Extension card · m-Polar
m-Polar Fuzzy PROMETHEE (Akram, Shumaiza and Alcantud, 2020)
This is the form of PROMETHEE for situations where every cell is rated separately from several independent viewpoints. Every cell is first reduced to a single score, and the pairwise comparison and net-flow calculation then proceed on these scores exactly as in crisp PROMETHEE.
Base method
PROMETHEE →
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?
Three things change; the decision logic does not.
Cells. In crisp PROMETHEE every cell is a single number. Here every cell holds m numbers, one from each independent viewpoint. A criterion assessed from three viewpoints, for example, 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. Where several experts are involved, their m-polar scores are first merged by a simple average. Weights remain crisp numbers; PROMETHEE does not generate weights, it takes them from outside. The seminal paper derives these weights with AHP and requires the consistency ratio to stay below 0.10.
Scale equalisation. The m-polar cell is reduced to a single number in the very first step. DecisionMind uses the score function for this: it takes the average of the m viewpoints. Three viewpoints of (0.7; 0.85; 0.75) give a score of 0.767. Once the score function has been applied, the remaining steps of crisp PROMETHEE run on these numbers unchanged; no vector normalisation is needed either, because the scores already lie between 0 and 1.
Preference function and flows. Once the score function is finished, the calculation follows exactly the same path as crisp PROMETHEE. One of six generalised preference functions is chosen for every criterion, pairwise differences are passed through these functions, combined with the weights, and the entering and leaving flows, then the net flow, are found. In the seminal paper's example, all six criteria run with six different function types; this is not a requirement to use a single type.
DecisionMind fixes, for this extension, the score function and the crisp PROMETHEE steps that follow it (preference functions, entering/leaving/net flow). Weights are taken from outside as crisp numbers; the method does not generate weights.
How to Read the Output
The output is a net flow and a complete ranking, as in crisp PROMETHEE, and it is read the same way. The entering and leaving flows should be shown separately, because only these tell you whether an alternative stands out by winning a great deal or by losing little.
The difference is here. Beneath the score lies an input from m independent viewpoints, but this input has already dissolved into a single number in the very first step, through the score function. Two alternatives can reach the same score through very different combinations of viewpoints. Viewpoints of (0.8; 0.8; 0.8) and of (0.95; 0.95; 0.55) give the same average (0.8); the first is complete agreement, the second carries a clear reservation in one viewpoint.
Thus instead of writing:
"m-polar PROMETHEE gives a richer result because it assesses several viewpoints together"
the report should read:
"The viewpoints have already been reduced to a single number in the first step by the score function; the net flow carries only the difference between these single numbers, and does not show whether the viewpoints agreed or conflicted"
When to Prefer This over the Base Method
Use this where a criterion's value genuinely comes from several independent sources, and preserving these sources without reducing them to an average, at least at the first step, carries information that is wanted. A typical case is a decision where stakeholder groups, regions or experts independently rate the same criterion, and reducing these ratings too early to a single number could hide a reservation.
Returning to base PROMETHEE is called for where the criterion is measured from a single source, with a single number. Where the viewpoint is single, forcing the m-polar structure adds an artificial layer; as the data-type card explains, every viewpoint must have its own body of evidence.
The exit condition is the same as for crisp PROMETHEE: where the number of alternatives is very large, or an honest partial relation is sufficient, this extension carries the same limits.
Mistakes Specific to This Extension
Forgetting to flip the sign on a "lower is better" criterion. The preference function is defined on increasing differences; on a "lower is better" criterion the difference must first be reversed. Skip this step and a cost-like criterion's contribution silently reverses, with the net flow coming out wrong without throwing any error. This card's validation example checks precisely this step.
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. The two are different data structures; one is not a special case of the other.
Not checking the consistency of AHP weights. Where weights come from AHP, the consistency ratio must stay below 0.10. Skip this check and a bias from inconsistent pairwise comparisons carries silently into the weights, and from there into the net flow.
The governing principle is this:
m-polar PROMETHEE reduces independent viewpoints to a single number with the score function and runs the rest of crisp PROMETHEE on these numbers; any application that skips the sign flip, copies viewpoints from a single source, or uses inconsistent weights makes this reduction unreliable.
Cases
The first case is drawn from the literature: the hydroelectric-plant site-selection example from Akram, Shumaiza and Alcantud (2020). The second case is an illustrative construction.
1. Energy: Choosing among seven candidate sites for a hydroelectric plant (Akram, Shumaiza and Alcantud, 2020)
Two site experts have scored seven candidate sites (R1-R7) on six criteria from three viewpoints (three sub-indicators): infrastructure (higher is better), land quality (higher is better), government incentives (lower is better), social infrastructure (higher is better), climate-change risk (lower is better) and cost (lower is better). The two experts' scores have been averaged with equal weight, and weights have been derived with AHP (consistency ratio 0.087, below the acceptable limit): infrastructure 0.45, land quality 0.11, incentives 0.03, social infrastructure 0.17, climate 0.18, cost 0.06. A different preference-function type is used for every criterion.
