Extension card · m-Polar

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

This is the form of ELECTRE II for situations where performance scores are rated separately from several independent viewpoints. The viewpoints are preserved without being reduced to an average, and the output remains a complete ranking.

Base method

ELECTRE II →

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 logic of forward/backward distillation and incomparability does not.

Cells. In the family's crisp member 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 weighted-average operation. Criterion weights can likewise be collected in m-polar form; DecisionMind reduces these to a single, normalised score degree (γ) before use. This is the point where this member differs most from the family's m-polar ELECTRE IV member: that member is entirely unweighted, whereas this one uses weights.

Weighting. Every pole of the merged cell is multiplied by its own criterion's normalised weight. Comparisons then run on the score degree drawn from these weighted cells, the average of the m poles.

Three sets and two matrices instead of concordance/discordance. In crisp ELECTRE II every pair carries a single concordance index and a single discordance index. Here every criterion is first placed into one of three sets: one alternative outranks the other, falls behind it, or ties with it; the concordance matrix is the sum of these sets' weights, tallied by the relative importance the user has set. The discordance matrix is the ratio of the largest gap among the criteria where the alternative falls behind to the largest gap across all criteria; the measure of gap here is a distance built jointly from the m poles, not a single-number difference.

The form of the thresholds. In crisp ELECTRE II, two concordance thresholds (s1, s2) are chosen by the analyst. Here five thresholds are chosen: three concordance thresholds (c⁻, c°, c*) and two discordance thresholds (d°, d*). Strong outranking demands the strictest threshold pair, weak outranking the loosest. What DecisionMind fixes here is that the strong and weak outranking graphs are built separately and then combined through the average of forward distillation (downward from the best) and backward distillation (upward from the worst); this is exactly crisp ELECTRE II's own distillation logic.

How to Read the Output

What stays the same as the base method is this: the output is a position, not a score; strict preference, indifference and incomparability are three separate relations.

The difference lies here: the concordance and discordance that determine this position rest on score degrees drawn from the average of m independent viewpoints. Even where two candidates' scores on a criterion look close, how consistent or scattered the m viewpoints are beneath that score needs separate examination; the final ranking depends not only on the score but also on the five chosen thresholds.

Thus instead of writing:

"m-Polar ELECTRE II found x3 to be the clear winner"

the report should read:

"With the chosen thresholds, x3 outranks both x1 and x2 in both the strong and the weak relation; this result rests on the criterion weights and the five thresholds"

When to Prefer This over the Base Method

Use this when a criterion's value genuinely comes from several independent sources that should not be reduced to an early average, and a complete ranking, not merely a shortlist, is wanted. Where the viewpoint comes from a single source, forcing the m-polar structure adds an artificial layer; the family's crisp ELECTRE II member is then sufficient.

Where the criteria's relative importance cannot be reliably established, the family's m-polar ELECTRE IV member, entirely unweighted, should be preferred instead. Crisp ELECTRE II's exit condition, wanting a complete ranking together with non-compensatory logic, applies here in exactly the same way.

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.

Merging experts after weighting. The experts' m-polar matrices are first merged among themselves, and criterion weights are applied only afterwards, to this merged matrix. Reversing the order, weighting each expert first and merging afterwards, breaks the symmetry of the merging operation and produces a different concordance/discordance matrix.

Pairing the weak-outranking threshold wrongly. Which of the five thresholds (c⁻, c°, c*, d°, d*) is used for the strong relation and which for the weak carries a wording ambiguity in the source; using the stricter threshold (d°) for the weak relation does not reproduce the published case study's own outranking matrix. The looser threshold (d*) must be used for the weak relation.

The governing principle is this:

In m-polar ELECTRE II, independent viewpoints are carried through without being reduced to an average, but, as in crisp ELECTRE II, criterion weight and the five thresholds are still set by the user. The robustness of the ranking should be read against these thresholds.

Cases

The first case is drawn from the literature: the nuclear power plant site-selection example from Akram and Adeel's (2023) book. The second case is an illustrative construction.

