Intuitionistic fuzzy PROMETHEE (Liao and Xu, 2014)

Four things change; the logic of the preference function and the flows does not.

Cells. In crisp PROMETHEE every cell is a single number. Here every cell consists of two degrees: support (μ) and rejection (ν), whose sum cannot exceed 1. Weights remain crisp and are taken from outside, summing to 1; the method does not generate weights. Unlike the family's other three extensions, this entry directly supports a group decision: the (μ, ν) matrices given separately by several decision-makers are combined into a single matrix using the intuitionistic fuzzy weighted average (IFWA) with the decision-makers' own weights; with a single decision-maker this step is skipped without changing anything.

Complementing a cost criterion. In crisp PROMETHEE the difference on a cost criterion is taken with the sign reversed. Here, every (μ, ν) pair in the cost columns is first converted to its complement, (ν, μ); this reduces every criterion to a single "higher is better" representation, and the pairwise comparison proceeds from there.

A difference reduced to a score. In crisp PROMETHEE the difference is a direct subtraction. Here, every (μ, ν) pair is first reduced to a single real number with the Chen-Tan (1994) score (μ minus ν), and the difference is taken between these scores. DecisionMind performs this scoring early, at the third step: in Liao and Xu's (2014) eight-step algorithm in the literature, the preference relation and the incoming and outgoing flows are kept as intuitionistic fuzzy values all the way to the end, and the score function (Szmidt-Kacprzyk) is applied only at the last step, to finalise the ranking. The two routes are not mathematically identical; DecisionMind records this as an engineering choice that reduces computational load, and this card does not hide it.

The preference function and the flows. The scored difference is processed with one of the same preference functions as crisp PROMETHEE (default: linear with thresholds, indifference threshold 0, preference threshold 1), multiplied by the criterion weight and summed; the incoming flow (φ⁺) and outgoing flow (φ⁻) are calculated with the same averaging definition as crisp PROMETHEE, and from this point on both are real numbers, not intuitionistic fuzzy values.

DecisionMind fixes the Chen-Tan score and early scoring in this entry. If a different score function is to be used, such as one of the other functions Xu (2007) proposes, this choice must be stated in the report; the ranking can change with a different score function.

The net flow is a value between −1 and +1, as in crisp PROMETHEE, and is read the same way: it is assessed together with the incoming and outgoing flows, and is not compared with another method's score.

What differs is this: this net flow has already reduced the support and rejection degrees to a single score as early as the third step; the hesitancy margin (1 − μ − ν) does not appear in this score, only the difference between μ and ν does. Two different (μ, ν) pairs can give the same difference (μ−ν) while carrying different hesitancy margins; in that case the net flow does not distinguish between them. The report should therefore state, alongside the net flow, which support-rejection pairs the flows were built from, and flag the cells with a large hesitancy margin.

Thus instead of writing:

"According to intuitionistic fuzzy PROMETHEE, A1 is clear, the flows still carry uncertainty"

the report should read:

"Support and rejection degrees have been reduced to a single score with the Chen-Tan (1994) score and processed with PROMETHEE's preference function; A1's net flow is the highest, and this flow carries only the support-rejection difference, not the hesitancy margin"

Use this entry when a criterion can be converted into a judgement, and the evidence for and against that judgement comes from separate sources: when the board's support and reservations for a proposition such as "this investment creates value in the long run" can be collected separately. This entry is particularly preferred when several decision-makers' separate intuitionistic fuzzy assessments need to be combined; base PROMETHEE and the family's other three extensions do not support this combination directly.

If the rejection degree is calculated only as 1 minus support (ν = 1 − μ), using intuitionistic fuzzy data adds nothing; it carries the same information as crisp data. Stay with the base method where criteria are measured. Turning a measured score directly into μ and writing ν as 1 − μ does not produce intuitionistic fuzzy data; it zeroes the hesitancy margin in every cell. DecisionMind requires a single data type if the table is mixed. Crisp PROMETHEE's exit conditions apply here too: if the number of alternatives exceeds fifty, the burden of pairwise comparison grows, and if a full order is not needed but an honest partial relation is, PROMETHEE I's reading or the ELECTRE family should be considered.

Forgetting to take the complement on a cost criterion. If the (μ, ν) pair in cost columns is not converted to (ν, μ), the direction reverses and the result becomes meaningless; this is the family's most common mistake.

Taking the absolute value of a signed difference. The preference function is designed to return zero for a difference of zero or below; taking the absolute value of the difference makes π(a,b) equal π(b,a) and breaks PROMETHEE's asymmetric comparison logic.

Assuming the flows are still intuitionistic fuzzy values. Because DecisionMind takes the score at the third step in this entry, the incoming and outgoing flows are real numbers; trying to reinterpret them under a constraint such as μ+ν≤1, or confusing them with the intuitionistic fuzzy flows that Liao and Xu's (2014) algorithm in the literature keeps until the final step, is a mistake.

The governing principle is this:

Intuitionistic fuzzy PROMETHEE's contribution is to keep support and rejection information separate until it is converted into a score; if the complement is forgotten, the difference is taken as an absolute value, or a scored flow is read as still having two components, this contribution is lost.