Methods · Defuzzification
Centroid Defuzzification (Centre of Gravity)
Centroid finds the centre of gravity of the area beneath a fuzzy number and reports that point as a single crisp figure; this makes it the defuzzification rule most often used in fuzzy control and fuzzy decision methods.
Base method's data type: Fuzzy
What Is the Method?
Defuzzification is the final step that reduces a fuzzy result to a single crisp number. A fuzzy method (fuzzy SAW or fuzzy TOPSIS, say) produces a three-cornered fuzzy score for every alternative; to place these scores in a ranking, each must first be reduced to one number. Centroid performs this reduction by borrowing the idea of a physical centre of gravity: think of the area beneath the fuzzy number as a flat plate, and centroid finds the point at which that plate would balance. Its output is a single crisp score; it produces no weights and eliminates no alternatives, only converting a fuzzy number into a number. Centroid has been the standard defuzzification rule of fuzzy systems since Mamdani and Assilian's (1975) original fuzzy controller experiment. Most fuzzy multi-criteria methods use this rule at their final step.
The Philosophy Behind It
The idea behind centroid is "take the average," but not an ordinary average: an average weighted by area. Every point of the fuzzy number carries a "weight" proportional to its own degree of membership, and centroid finds the point at which these weights balance. For a triangular fuzzy number this equals the arithmetic mean of the three vertices; if the number is symmetric, this point already coincides with the peak, and if it is skewed, it departs from the peak and shifts towards the longer tail.
This philosophy carries a consequence: centroid takes the entire shape of the fuzzy number into account, both tails included. Where an application calls for "a balanced summary representing the whole range of the uncertainty," centroid answers the right question. Where only "the most likely value" or only "the fair midpoint of the area" is wanted, other rules such as MOM or the bisector answer a different question.
How It Works
The method proceeds through three steps.
First, establish the general idea of a centre of gravity. The area beneath the fuzzy number's membership curve is treated as a mass; the point sought is that mass's centre of gravity on the x-axis.
Second, apply the closed-form formula for a triangular fuzzy number. Because a triangle's area can be computed as a triangle's area, its centre of gravity also equals the arithmetic mean of the three vertices. This is a practical shortcut that makes centroid calculable by hand.
Third, apply the closed-form formula for a trapezoidal fuzzy number. A trapezoidal fuzzy number's area consists of two triangles plus a rectangle; its centre of gravity is found by taking the area-weighted average of these three sub-areas' own centres of gravity. The triangular formula (the mean of three vertices) cannot be applied directly here.
The formulas behind each step, the intermediate tables and the citation formats are given on the DecisionMind method page; this card carries no formulas.
How to Read the Output
The figure centroid returns is the area-weighted average of the fuzzy score. This figure is not "the most likely value"; if the fuzzy number is skewed, centroid departs from the peak (the most likely value) and shifts towards the longer tail. If two alternatives' centroid scores sit close together, this shows that the two fuzzy scores' total areas balance in a similar way, but the shape of these two scores (the length of the tails, the position of the peak) may differ considerably. Centroid alone does not reveal this difference in shape.
Thus instead of writing:
"The defuzzified score is the most likely value of the fuzzy number"
the report should read:
"The defuzzified score is the centre of gravity of the fuzzy number's area; it coincides with the most likely value when the number is symmetric, and shifts towards the longer tail when it is skewed"
Data Type and Inputs
Centroid works with fuzzy data: a triangular or trapezoidal fuzzy number. DecisionMind holds no independent extension for this building block on its own; centroid is not a family in its own right but a component used at the final step of other fuzzy methods (fuzzy SAW, fuzzy TOPSIS, fuzzy VIKOR, and the like). You need a valid triangular fuzzy number (three vertices, ordered from smallest to largest) or trapezoidal fuzzy number (four vertices, ordered from smallest to largest). This number is usually the aggregated fuzzy score emerging from a host method's weighted-sum or distance step; centroid defuzzifies this score and readies it for ranking. There is no limit on the number of alternatives or criteria.
