m-Polar Fuzzy ELECTRE I (Akram, Waseem and Liu, 2019)

Three things change; the family's concordance/discordance logic does not.

Cells. In the family's crisp members, every cell is a single number. Here every cell consists of m numbers coming from m separate viewpoints. A criterion assessed from three viewpoints is three numbers, such as (0.7; 0.85; 0.75); each number comes from its own source and is not derived from the others. Weights remain crisp numbers; the ELECTRE family does not generate weights, it takes them from outside.

Weighting and pole summation. The family's crisp members first divide columns by their magnitude and then multiply by the weight. Here there is no normalisation step; the criterion weight is applied directly to each cell's m poles. To build the concordance and discordance sets, a cell's weighted m poles are summed to obtain a single comparison value. This sum exists only to answer "who outranks whom"; it is not the cell itself, which remains m separate numbers.

Concordance, discordance and the core set. The concordance set (the criteria on which p outranks q) and the discordance set (the criteria on which p falls behind q) are built from these comparison values. The concordance index is the sum of the weights of these criteria, exactly as in the family's crisp member. The discordance index, however, is here a normalised Euclidean distance taken over all m poles together; rather than the single largest gap on one criterion, the distance formed jointly by all m poles is measured. DecisionMind, following the family's default practice, sets the concordance and discordance thresholds from the data set's own averages (f̄, ḡ); pairs that are concordant above these averages and discordant below them are linked by an outranking relation, and those outranked by no other alternative make up the core set.

DecisionMind fixes, for this extension, the pole-summation comparison and the Euclidean-based discordance ratio. Weights are taken from outside as crisp numbers; the method does not generate weights.

The output, as with the family's ELECTRE I member, is not a ranking but an outranking relation: who outranks whom. An alternative outranked by no other belongs to the core set. If two alternatives outrank each other mutually, there is no definite order between them, and both remain in the set.

The difference is here. Before the concordance and discordance test, the m-polar input has already been reduced to a single comparison value by pole summation, but this sum exists only for comparison, it is not a reported score. DecisionMind's own net outranking count (how many alternatives it outranks minus how many outrank it) makes the reading easier, but this number is not the family's own output; it is a summary derived from the outranking graph.

Thus instead of writing:

"MPF-ELECTRE-I found A2 to be ranked first"

the report should read:

"A2 outranks all four of the other alternatives under these thresholds; A1 and A3 outrank each other mutually, so there is no definite order between them"

grounding the statement in the outranking graph rather than in a rank.

Use this extension when a criterion's value genuinely comes from more than one independent source, and these sources should not be reduced to an early average. It is also suitable when a very poor viewpoint on one criterion should not be offset by the criterion's other viewpoints, because the ELECTRE family's non-compensatory logic continues to operate at the criterion level, not at the viewpoint level.

The case for returning to the family's crisp member is when a criterion is measured from a single source with a single number. If there is only one viewpoint, forcing an m-polar structure adds an artificial layer.

The exit condition is the same as for the family's ELECTRE I member: if the number of alternatives is below three, or a complete ranking is required, this member is not appropriate.

Mistaking the pole sum for the cell itself. The sum built for the concordance and discordance test is only an intermediate comparison value. Reporting this sum as if it were the cell's real value hides the m independent viewpoints.

Confusing m-polar with bipolar fuzzy. In an m-polar cell, every number is an independent viewpoint between 0 and 1. In a bipolar cell, the second number is the opposite-direction effect and lies between −1 and 0. The two are different data structures.

Mistaking an empty or very wide outranking graph for a method error. A case where no alternative outranks any other, or all alternatives remain in the core set, is a sign that the weights are very balanced or the pole values are tightly clustered. The weighted matrix and pole sums should be examined first; the method should not be said to have "failed."

The governing principle is this:

m-Polar ELECTRE I combines independent viewpoints only through a comparison-purpose sum; any reading that mistakes this sum for the cell itself, confuses the m-polar structure with the bipolar one, or mistakes a wide core set for an error misinterprets the outranking relation.