m-Polar Fuzzy ELECTRE III (Akram and Adeel, 2023)

Five things change; the idea of graduated comparison with indifference, preference and veto thresholds does not.

Cells. In crisp ELECTRE III every cell is a single number. Here every cell holds m numbers, one from each independent viewpoint: three if three experts assess it, four if four regions measure it. Each expert's own matrix is aggregated separately and then merged into a single m-polar matrix through a weighted-average operation (mFWAO).

Weights do not come from outside; they are derived from the data. In crisp ELECTRE III, weights come from the user; the method does not generate them. Here the opposite holds: DecisionMind reduces every cell of the merged matrix to a single score, using the average of the m poles, ratios these scores to the column total, and measures each criterion's discriminating power (divergence) with Shannon's (1948) entropy formula. A criterion with high discriminating power, one that genuinely separates the alternatives, receives a higher weight. The user does not supply a weight from outside; doing so would skip this step and disable the method's most important contribution.

Thresholds still come from outside. As in crisp ELECTRE III, the indifference (q), preference (p) and veto (v) thresholds for every criterion are taken from the user; unlike the entropy weights, these three are not derived from the data. Full concordance holds when the score difference is below q, none once it exceeds p, with a graduated zone in between. If the difference exceeds v, that criterion alone applies a veto.

Credibility and outcome mechanics. Crisp ELECTRE III pulls concordance down with discordance, then runs a two-way (ascending/descending) distillation and declares non-intersecting pairs incomparable. Here too a credibility index (β) is built: every criterion whose discordance exceeds its concordance pulls the concordance down. But the outcome mechanics differ. DecisionMind uses Li and Wang's (2007) net credibility method: each alternative's power to outrank the others (γ⁺) is added to its power to be outranked by them (γ⁻), the difference (γ) is taken, and the alternatives are ranked from highest to lowest on this single number.

DecisionMind fixes Shannon entropy weighting and Li-Wang net-credibility ranking for this extension. Combining the two differs from crisp ELECTRE III's two-way distillation-plus-intersection mechanics: because γ is a single real number for every alternative, the result is always a complete ranking. The incomparable pairs found in crisp ELECTRE III are not produced here.

What stays the same as the base method: an alternative's superiority comes not from being best on every criterion but from the graduated concordance-discordance calculation, and a single criterion that exceeds the veto threshold can block it on its own.

The difference lies here. Crisp ELECTRE III can leave some pairs incomparable, saying "the data could not separate these." This extension gives no such result; net credibility (γ) is always a real number, and the alternatives are ranked from top to bottom on this number. The weights, moreover, reflect the data's own discriminating power rather than the user's preference; where alternatives are very similar to one another on a criterion (low entropy separation), that criterion automatically receives a low weight, and the user cannot change this.

Thus instead of writing:

"m-Polar ELECTRE III chose x1 first because it is balanced across all the criteria"

the report should read:

"With weights derived from the data's own discriminating power (the heaviest criterion carrying 54 per cent), x1's net credibility comes out highest; this superiority should be read bearing in mind that the weights come from the data and that one criterion dominates"

Use this extension where a criterion's value genuinely comes from several independent sources (an expert panel, a region, a period) that should not be reduced to an early average. It is also directly suitable where the weights are wanted from the data's own discriminating power rather than the user's preference, that is, where an objective answer is sought to "which criterion genuinely separates the alternatives."

Crisp ELECTRE III's threshold condition still applies here: an expert opinion must exist to set the indifference, preference and veto thresholds. Where a criterion has a single source, forcing the m-polar structure is unnecessary; the family's crisp member should be used instead. Where seeing incomparable pairs separately matters, that is, where the information "the data could not separate these two" must not be lost, this extension is not sufficient; net credibility always produces a ranking.

Confusing m-polar with bipolar. m-Polar membership lies in [0,1]^m, m independent viewpoints; in the bipolar structure the second number is the opposite-direction effect, in [-1,0]. A bipolar dataset is not an m-polar dataset with m=2; the two are different value spaces.

Breaking the threshold order. Every criterion must satisfy 0 ≤ q < p ≤ v. Setting q=p empties the weak-preference band, and the graduated transition starts behaving like a switch (0/1), losing the method's main contribution.

Trying to supply a weight from outside. This extension generates its own weight through Shannon entropy; imposing the user's own preference weight disables the method's data-based weighting advantage, its principal difference from m-polar ELECTRE II.

Confusing a scalar weight with a pole-based weight. In this extension the entropy weight is scalar; the same number applies to every pole of a criterion. In m-polar ELECTRE II, by contrast, the weight is a separate vector per pole. Using one method's weighting step in place of the other's produces an incorrect weighted matrix.

Passing over flat (weakly discriminating) data unnoticed. If every alternative's score on a criterion sits very close to the others, entropy approaches 1, discriminating power approaches 0, and that criterion receives almost no weight. Whether the criterion itself is weak or the data merely looks flat should then be questioned.

The governing principle is this:

In m-polar ELECTRE III the weight comes from the data, the threshold from the user; confusing the two, or imposing the weight from outside, defeats the method's claim to objectivity. The result is always a complete ranking and produces no incomparable pair.