Extension card · m-Polar
m-Polar Fuzzy ELECTRE IV (Akram and Adeel, 2023)
This is the member of the ELECTRE family that both assigns no weight to criteria at all and rates every cell from several independent viewpoints. Its output is not a complete ranking but a partial pre-order emerging from a two-way distillation.
Base method
ELECTRE →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
m-Polar →
What this data type is, when to use it, how to write it in a cell: the family's full account is here.
What Changes from the Base Method?
Four things change; the family's non-compensatory logic does not.
Cells. In the family's crisp members every cell is a single number. Here every cell holds m numbers, one from each independent viewpoint; where several experts are involved, each expert's own m-polar matrix is first aggregated separately and then merged through a single weighted-average operation. This merged cell is reduced to a single score degree, the simple average of the m poles. This average is not a weighting; it is only a summarising step that reduces m independent viewpoints to a single comparison number.
Weight disappears entirely. The family's m-polar ELECTRE I and II members take criterion weight from outside. Here there is no criterion weight at all; every criterion is defined solely by its own three thresholds (indifference threshold q, preference threshold p, veto threshold v). These thresholds are entered on the score-degree scale and come from outside, not derived from the data.
Five nested classes instead of concordance/discordance. For every pairwise comparison, it is counted which threshold band the score difference on every criterion falls into. From these counts, as in crisp ELECTRE IV, five nested outranking classes (quasi, canonical, pseudo, sub, veto) are built, and an ordered credibility level (1; 0.8; 0.6; 0.4; 0.2) is assigned according to the strongest class reached.
Outcome and defuzzification. Two-way (Belton-Stewart) distillation is run on these credibility levels: once downward from the strongest alternative, once upward from the weakest. DecisionMind fixes the discrimination threshold for this classical m-polar ELECTRE IV: on Akram and Adeel's decision, this threshold is a fixed number, 0.1, and is not computed with the family's more general Vallée-Zielniewicz formula (graduated, data-sensitive). The intersection of the two directions gives a partial pre-order.
How to Read the Output
What stays the same as the base method: the output is a position, not a score; strict preference, indifference and incomparability are three separate relations.
The difference lies here: what determines this position is not a weighted concordance percentage but which threshold band the score, drawn from the average of m independent viewpoints, falls into. A credibility level of 1.0 (the quasi class) is the strongest relation; 0.2 (the veto class) is the weakest, still counted as a relation.
Thus instead of writing:
"m-Polar ELECTRE IV found A2 to be the clear winner"
the report should read:
"A2 outranks all four other candidates with the strongest class (quasi dominance); this classification rests not on criterion weight but on the average of m independent viewpoints and the chosen thresholds"
When to Prefer This over the Base Method
Use this when a criterion's value genuinely comes from several independent sources, and the criteria's relative importance cannot at the same time be reliably established. Where no source for a weight exists, and treating the criteria as equally weighted also looks like an artificial assumption, this member, which never uses weight, is the right choice.
Where the viewpoint comes from a single source, forcing the m-polar structure adds an artificial layer; the family's crisp ELECTRE IV member is then sufficient. Where the criteria are known to genuinely carry different importance, the family's m-polar ELECTRE I or II member, which uses weight, should be preferred.
Mistakes Specific to This Extension
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, between −1 and 0.
Interpreting the absence of weight without noticing it. Unlike the family's m-polar ELECTRE I and II members, this method uses no criterion weight whatsoever. Using this method on a problem where the criteria genuinely carry different importance equates an important criterion's voice with an unimportant one's.
Taking the discrimination threshold as the wrong fixed value. This method uses a fixed value of 0.1; carrying over the family's more general graduated formula (Vallée-Zielniewicz's default α=−0.15, β=0.30) here coarsens the distillation's cut-off points and can produce an earlier, cruder pre-order.
The governing principle is this:
m-Polar ELECTRE IV abandons the assumption that both criterion weight and every cell come from a single source. The score is the average of m independent viewpoints; the classification rests on this average and the chosen thresholds, not on weight.
Cases
The first case is drawn from the literature: the Qazvin Innovation Park contractor-selection example from Akram and Adeel's (2023) book. The second case is an illustrative construction.
1. Construction: Full ranking of five contractors for an innovation park project (Akram and Adeel, 2023)
For a university's innovation park project in Iran, five candidate contractors (A1-A5) have been assessed by three experts (with roughly equal weights) on five criteria: financial soundness, technical competence, management capacity, occupational health and safety, and reputation; all four sub-viewpoints (m=4) are rated so that "higher is better." Once the three experts' matrices have been merged, the score degrees drawn from every cell (the average of the four viewpoints) are as follows:
| Contractor | Financial soundness | Technical competence | Management capacity | OHS | Reputation |
|---|---|---|---|---|---|
| A1 | 0.516 | 0.588 | 0.571 | 0.459 | 0.620 |
| A2 | 0.792 | 0.826 | 0.783 | 0.654 | 0.786 |
| A3 | 0.640 | 0.623 | 0.637 | 0.485 | 0.721 |
| A4 | 0.774 | 0.726 | 0.726 | 0.550 | 0.806 |
| A5 | 0.593 | 0.726 | 0.710 | 0.591 | 0.682 |
No weight is used; indifference, preference and veto thresholds are set separately for every criterion (indifference thresholds 0.05-0.08, preference thresholds 0.10-0.15, veto thresholds 0.15-0.20). The method tries all five outranking classes with these thresholds for every pairwise comparison, chooses the strongest, and runs the two-way distillation.
