Pythagorean fuzzy ELECTRE II (Akram, Ilyas and Garg, 2021)

Five things change; the idea of building the order from two directions does not.

Cells. In crisp ELECTRE II every cell is a single number. Here every cell is a support (μ) and a rejection (ν) degree; μ² + ν² ≤ 1 must hold. Criterion weights are also given as Pythagorean fuzzy numbers and reduced to a single number. This extension also supports group decisions: several decision-makers can fill in the same table with their own PFN scores, and even the decision-makers' own importance can be reported as a PFN.

Combining the decision-makers. Crisp ELECTRE II starts from a single decision table. Here, every decision-maker's importance is first converted from the PFN given to them into a weight. All the decision-makers' score matrices are then combined into a single matrix, with these weights, through the Pythagorean fuzzy weighted average operation (PFWA). In an application with a single decision-maker, this step leaves the one matrix unchanged; its contribution only shows up when there is more than one decision-maker.

Scale equalisation and weighting. On a cost criterion the support and rejection degrees swap places; crisp ELECTRE II's division by column magnitude does not exist here. Criterion weights are also reduced to a single PFN through the same PFWA logic, and this weight is then applied to every cell through an exponentiation operation.

Concordance and discordance. In crisp ELECTRE II, a single concordance set is built where one alternative is "at least as good as" another. Here there are three concordance sets: strong (where an alternative's squared support is larger and its squared rejection smaller), medium (where its squared support is larger but only a score advantage holds), and weak (where only a score advantage holds). Each set's contribution to the criterion weights is measured by its own set-weight; these set-weights (values such as 1, 2/3, 1/3, 1/4 in the paper) are supplied from outside. Discordance is the ratio of the largest Pythagorean fuzzy distance between two alternatives to the largest such distance across all criteria.

Two-directional ranking. Crisp ELECTRE II builds two separate graphs from certain and possible outranking, derives a descending and an ascending order, and declares disagreeing pairs incomparable. This extension also builds two outranking graphs, strong and weak, and derives one ranking in the forward direction and one in the reverse direction. But DecisionMind does not take an intersection here; it averages the two directions' rank numbers and ranks alternatives by this average. Unlike the base ELECTRE II, the result is always a complete order; no pair is declared incomparable, and pairs that come out with an equal average are separated by a fixed rule.

DecisionMind fixes, in this extension, the forward+reverse average ranking, and the combination of the four concordance / three discordance sets through set-weights. The concordance and discordance thresholds (ψ⁻, ψ⁰, ψ*, δ⁰, δ*) and the set-weights are exposed to the user; the default values are taken from the founding paper's supplier example.

What is the same as the base method: the ranking comes from an outranking relation; it is not a direct percentage of the alternatives' performance on the criteria.

The difference lies here. Crisp ELECTRE II's option of incomparability does not exist here; even where the forward and reverse directions disagree, an average is taken and a rank number results. This does not mean the base method's honesty, "say so if the data cannot separate two alternatives", is lost, but it is a different form of honesty: a rank always exists, and its robustness must be tested separately by looking at the thresholds and the set-weights.

Thus instead of writing:

"Pythagorean fuzzy ELECTRE II placed A2 in first place with certainty"

the report should read:

"With the chosen thresholds and set-weights, A2 comes out on top once its forward and reverse rank numbers are averaged; this rank number carries no option of incomparability, as crisp ELECTRE II does, and its robustness must be separately tested by changing the thresholds"

This extension is used where criterion scores rest on a judgement, and that judgement is given both strong support and marked reservation, that is, where their sum exceeds what the intuitionistic fuzzy structure allows. It provides an additional contribution where the decision comes from more than one expert, and where the experts' own opinion weights can also be expressed as a judgement (a PFN); in a single-expert application this contribution does not come into play.

The base ELECTRE II's exit condition holds here too: this family is suited to cases wanting a full order and a non-compensatory (veto-like) logic, where graded pseudo-criterion thresholds are not required. Where it matters that incomparable pairs be shown separately, that is, where the information "the data cannot separate these two" must not be lost, this classical extension is not sufficient; it always resolves the result into a complete order.

Domain violation. In every cell, μ ∈ [0,1], ν ∈ [0,1] and μ² + ν² ≤ 1 must hold; neither each decision-maker's PFN nor the post-aggregation matrix should enter the calculation without this check.

Breaking the threshold ladder. ψ⁻ < ψ⁰ < ψ* and δ⁰ > δ* must hold strictly. If the thresholds are given equal to one another or in reverse order, the three-tier concordance distinction (strong/medium/weak) loses its meaning.

Leaving the set-weights at 1 throughout. If the strong, medium and weak concordance sets' set-weights are all given as 1, the structure reverts to Devadoss and Rekha's (2017) simpler intuitionistic fuzzy ELECTRE II form, and the graded distinction never comes into play. In the illustrative example below, these weights are left at 1 for simplicity (see the verification notes); in a real application, graded set-weights as in the founding paper (values such as 1, 2/3, 1/3, 1/4 and 1, 3/4, 2/4) are recommended.

Expecting incomparability and not finding it. Unlike the base ELECTRE II, this extension never declares a pair incomparable; even where the forward and reverse directions disagree, it produces an averaged rank number. Mistaking this for a fault and saying "the method did not work" is wrong; the robustness of the ranking must be separately tested by changing the thresholds and the set-weights.

The governing principle is this:

Pythagorean fuzzy ELECTRE II carries support and rejection without clipping them, and combines the two directions through an average rather than an intersection. Leaving the set-weights at 1 without questioning them, or treating its always producing a complete order as a shortcoming, misses the point of how the method was built.