MPF DOMBI WA (m-Polar Fuzzy Dombi Weighted Average)

Unlike other aggregation and voting methods, MPF-DOMBI-WA does not combine the rankings of several people or methods; it combines a SINGLE expert's multi-criterion assessment of one alternative. The assessment on each criterion is not a single number but an m-polar fuzzy number carrying several "poles" (independent sub-dimensions); for instance, a "location" criterion might be expressed at once through three separate poles such as "proximity to market," "proximity to a water source" and "access to transport." The method combines an alternative's multi-polar values across all criteria, using weights, into a single m-polar summary value per alternative; from this summary value a single score is then extracted to form the basis of the ranking. MPF-DOMBI-WA was proposed by Akram, Yaqoob, Ali and Chammam in 2020, and is one of six aggregation operators using the Dombi operator on m-polar fuzzy sets.

The idea behind m-polar fuzzy sets is that some judgements are, by their nature, better described through several independent dimensions than through a single number. A plot of land's fertility is not simply "good" or "bad"; it requires the joint assessment of several independent properties such as soil pH, nutrient level and water-retention capacity. An m-polar fuzzy number carries these several independent properties in a single cell without losing them.

The Dombi operator is itself the mathematical tool used to combine these multi-polar values, and it carries an adjustable parameter (k). This parameter determines how "generous" or "cautious" the combination will be. MPF-DOMBI-WA works in arithmetic form and tends to reward high pole values; its sister operator, MPF-DOMBI-WG, works in geometric form and is more sensitive to low pole values. The philosophical consequence of this is that MPF-DOMBI-WA is compensatory, but the strength of this compensation is not fixed; it has an adjustable dial through the k parameter.

The method proceeds through four steps.

First, accepting the inputs. Every alternative's assessment on every criterion is taken as an m-polar fuzzy number; criterion weights are set so as to sum to 1, and the Dombi parameter (k, at least 1) and the operator type (arithmetic or geometric) are chosen.

Second, combining by row. An alternative's m-polar values across all criteria are combined, using the chosen Dombi operator and the criterion weights, into a single m-polar summary value for that alternative. This step is carried out separately for each pole, so the summary value also remains m-polar.

Third, computing the score. The average is taken across all poles of the summary m-polar value, giving a single score, between 0 and 1, on which the ranking is based. If two alternatives' scores tie, a second accuracy indicator, measuring imbalance between the poles, is consulted.

Fourth, ranking. Alternatives are ranked from the highest score to the lowest; the highest-scoring alternative comes first.

The formulas behind each step and the intermediate tables are given on the DecisionMind method page; this card carries no formulas.

The score is a general summary of an alternative's multi-polar assessment across all criteria; it lies between 0 and 1 and is not a percentage. A score such as 0.75 does not mean "75 per cent suitable"; it means only that the average of this alternative's poles is 0.75. This score cannot be compared with a score from a different analysis, because the criterion set, the weights and the chosen Dombi parameter may differ between analyses.

An important caution is that the result is sensitive both to the chosen operator, arithmetic or geometric, and to the Dombi parameter. The same decision table can give a different order when the k parameter is changed; this is not a faulty calculation but a flexibility built into the method's nature, and the report should show this sensitivity.

Thus instead of writing:

"MPF-DOMBI-WA found the best alternative"

the report should read:

"With these weights, this operator and this Dombi parameter, the alternative with the highest summary score is this one; the order may change if the parameter is changed"

MPF-DOMBI-WA works with m-polar fuzzy data: each cell carries not a single number but m independent pole values, each between 0 and 1. Criterion weights must be supplied from outside and must sum to 1; the method produces no weights, it asks for them. The Dombi parameter (k, at least 1) and the operator type (arithmetic or geometric) are set by the user. The method processes a single decision-maker's assessment; where several experts' opinions exist, these opinions must be combined in a separate step before entering MPF-DOMBI-WA. DecisionMind currently holds no further member alongside this base method.

MPF-DOMBI-WA is suitable where the judgement under a criterion naturally carries several independent sub-dimensions and you want to assess these dimensions without collapsing them into a single number. If your data is already expressed as single (crisp) numbers, this method's multi-polar structure is unnecessary and a simpler aggregation operator will do. If you need to combine several experts' opinions at once, MPF-DOMBI-WA is not sufficient on its own; the expert opinions must first be combined, and only then should this method be applied.

Criterion judgement is multi-polar (several independent sub-dimensions) → MPF-DOMBI-WA

Criterion judgement is a single number, crisp data → classical weighted-average aggregation

Several experts' opinions are to be combined → first a group-aggregation operator, then MPF-DOMBI-WA

Judgement to be expressed three ways, as positive, neutral and negative → PIF-DOMBI (picture fuzzy Dombi)

MPF-DOMBI-WA's most important advantage is that it carries a criterion's several independent sub-dimensions without losing them; this offers richer information than methods that collapse everything into a single number too early. Thanks to the Dombi parameter, how compensatory the combination will be can be adjusted; this flexibility provides a tool adaptable to different decision contexts. The method is idempotent and preserves boundedness, meaning that if all criteria carry the same value, the summary value also equals that value.

Its limitations come mainly from parameter sensitivity and the single-decision-maker assumption. First, when a pole value is exactly 0 or exactly 1, a division error arises in the Dombi operation; these extreme values need to be clipped by a small amount. Second, the arithmetic (MPF-DOMBI-WA) and geometric (MPF-DOMBI-WG) operators can give a different order on the same data; which operator was chosen should be stated clearly in the report. Third, the method processes only a single decision-maker's matrix; a group decision with several experts requires a separate group-aggregation step first (Akram, Yaqoob, Ali and Chammam, 2020). Fourth, criterion direction, whether lower or higher is better, lies outside the method; cost criteria must be complemented separately before entering the method.

The most common mistake is entering a cost-direction criterion (lower is better) directly without complementing it first; the method takes every criterion as benefit-oriented, and the direction conversion is the user's responsibility. A second mistake is choosing the Dombi parameter and the operator type without stating them in the report; because the same data can give a different order with different parameters, this choice must be transparent. A third mistake is entering a pole value at a boundary point (0 or 1) into the calculation without clipping it; this leads to a computational error. A fourth mistake is running several experts' matrices through MPF-DOMBI-WA one by one and then averaging the results afterwards; the correct approach is to combine the matrices first and aggregate only once.

The governing principle is this:

An MPF-DOMBI-WA result is a summary score jointly produced by the chosen weights, the operator type and the Dombi parameter; the order can change when the parameter changes, and this sensitivity must be shown in the report.