When to Use It, When Not To
Centroid is a fitting choice whenever a balanced figure is needed that summarises the aggregated fuzzy score's general tendency, that is, both its peak and its tails together. Being the best known and the most easily hand-checked of the defuzzification rules, it can also serve as the default choice absent a reason to prefer another. Where the application needs only the "most likely/typical value," meaning the position of the peak matters and not the width of the tails, MOM answers the more accurate question. Where the "fair midpoint of the area" is wanted (equal area to the left and right), the bisector is preferred.
A single figure summarising the general tendency (peak and tails together) → Centroid
"Most likely/typical value" is needed → MOM
"The point that divides the area fairly" is needed → Bisector
The width of the uncertainty must also be reported → Type reduction for Type-2 fuzzy numbers
Data that is intuitionistic (membership/non-membership pair) rather than fuzzy → Score function
Strengths
Centroid's clearest strength is the ease of its computation: for a triangular fuzzy number it amounts to no more than averaging three vertices, and it can be checked by hand with ease. Being the most widely known and most published defuzzification rule, it gives a reader wishing to compare across different studies a common point of reference. Because it takes the entire shape of the fuzzy number into account, both tails included, it carries more information than MOM, which looks only at the peak.
Weaknesses
Centroid's limitations arise from this same inclusiveness. Runkler (1997), comparing different defuzzification methods, showed that centroid can be disproportionately affected by a long tail, meaning a single extreme value can shift the result markedly. Talon and Curt (2017), in a genuine engineering application (assessing dam performance), showed that centroid and other defuzzification rules produce different scores from the same fuzzy input, and that this difference must be reported. For trapezoidal fuzzy numbers the triangular formula (the mean of the three vertices) cannot be applied directly; a separate formula is required, and applying the wrong one without remembering this is a common error. Finally, because centroid reports the balance of the area rather than the peak itself, it can mislead a reader expecting a "most likely" interpretation.
Common Mistakes
The most common mistake is applying the triangular fuzzy number formula (the mean of three vertices) to a trapezoidal fuzzy number without checking, which produces an incorrect result. A second mistake is reporting the centroid score as "the most likely value," which is true only for symmetric fuzzy numbers. A third is assuming, whenever two alternatives' centroid scores come out close together, that the shapes of their fuzzy scores are also similar; very differently shaped fuzzy numbers (narrow and symmetric, or wide and skewed) can share the same centroid.
The governing principle is this:
Centroid gives the area-weighted average of a fuzzy number; this figure summarises but does not display the number's shape, that is, the width of its tails, so the report must state clearly which defuzzification rule was used.
Cases
Each case begins with the aggregated fuzzy score produced by a host method (usually fuzzy SAW), describes in words what different defuzzification rules do to it, and shows how to read the result.
1. Public Transport: the combined score of two bus routes under fuzzy SAW (illustrative example)
A municipality has used fuzzy SAW to choose between two routes for a new bus line. Experts gave each route a triangular fuzzy score on the criteria of passenger potential, operating cost and traffic risk, and SAW multiplied these scores by weights and summed them. The resulting aggregated fuzzy scores are as follows.
| Route | Aggregated fuzzy score (a, b, c) |
|---|---|
| Route A | (0; 5; 6) |
| Route B | (2; 4; 10) |
Both scores are skewed, but in different directions: Route A's left tail is long (0 to 5), Route B's right tail is long (4 to 10). Defuzzification reduces these two scores to a single rankable number.
Applying centroid gives Route A (0+5+6)/3 = 3.67 and Route B (2+4+10)/3 = 5.33; Route B appears ahead. Applying MOM (mean of maxima) gives Route A 5 and Route B 4; this time Route A appears ahead.
| Route | Centroid score | Rank (centroid) | MOM score | Rank (MOM) |
|---|---|---|---|---|
| Route A | 3.67 | 2 | 5 | 1 |
| Route B | 5.33 | 1 | 4 | 2 |
The result reads as follows. Route B's most likely value (its peak) is lower than Route A's (4 against 5). But Route B's wide tail extending to the right pulls the centroid calculation upward and brings Route B ahead. MOM, looking only at the peak, does not see this wide tail and brings Route A ahead instead. The two rules produce opposite rankings from the same fuzzy data.
