered, that is, more consistent. The report should therefore state not only the scores but also which alternative was judged more consistently on which criterion.

Thus instead of writing:

"HF-ELECTRE I found A1 to be the best"

the report should read:

"On two of the three criteria, A1 outranks the other candidates through both a higher score and a more consistent judgement; this outranking rests not only on the average score but also on the consistency of the judgement"

Use this extension when a performance judgement is made up of several plausible values that need to be preserved rather than collapsed into an early average. This can be one expert's own hesitation, or it can be several assessors' opinions carried separately; the method naturally supports a group decision. If a performance has actually been measured, opening it out into a hesitant set does not model uncertainty, it manufactures it; DecisionMind asks for a single data type, and a measured cell is written as the set's sole element.

Crisp ELECTRE I's exit condition applies here too: this method is suitable when a core set is enough, rather than a full ranking. If a complete ranking is required, the family's other member for this data type, Hesitant ELECTRE II, should be considered instead.

Confusing the pessimistic and optimistic length-equalisation rules. This method extends a short HFE by repeating its own smallest value (the pessimistic rule). The family's Hesitant ELECTRE II member uses the rule that repeats the largest value (the optimistic rule) instead. Confusing the two can make the alternative with the shorter set look better or worse than it is.

Assuming the thresholds (C̄, D̄) are fixed numbers chosen by the analyst. They are the data's own mean, a feature this method shares with the family's classical Fuzzy ELECTRE I member. Writing a sentence such as "the threshold was fixed at this value" in the report is wrong.

Ignoring the deviation (consistency) and looking only at the score. The concordance and discordance sets are built by looking at both score and deviation. Comparing scores alone loses the information that, even when two alternatives share the same average, one may have been judged far less consistently than the other.

The governing principle is this:

In Hesitant ELECTRE I, an alternative's outranking rests on both its average performance and the consistency of that performance; the thresholds here are not chosen, they derive from the data itself.