| Rank | Contractor |
|---|---|
| 1 | A2 |
| 2 | A4 |
| 3 | A5 |
| 4 | A3 |
| 5 | A1 |
The result reads as follows. A2 holds the highest score on four of the five criteria and outranks the other four contractors with the strongest class (quasi dominance, credibility level 1.0). Forward and backward distillation agree entirely in this case; no pair of contractors remains incomparable.
The board's hesitation is this: the second-versus-third place between A4 and A5 rests on nearly identical scores on the technical-competence criterion (0.7262 against 0.7260); the real separation comes from the financial-soundness and reputation criteria. In an independent check carried out by this card's author, when the expert weights (0.33/0.33/0.34) were shifted noticeably in favour of one expert, and the discrimination threshold was computed with the family's more general (graduated) formula instead, the ranking DID NOT CHANGE; this five-contractor ranking is robust to both of these parameters.
In the report: "With the thresholds set, the ranking is A2, A4, A5, A3, A1, and every consecutive relation is established with the strongest outranking class. The order between A4 and A5, despite their near-equal scores on technical competence, is settled by the gap coming from the financial-soundness and reputation criteria."
Source: Akram, M., & Adeel, A. (2023). MCDM Methods with Multi-polar Fuzzy Information, Chapter 6, §6.3 (pp. 324–332, the Qazvin Innovation Park case study, Table 6.14). The score degrees, outranking classes and sensitivity scenarios (expert-weight shift, discrimination-threshold change) were computed by this card's author running the DecisionMind engine independently; the baseline result matches the book's own ranking exactly (A2, A4, A5, A3, A1).
2. Sports facility: Preliminary comparison of three candidate operators for a municipal sports complex
A municipality will hand the operation of a new sports complex to one of three candidate firms. Three criteria apply: facility-maintenance capacity, event-organisation experience, and financial guarantee; all three are "higher is better." Each firm has been rated independently by three separate evaluation boards (a technical board, a financial board, a sports-federation representative); these three viewpoints are held together without being reduced to an average. Because the municipal council could not agree on the criteria's relative importance, no weight has been assigned, only thresholds for every criterion.
The method compares the three firms pairwise, chooses the strongest of the five outranking classes, and builds a partial order through the two-way distillation. Suppose the result showed the firm with the highest financial guarantee outranking the other two, while the remaining two firms failed to outrank each other and remained incomparable.
The council's hesitation is this. One of the two incomparable firms scored higher than two of the three boards on event-organisation experience and lower than one; the average of these three viewpoints comes out very close to the other firm's. If one board's score is reviewed, this incomparability could turn into a firm preference.
In the report: "The firm with the highest financial guarantee outranks the other two; the remaining two firms have failed to outrank each other and remain incomparable. This is sensitive to the divergence in evaluators' opinions on event-organisation experience."
3. What Not to Do
Had the four viewpoints in A1's financial-soundness cell in the innovation-park table been reduced to a single average from the outset and only this single number reported, the information about which expert gave which value would have been lost, and the threshold comparisons across criteria could have been rebuilt in the same way, but the sensitivity analysis, which expert's opinion was decisive, could not have been carried out. The second mistake is trying to add criterion weight to this method; m-polar ELECTRE IV is unweighted by definition, and no weight, not even an equal one, is assigned. The third mistake is reading A2's quasi-dominance level (1.0) as "A2 is 100 per cent better"; this level is not a percentage but the label of the strongest of five classes.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/mpf-electre-iv
Akram, M., & Adeel, A. (2023). MCDM Methods with Multi-polar Fuzzy Information. Studies in Fuzziness and Soft Computing, vol. 430, Springer Nature. DOI: 10.1007/978-3-031-43636-9
Roy, B., & Hugonnard, J. C. (1982). Ranking of suburban line extension projects on the Paris metro system by a multicriteria method. Transportation Research Part A: General, 16(4), 301–312. DOI: 10.1016/0191-2607(82)90057-7
Vallée, D., & Zielniewicz, P. (1994). ELECTRE III-IV, version 3.x, Aspects méthodologiques (Tome 1). Document du LAMSADE n° 85, Université Paris-Dauphine. (no DOI)
Chen, J., Li, S., Ma, S., & Wang, X. (2014). m-Polar fuzzy sets: An extension of bipolar fuzzy sets. The Scientific World Journal, 2014, 416530. DOI: 10.1155/2014/416530
Akram, M., Waseem, N., & Liu, P. (2019). Novel approach in decision making with m-polar fuzzy ELECTRE-I. International Journal of Fuzzy Systems, 21(4), 1117–1129. DOI: 10.1007/s40815-019-00608-y