PROMETHEE III (Brans, Vincke & Mareschal, 1986)

Two things change: how the net flow is computed, and the ranking itself.

Mean-flow computation. In base PROMETHEE, the incoming and outgoing flow for every alternative is found by averaging the comparisons against the other m-1 rivals, and the net flow is the difference between the two. PROMETHEE III averages the signed differences (π(a,a′) − π(a′,a)) for every alternative over ALL m alternatives (including the comparison with itself, which contributes a term of zero); the divisor is m, not m-1. This small difference affects the result once combined with the standard-deviation computation below.

Building the interval. Base PROMETHEE uses this mean flow directly for ranking. PROMETHEE III additionally computes how spread out these same m terms are (the standard deviation, σ) and builds, for every alternative, an interval of the form [mean flow − α·σ, mean flow + α·σ]. α is a coefficient the decision maker fixes in advance (typically between 0.10 and 0.30 in the literature); in DecisionMind this value is taken from the user, it has no default, and it must be stated explicitly with every run.

Ranking rule. Two alternatives are ranked only if one's interval lower bound is strictly greater than the other's upper bound. If the intervals overlap at any point, the two alternatives are treated as indifferent; even if base PROMETHEE's mean flows differ, an overlapping interval means this difference is accepted as "within the noise." For the table view, DecisionMind also renders this interval relation as a rank ordered by the size of the mean flow, but the underlying real output is the interval relation; even if two neighbouring alternatives appear consecutively in the table, there is no strict dominance between them if their intervals overlap.

DecisionMind takes the choice of preference function, the threshold values and the weights from the user in this extension; these work as in base PROMETHEE. What changes is the interval construction after the fifth step and the decision rule in the sixth step.

The order shown in the table is a view built on the midpoint of every alternative's interval; it is read with the same logic as base PROMETHEE's net-flow order, but it is not sufficient on its own. The real result is which pairs' intervals overlap and which do not. An alternative whose interval does not overlap with any other alternative's genuinely stands apart; an alternative whose interval overlaps with its neighbours' is treated as indifferent to them despite its position in the table, and this must be stated separately in the report. As α is increased, the intervals widen and more pairs become indifferent; as α is decreased, more pairs are strictly separated. The choice of α is therefore itself an analytical decision, not merely a computational detail.

Thus instead of writing:

"According to PROMETHEE III, A5 is first, A2 second, A4 third"

the report should read:

"A5's interval is strictly higher than every other alternative's interval, and its first place is robust. A2's and A4's intervals, however, overlap (with α = 0.16); there is no strict dominance between the two, and their appearing second and third in the table comes only from the midpoint comparison"

Use this when net flows come out close to one another and the question of whether this closeness is "a genuine difference or computational noise" matters. PROMETHEE III restrains the decision maker from speaking too precisely (the net flow always gives a complete ranking) and shows which differences are meaningful relative to the width of the interval. If a complete ranking is required and overlapping intervals are not acceptable, base PROMETHEE (net flow) should be used; PROMETHEE I's partial relation offers a similar honesty but rests directly on the φ⁺/φ⁻ comparison rather than an interval. Crisp PROMETHEE's exit conditions (the computational burden once the alternative set exceeds fifty) apply here too.

Forgetting that the divisor is m, not m-1, and carrying base PROMETHEE's flow formula over here. PROMETHEE III's mean signed flow is divided by m (the comparison with itself is included as a zero term); this differs from base PROMETHEE's φ⁺/φ⁻ computation, and mixing the two makes the numbers fail to reconcile.

Trying several values of α as if it were a sensitivity-scanning parameter and picking the one that "looks best." α is an analytical decision that must be fixed once before the study begins and explained in the report; typical values in the literature lie between 0.10 and 0.30. Searching for an α that steers the result in a desired direction undermines the method's honesty from the outset.

Mistaking the consecutive order in the table for strict dominance. As explained above, the table is built on the midpoint; the method makes no dominance claim between neighbouring alternatives whose intervals overlap.

Mistaking the interval width (σ) for a sign of poor data quality. σ shows how INCONSISTENT that alternative's signed differences against every other alternative are (far ahead of some rivals, far behind others); it has nothing to do with measurement error.

The governing principle is this:

PROMETHEE III's contribution is to show whether a small difference in the net flow is meaningful relative to the width of the interval; a ranking produced without α being chosen and explained in the report does not carry this contribution.