Fuzzy ELECTRE III (Montazer, Qahri Saremi & Ramezani, 2009)

The input is fuzzy; the workings become crisp right at the start, with the same logic as in Fuzzy ELECTRE II.

Cells. Performance scores, in Montazer and colleagues' own example the output of a fuzzy expert decision-support system, take the form of triangular fuzzy numbers. The thresholds, indifference (q), preference (p) and veto (v), may also be given as triangles.

Where defuzzification happens: here too, at the start. The order DecisionMind fixes in this classical Fuzzy ELECTRE III is the same as in Fuzzy ELECTRE II. Everything, the performance scores and all three thresholds included, is reduced in one step to a crisp number, by the centre of gravity ((l+m+u)/3), at the second step of the calculation. After this step there is a crisp decision matrix and a crisp table of three thresholds. None of the remaining ten steps, that is threshold-order verification, criterion-by-criterion graded concordance and discordance, the credibility degree, two-way distillation, and intersection, sees a fuzzy number again.

Concordance, discordance and credibility. On the now-crisp performance and thresholds, the graded transition described on the ELECTRE III card operates unchanged: if the difference is below the indifference threshold, concordance is complete; if it is above the preference threshold, there is no concordance at all; in between, concordance is graded. Discordance is likewise graded, between the preference and veto thresholds. The credibility degree is built by pulling the overall concordance down where criteria show high discordance; all of this is done on crisp numbers.

Result and defuzzification. In this manifest DecisionMind fixes Roy's (1978) canonical two-way distillation procedure, the intersection of the descending and ascending distillations, rather than Montazer and colleagues' own net-strength (Yager) formula; the result is a partial order of strict preference, equality and incomparable relations, as in crisp ELECTRE III. Because defuzzification already finished at the second step, the ranking itself requires no further defuzzification.

What stays the same as the base method: an alternative coming "first" does not mean it is best on every criterion; incomparable pairs are not a fault, they are a signal that the data cannot separate the two.

What differs: this ranking is built after the fuzzy scores produced by the expert system have been made crisp right at the start. How robust the ranking is, that is, which alternatives' relative position is sensitive, should be read not from which fuzzy interval a score came from, but from the choice of the three thresholds, especially the veto threshold. When the veto threshold is narrowed, that is, when the line marking "a difference this large can no longer be accepted" is pulled down, even a small, now-crisp difference can become decisive on a single criterion by itself.

Thus instead of writing:

"Fuzzy ELECTRE III found V3 to be the best alternative, and this result demonstrates the robustness of the fuzzy input"

the report should read:

"Once the expert system's fuzzy scores were made crisp, and with the chosen indifference/preference/veto thresholds, V3 leads in both the descending and the ascending distillation; how robust this ranking is on any given criterion should be tested separately against that criterion's veto threshold"

Use it when the performance judgement comes from an expert system or a verbal assessment, and converting it into a triangle with a rule declared in advance, then feeding it into a single crisp ELECTRE III run with graded thresholds, 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).

Crisp ELECTRE III's exit condition applies here too: expert judgement must exist to set the thresholds (indifference, preference, veto); these thresholds come from outside and are not estimated from the data. If the aim is to carry the uncertainty through to the end, avoiding early defuzzification, this classical form is not sufficient.

Choosing the three thresholds (q, p, v) too close to one another, or "standard" without looking at the data. This mistake in crisp ELECTRE III applies here just the same; the benefit of the graded transition only appears if the thresholds are discriminating.

Setting the veto threshold too tight and then being surprised that "the method could not separate any alternative". Even a small, now-crisp difference can become decisive on its own if the veto threshold is squeezed too far, and almost every pair can come out incomparable.

Using only one of the descending and ascending distillations. The final result is the intersection of both; a one-directional distillation gives an incomplete picture. This is a mistake inherited from crisp ELECTRE III.

Treating early defuzzification as a "loss". In this method defuzzification is not a shortcoming but the design itself; performance and thresholds are both made crisp at the same second step. Making only performance crisp while leaving the thresholds fuzzy, or the reverse, leads to a comparison across two different scales.

The governing principle is this:

In Fuzzy ELECTRE III fuzziness is only the expert system's way of producing scores; DecisionMind makes it crisp at the second step and runs everything else with crisp ELECTRE III's graded threshold logic. Which criterion the ranking is fragile on should be tested against that criterion's veto threshold.