PIF DOMBI (Picture Fuzzy Dombi Aggregation Operator)

PIF-DOMBI, like MPF-DOMBI-WA, is an aggregation operator; it does not combine several people's rankings but a single expert's assessments of an alternative across several criteria. Its difference lies in how the assessments are expressed: through a three-component structure called a picture fuzzy number. Every cell holds a positive degree, a neutral (undecided) degree and a negative degree; the sum of these three degrees cannot exceed 1. Unlike classical "yes or no" fuzzy numbers, or ones that state only "how much yes," this structure also allows for saying "I am undecided" or "I abstain," much like the distinction between a yes, no, abstention and blank vote in a voting system. The method combines an alternative's picture fuzzy values across all criteria using weights to produce a single summary picture fuzzy value, and from that a single score for ranking. PIF-DOMBI was proposed by Jana, Senapati, Pal and Yager in 2019.

The idea behind picture fuzzy sets is that human judgement can sometimes be expressed not only through "I agree" or "I disagree" but also through a third state of "I am partly undecided." A classical fuzzy number asks only "how much do I agree"; an intuitionistic fuzzy number adds "how much do I disagree." A picture fuzzy number adds a third dimension: "how undecided/neutral am I." This lets an expert state a richer judgement, "partly positive, somewhat undecided, and to a small degree also negative," rather than a flat "definitely positive" or "definitely negative" on a criterion.

The Dombi operator is the adjustable mathematical tool used here to combine these three components, carrying a tunable parameter, usually denoted ℜ. This carries a philosophical consequence: PIF-DOMBI is a compensatory method that both preserves the three-way (positive, neutral, negative) judgement and lets the intensity of aggregation be tuned by the parameter. As the parameter grows, the aggregation behaves more sharply; as it shrinks, more gently.

The method proceeds through four steps.

First, accepting the inputs. Every alternative's assessment on every criterion is taken as a picture fuzzy number (a positive, neutral, negative triple); criterion weights are set summing to 1, and a Dombi parameter (at least 1) and an operator type (arithmetic or geometric) are chosen.

Second, row combination. Every alternative's picture fuzzy values across all criteria are combined, using the chosen Dombi operator and the criterion weights, into a single summary picture fuzzy value for that alternative; the positive, neutral and negative components are combined separately.

Third, score calculation. A score between 0 and 1 is computed from the summary value's positive and negative components, without the neutral component entering the calculation. Where two alternatives' scores come out equal, a second accuracy indicator, showing the difference between the positive and negative components, 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 summarises the balance between an alternative's positive and negative judgements; it lies between 0 and 1 and is not a percentage. It is important to know that the score does not directly reflect the neutral (undecided) component; two alternatives can share the same score while one reaches it with far less indecision and the other with far more. The report should address not only the score but the underlying level of indecision too.

The result is sensitive to the chosen operator and the Dombi parameter; the same decision table can give a different ranking under a different parameter. This sensitivity is not a flaw in the calculation but a natural consequence of the method's adjustable structure, and it should be shown in the report.

Thus instead of writing:

"PIF-DOMBI found the best alternative"

the report should read

"With these weights and this Dombi parameter, the alternative with the best balance between positive and negative judgement is this one; the ranking can change if the parameter changes, and the level of indecision should be examined separately"

PIF-DOMBI works with picture fuzzy data: every cell holds a positive, a neutral and a negative degree; the sum of these three degrees cannot exceed 1. Criterion weights must be supplied from outside and must sum to 1; the method does not produce weights, it asks for them. The Dombi parameter and the operator type are set by the user. The method processes a single decision-maker's assessment; where several experts' opinions exist, they must first be combined in a separate step. DecisionMind currently holds no further member alongside this base method.

PIF-DOMBI is suitable if expert judgement is expressed not only through "how positive" but also through "how undecided," and you want to assess this without losing that third dimension. Where judgements are expressed only along a binary line between "positive and negative" (the indecision dimension is not meaningful), a simpler intuitionistic fuzzy or crisp method is sufficient. Where you need to combine several experts' opinions at once, PIF-DOMBI alone is not enough; the opinions must first be combined, then this method applied.

Judgement to be expressed three ways, positive, neutral, negative → PIF-DOMBI

Judgement only binary, between positive and negative → intuitionistic fuzzy or crisp aggregation

Criterion judgement carries several independent sub-dimensions (poles) → MPF-DOMBI-WA (m-polar fuzzy)

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

PIF-DOMBI's greatest strength is adding a third, indecision dimension to the classical positive-negative duality; this lets it also carry the "I am not entirely sure" state of genuine expert judgement. The Dombi parameter allows the intensity of aggregation to be tuned. The method is idempotent; if every criterion carries the same value, the summary value equals that same value too.

Its limitations stem mainly from parameter sensitivity and the single-decision-maker assumption. First, when one of the components (positive, neutral or negative) is exactly 0, a division error occurs in the Dombi operation; these extreme values need clipping by a small amount. Second, the arithmetic and geometric operators can give different rankings, and this gap can widen especially where the neutral component's distribution is skewed. Third, the score calculation does not use the neutral component directly; this is consistent with the scoring logic of intuitionistic fuzzy numbers, but it pushes the indecision information outside the report, and this information needs showing separately. Fourth, some early aggregation studies in the picture fuzzy sets literature have since been retracted (Wei, 2017); this shows the field is still maturing and that care is needed in source selection.

The most common mistake is entering data in which the sum of the three components (positive, neutral, negative) exceeds 1; this contradicts the definition of a picture fuzzy number and must be checked first. A second mistake is feeding a component that is exactly 0 into the calculation without clipping; this leads to a division error in the Dombi operation. A third mistake is presenting the result without stating the Dombi parameter and operator type in the report. A fourth mistake is reporting a highly-scored but highly-undecided alternative with the same confidence as a low-indecision alternative; even where the score is equal, the underlying level of indecision can differ.

The governing principle is this:

A PIF-DOMBI result summarises the balance between positive and negative judgement; this summary does not show the level of indecision, so the report must state the size of that indecision alongside the score.