Two examples from the averaged three-polar scores: R6's infrastructure cell (0.675; 0.635; 0.615), R2's land-quality cell (0.725; 0.765; 0.725). The score function (the average of the three viewpoints) reduces every cell to a single number; the full matrix is on the decisionmind.app method page.
| Site | Infrastructure | Land quality | Incentives | Social infrastructure | Climate | Cost |
|---|---|---|---|---|---|---|
| R1 | 0.448 | 0.547 | 0.487 | 0.467 | 0.607 | 0.567 |
| R2 | 0.553 | 0.738 | 0.422 | 0.450 | 0.738 | 0.717 |
| R3 | 0.508 | 0.557 | 0.503 | 0.580 | 0.528 | 0.632 |
| R4 | 0.595 | 0.533 | 0.528 | 0.512 | 0.523 | 0.578 |
| R5 | 0.545 | 0.603 | 0.512 | 0.508 | 0.543 | 0.542 |
| R6 | 0.642 | 0.592 | 0.508 | 0.530 | 0.603 | 0.528 |
| R7 | 0.533 | 0.567 | 0.633 | 0.603 | 0.545 | 0.522 |
From these scores the method computes every pair of sites' difference on every criterion, reverses the difference on the "lower is better" criteria (incentives, climate, cost), passes it through the six different preference functions, and combines the results with the weights. Every site's entering flow (Φ⁺, how far it leaves the others behind) and leaving flow (Φ⁻, how far it lags behind) are found.
| Site | Φ⁺ | Φ⁻ | Net flow | Rank |
|---|---|---|---|---|
| R6 | 0.619 | 0.157 | 0.462 | 1 |
| R7 | 0.367 | 0.174 | 0.194 | 2 |
| R4 | 0.367 | 0.228 | 0.139 | 3 |
| R5 | 0.294 | 0.235 | 0.059 | 4 |
| R3 | 0.267 | 0.303 | -0.036 | 5 |
| R2 | 0.238 | 0.494 | -0.256 | 6 |
| R1 | 0.114 | 0.674 | -0.561 | 7 |
The result reads as follows. R6 is strong on infrastructure (0.642) and on climate risk (0.603, low risk); being superior on the two heaviest criteria carries it to first place. R7 is not the best on any single criterion but is strong on incentives (0.633) and social infrastructure (0.603) and finishes second. R1, weak on infrastructure (0.448) and on cost, finishes last. This ranking matches Table 12 of Akram, Shumaiza and Alcantud's (2020) paper exactly (R6, R7, R4, R5, R3, R2, R1); the net-flow values match to within a small decimal-place difference (at most 0.03), arising from rounding in the paper's manual calculation.
The board's hesitation is this: does the ranking change if the infrastructure weight is lowered from 0.45 to 0.20 and social infrastructure raised from 0.17 to 0.42? When the same calculation is re-run independently in Python, R7 (net flow 0.470) moves ahead of R6 (0.330). The infrastructure weight is the main support behind R6's first place; once this weight is shifted to social infrastructure, the lead changes hands.
In the report: "With the AHP-derived weights (infrastructure 0.45 the highest), R6 is clearly the most preferred site (net flow 0.462); if the infrastructure weight is shifted to social infrastructure (0.20 against 0.42), R7 moves ahead. The ranking is sensitive to the relative weight of infrastructure and social infrastructure."
Source: Akram, M., Shumaiza, & Alcantud, J. C. R. (2020). An m-Polar Fuzzy PROMETHEE Approach for AHP-Assisted Group Decision-Making. Mathematical and Computational Applications, 25(2), 26, Tables 2–12. The score matrix, preference-function calculations and net flows were reproduced by this card's author running the DecisionMind engine independently; the ranking has been verified against the paper's result. The weight-change scenario was likewise computed independently.
2. Livestock farming: A cooperative's choice of new dairy-cattle breed
An agricultural cooperative will choose one of three candidate dairy-cattle breeds to renew its herd. Four criteria apply: milk yield (higher is better), feed-conversion ratio (higher is better), disease resistance (higher is better) and climate adaptation (higher is better). The cooperative has rated every breed from three independent sources: the regional veterinarian, neighbouring farms' field experience, and breed-improvement records. The cooperative has given milk yield the highest weight.
The method reduces every breed's three-sourced score on every criterion to a single number with the score function, passes the pairwise differences through the preference functions, and finds the net flows. Suppose the breed that looks strongest on milk yield came out weak on disease resistance and still finished first, because its superiority on milk yield was won on the heaviest criterion.
The cooperative's hesitation is this. The veterinarian rated one breed's disease resistance at 0.8, while the field experience and the breed-improvement records rated the same breed at around 0.4. The score function averages these three sources into a middling figure, but this wide divergence between the sources should be shown in the report; otherwise the cooperative could unknowingly take on a risk that rests on a single expert's opinion.
In the report: "With the highest weight given to milk yield, the breed judged strong on this criterion clearly stands out; on disease resistance the divergence between the three sources is wide, and this divergence should be assessed separately before the herd-health decision."
3. What Not to Do
Had the sign-flip step for the incentives, climate and cost criteria in the illustrative table been skipped and the difference used directly, the sites with a high value, in fact a bad one, on these three criteria would be counted as advantaged, and the ranking would reverse. The second mistake is reducing R6's three-viewpoint infrastructure cell (0.675; 0.635; 0.615) to a single "high" label and ignoring the 0.06 gap within it; this gap shows how much the viewpoints agree with one another. The third mistake is reporting R6's net flow of 0.462 as "46 per cent better"; the net flow only ranks these seven sites relative to one another.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/mpf-promethee
Akram, M., Shumaiza, & Alcantud, J. C. R. (2020). An m-Polar Fuzzy PROMETHEE Approach for AHP-Assisted Group Decision-Making. Mathematical and Computational Applications, 25(2), 26. DOI: 10.3390/mca25020026
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
Brans, J. P., & Vincke, Ph. (1985). A preference ranking organisation method (The PROMETHEE method for multiple criteria decision-making). Management Science, 31(6), 647–656. DOI: 10.1287/mnsc.31.6.647
Brans, J. P., Vincke, P., & Mareschal, B. (1986). How to select and how to rank projects: The PROMETHEE method. European Journal of Operational Research, 24(2), 228–238. DOI: 10.1016/0377-2217(86)90044-5