Fuzzy Delphi

Fuzzy Delphi is an extension of classical Delphi and a method that converts expert opinion on criterion importance into a weight. Its output is a weight vector summing to one; it does not rank alternatives, and it works independently of ranking. In classical Delphi, every expert gives a single score for a criterion; in Fuzzy Delphi, an expert may instead give this score as a range, because even the uncertainty inside a single expert's own mind cannot be fully captured by one number. Introduced in Kaufmann and Gupta's 1988 book, the method has a wide range of applications, from criterion weighting to technology foresight, from education-programme evaluation to health-policy prioritisation.

The problem classical Delphi solves is disagreement between experts: as rounds proceed, experts converge on one another. Fuzzy Delphi adds a second layer of uncertainty to this. An expert may say, about a criterion, not "exactly 7" but "between 6 and 9, probably 7"; this is not the expert's indecision but an ambiguity inherent in the nature of judgement itself. The method handles these two kinds of uncertainty separately. It first pools each expert's own triangular range, then looks at how wide the range between experts is (whether they have converged), and finally reduces a wide triangle to a single number.

This carries a consequence: Fuzzy Delphi keeps the uncertainty in expert opinion hidden at the start of the calculation, carries it right to the end, and reduces it to a single number only when converting it into a weight. Reversing this order, reducing each expert's range to a single number first and then averaging, defeats the method's actual purpose, which is not to lose uncertainty too early.

The method proceeds through four steps.

First, expert assessment. Every expert gives every criterion a triangular fuzzy number: a worst case, a most likely value, a best case. If an expert feels no uncertainty, all three can be made the same number; in that case the triangle collapses to a single point and behaves like a crisp score.

Second, combining across experts. For every criterion, the smallest of all experts' worst-case values and the largest of their best-case values are taken; the most-likely values are combined through a geometric mean. This produces a single group triangle for each criterion. If the experts have given a single crisp number (no triangles), this step reduces to a geometric mean.

Third, checking for consensus. If the width of the group triangle (the gap between its largest and smallest end) falls below a pre-set threshold, consensus is considered reached. If the threshold is exceeded, a new Delphi round is needed; this card describes the single-round combination step.

Fourth, defuzzification and conversion to weight. The group triangle's three corners are averaged to reduce it to a single number (the centroid method). These figures are then divided by their own total to convert them into criterion weights summing to one.

The formulas behind each step are given on the DecisionMind Fuzzy Delphi method page; this card carries no formulas.

The weight is the experts' triangular ranges combined and reduced to a single number; this single figure no longer shows the uncertainty that lay beneath it. If two criteria's weights are close (even 0.365 against 0.365, exactly equal), this does not mean the experts truly saw these two criteria as equally important; the group triangles' widths may differ, and this width is lost in the final weight. A high weight does not mean "certainly important" but "the central value the experts reached in this round is high."

Thus instead of writing:

"Fuzzy Delphi measured this criterion's importance precisely"

the report should read:

"The centre of the experts' triangular ranges has given this criterion the highest weight; the width of the ranges (the tightness of the consensus) must be reported separately"

Fuzzy Delphi fundamentally works with fuzzy (triangular) data, but it also accepts a crisp number if an expert wants to give one; in that case the triangle collapses to a single point. In DecisionMind's Delphi family, Fuzzy Delphi is the fuzzy member alongside classical Delphi; within this two-member family, Fuzzy Delphi itself has no further extension. You need: at least two experts, at least two criteria, a triangular (or crisp) assessment from every expert for every criterion, and a consensus threshold. The method produces weights, it does not ask for weights from outside. Three to twelve criteria are typical, and at least five experts are typical for the statistical summary to be meaningful.

If experts struggle to give a single number for a criterion, say "roughly around this much, but I am not certain," and you want to factor this uncertainty in rather than ignore it, Fuzzy Delphi is a suitable choice. If experts already give clean, single-number scores and the only difference between them is group consensus, classical Delphi is sufficient; adding a fuzzy range adds unnecessary complexity. If experts can make a pairwise comparison in a single round, faster methods such as AHP, BWM or LBWA give a quicker result.

Expert uncertainty is expressed as a range, round-by-round consensus is sought → Fuzzy Delphi

Experts give a precise score, only group consensus matters → Classical Delphi

No time for round-by-round feedback, pairwise comparison is sufficient → AHP, BWM, LBWA

What is actually needed is not group consensus but the data's own variability → Entropy, CRITIC

Fuzzy Delphi's greatest strength is that it makes visible, separately, two different sources of uncertainty in expert opinion, disagreement between experts and an expert's own internal indecision. Classical Delphi measures only the first; Fuzzy Delphi carries both together and reduces them to a single number only at the last step. When an expert wants to say "I do not know exactly, but within this range," it does not force this into a single number, and this in turn allows a more honest collection of input, particularly on new or little-known topics.

The method's most contested point is the defuzzification step. When a fuzzy range is pooled and finally reduced to a single number, part of the information the range carried is lost in the report; some researchers criticise this as the method's "creating fuzziness at the start only to discard it at the end" (Saffie, Shukor and Rasmani, 2016). Second, the choice of the consensus threshold is subjective and carries the risk of being adjusted to fit the result if it is not fixed before the analysis begins. Third, taking the smallest and largest ends when combining across experts can let a single expert at one extreme widen the group range more than it should. Fourth, like classical Delphi, this method carries a cost in time and participant fatigue; when more than one round is required, the process can take weeks.

The most common mistake is collecting a single number instead of a triangular range from experts and then calling it "Fuzzy Delphi" after the fact; if uncertainty was never genuinely expressed as a range, the method's real contribution never comes into play. A second mistake is setting the consensus threshold after the analysis, by looking at the results. A third mistake is reporting the defuzzified weight without ever stating the width of the triangle beneath it (how tight the consensus is); two criteria can end up with the same weight even though one comes from a wide consensus and the other from a narrow one. A fourth mistake is leaving a single expert's extreme value untouched at the combination step, as is, without questioning how much it is skewing the group range.

The governing principle is this:

Fuzzy Delphi's weight is a central summary of the experts' triangular ranges; the wider the range behind this summary, the more fragile the weight, and the report must show this.