1. Energy: Full ranking of three candidate sites for a nuclear power plant (Akram and Adeel, 2023)

In China's Fujian region, three candidate sites (x1, x2, x3) have been assessed, by three experts (with roughly equal weights), in 3-polar form on eight criteria: geographical condition, meteorological features, ground condition, natural-disaster risk, social factor, effect on other activities, environmental factor, and economic factor; all of them are "higher is better." Once the three experts' matrices have been merged, the score degrees drawn from every cell (the average of the three viewpoints) are as follows:

Site	Geographical condition	Meteorological	Ground condition	Natural-disaster risk	Social factor	Effect on other activities	Environmental factor	Economic factor
x1	0.756	0.158	0.306	0.310	0.361	0.316	0.542	0.264
x2	0.786	0.154	0.305	0.339	0.371	0.310	0.527	0.275
x3	0.790	0.155	0.296	0.360	0.355	0.307	0.571	0.267
Weight (γ)	0.263	0.051	0.096	0.102	0.115	0.098	0.183	0.091

Geographical condition and environmental factor are the two heaviest criteria (γ together approximately 0.45). The method builds the three sets (outranks/falls behind/ties) for every pair and computes the weighted concordance matrix and the normalised discordance matrix: x3's concordance over x1 comes out at 0.64, x2's concordance over x1 at 0.57, and x3's concordance over x2 at 0.60. These values are compared against the five thresholds (c⁻=0.50, c°=0.55, c*=0.60, d°=0.50, d*=0.60) to build the strong and weak outranking graphs; the forward, backward and average rankings are then computed.

Rank	Site
1	x3
2	x2
3	x1

The result reads as follows: x3 holds the highest score on the two heaviest criteria (geographical condition and environmental factor) and outranks both x1 and x2 in the strong relation. Forward and backward distillation agree entirely in this case; no pair remains incomparable.

The board's hesitation is this: although x3's outranking of x2 (concordance 0.60) only just meets the strictest threshold (c*=0.60), this relation is in fact also satisfied by the looser threshold pair (c°=0.55, d°=0.50), so the relation does not break even if c* is raised slightly. In an independent check carried out by this card's author, when the discordance threshold d° is tightened from 0.50 to 0.35, the relations of x3 outranking x2 and x2 outranking x1 disappear entirely from both the strong and the weak graph, leaving only the relation of x3 outranking x1. Even so, the final average ranking (x3, x2, x1) does not change; the two-way distillation procedure produces the same ranking with only this single remaining relation.

In the report: "With the chosen five thresholds, the ranking is x3, x2, x1; x3's superiority comes from the two most heavily weighted criteria (geographical condition, environmental factor). Even when the discordance threshold is tightened substantially and most of the intermediate relations disappear, the final ranking does not change."

Source: Akram, M., & Adeel, A. (2023). MCDM Methods with Multi-polar Fuzzy Information, Chapter 4, §4.5 (pp. 267–277, the Fujian nuclear power plant site-selection case study, Tables 4.16–4.27). The score degrees, concordance/discordance matrices and threshold sensitivity were computed by this card's author running the DecisionMind engine independently; the baseline result matches the book's own ranking exactly (x3 the most suitable site, p. 277).

2. Furniture manufacturing: Full ranking of three raw-material suppliers for a manufacturer

A furniture manufacturer will fully rank three timber suppliers. Four criteria apply: timber quality, delivery-time reliability, sustainable forestry certification coverage, and unit-price competitiveness; all four are "higher is better." Each supplier has been rated independently by three separate departments (production, quality control, procurement); these three viewpoints are held together without being reduced to an average.

The method builds the three sets, computes the weighted concordance and normalised discordance matrices, forms the strong and weak outranking graphs with the five thresholds, and returns the forward, backward and average ranking. Suppose the result shows the supplier with the highest score on timber quality coming out first, but forward and backward distillation fail to agree on the second and third suppliers, leaving them incomparable.

The manufacturer's hesitation is this. One of the two incomparable suppliers scored high from quality control but low from procurement; the gap between these two viewpoints concentrates on the unit-price criterion. The reason for this disagreement between departments, for instance an unfinished price negotiation, should be stated in the report.

In the report: "The supplier that stands out on timber quality is clearly first; the relation between the second and third suppliers comes out incomparable with these thresholds, and this stems from the departments' differing assessments of unit price."

3. What Not to Do

Reducing the three experts' scores in the nuclear-plant table straight to a single average from the outset and running crisp ELECTRE II on it is the first mistake; this erases the information about which expert disagreed on which criterion. The second mistake is applying the criterion weights BEFORE merging the experts; the correct order is to merge the experts first and apply the weight afterwards. The third mistake is using the stricter discordance threshold (d°) for the weak-outranking relation; reproducing the book's own case study requires the looser threshold (d*) for the weak relation.

Sources

For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/mpf-electre-ii

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

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

Roy, B., & Bertier, P. (1973). La méthode ELECTRE II: une application au media-planning. In Operational Research '72: Proceedings of the Sixth IFORS International Conference on Operational Research (pp. 291–302). North-Holland. (no DOI)