The municipal team hesitates here: Route B's wide tail shows that the experts were less certain about this route's passenger potential, that is, it carries greater uncertainty. Whether the correct approach is to reward this uncertainty (centroid) or to disregard it and look at the most likely scenario (MOM) depends on the decision-maker's preference and must be stated clearly in the report.
In the report: "Under the centroid rule Route B (5.33) comes ahead, and under the MOM rule Route A (5) comes ahead; the ranking depends on the defuzzification rule because Route B's score carries a wide tail of uncertainty."
Source: The route scores in this case are constructed for illustrative purposes; the defuzzification formulas are anchored to Mamdani and Assilian (1975) and van Leekwijck and Kerre (1999), and the figures were calculated and verified in Python while this card was prepared. This serves as the validation example for DecisionMind's centroid and MOM building blocks.
2. Librarianship: defuzzifying collection-expansion priority
A public library is combining reader demand, existing stock shortfall and lack of digital access using fuzzy SAW to decide which subject heading should receive budget priority. The aggregated fuzzy score of two subject headings comes out narrow and symmetric for one, wide and right-skewed for the other.
When centroid is applied, the wide-tailed heading may come ahead because the tail pulls the average upward, even though this heading's peak (its most likely value) is lower than the other's. The library manager must decide whether the budget should be steered by "average expectation" or by "most likely demand."
The manager hesitates here: a wide tail can mean that some experts forecast very high demand for this subject heading while others forecast low demand. A single centroid figure conceals this disagreement, and the report must show it too.
In the report: "The subject heading's centroid score is high owing to a wide tail arising from disagreement among the experts; the budget decision should be presented together with a statement of this disagreement."
3. Food Safety: defuzzifying supplier inspection priority
A food inspection agency is combining past violation count, cold-chain risk and complaint density using fuzzy SAW to decide which supplier should be inspected first. The aggregated fuzzy scores of two suppliers have been calculated, one close to symmetric, the other markedly skewed.
When centroid is applied, the skewed supplier can move ahead or fall behind depending on the direction of its tail. The inspection team must distinguish whether this shift reflects a genuine increase in risk or merely a computational effect.
The team hesitates here: with limited inspection capacity, which supplier is visited first should not hinge on a small difference in the centroid score. If the scores are close, additional evidence, such as the past three months' complaint records, should support the ranking.
In the report: "The two suppliers' centroid scores came out close; the inspection order was based not on this score alone but also on recent complaint records."
4. What Not to Do
If Route B's score (2; 4; 10) from the public transport case were treated as a trapezoidal fuzzy number and computed with the wrong formula instead of the mean of three vertices, the result would be incorrect. A second error is reporting the centroid score as "the most likely value": Route B's centroid score is 5.33, but its most likely value (its peak) is 4, and the two are not the same thing. A third error is comparing Route A's and Route B's centroid scores and declaring the 1.66-point gap between them "certain and beyond dispute"; this gap arises from the two scores having very different shapes (one symmetric, one wide-tailed) and can reverse under another rule such as MOM.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/defuzz-centroid
Mamdani, E. H., & Assilian, S. (1975). An experiment in linguistic synthesis with a fuzzy logic controller. International Journal of Man-Machine Studies, 7(1), 1–13. DOI: 10.1016/S0020-7373(75)80002-2
van Leekwijck, W., & Kerre, E. E. (1999). Defuzzification: criteria and classification. Fuzzy Sets and Systems, 108(2), 159–178. DOI: 10.1016/S0165-0114(97)00337-0
Runkler, T. A. (1997). Selection of appropriate defuzzification methods using application specific properties. IEEE Transactions on Fuzzy Systems, 5(1), 72–79. DOI: 10.1109/91.554449
Talon, A., & Curt, C. (2017). Selection of appropriate defuzzification methods: Application to the assessment of dam performance. Expert Systems with Applications, 70, 160–174. DOI: 10.1016/j.eswa.2016.09.004