Methods · Defuzzification
Mean of Maxima Defuzzification (MOM)
MOM finds the point at which a fuzzy number's degree of membership is highest (or, where the peak is flat, the midpoint of that plateau) and reports that point as a single crisp figure; it never looks at the fuzzy number's tails.
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. MOM (Mean of Maxima) performs this reduction by the most direct route: it finds the region where the fuzzy number is "most likely" and produces a single figure from there. For a triangular fuzzy number this is simply the peak itself. For a trapezoidal fuzzy number, membership reaches its highest value (usually 1) over a flat interval; MOM returns the midpoint of that interval. Its output is a single crisp score; it is unconcerned with how wide or narrow the tails are, concerned only with where the peak lies. MOM is one of the three most frequently cited classical defuzzification rules alongside centroid and the bisector, and its origin lies in fuzzy control systems.
The Philosophy Behind It
The idea behind MOM is "take the most likely value, and set the rest aside." A fuzzy number distributes different degrees of belief across a range of values; MOM is concerned not with the whole of this distribution but only with the point where the highest belief is concentrated. This stands opposite to centroid's philosophy of "take the whole area into account." According to MOM, however long or short a fuzzy number's tails may be, what matters for the decision is the "most likely" value; the tails only show the uncertainty surrounding that value, and should not change the decision itself.
This philosophy carries a consequence: MOM gives a different result from centroid for skewed (asymmetric) fuzzy numbers, because centroid takes the long tail into account and shifts in that direction, whereas MOM never shifts at all. Where an application calls for the "typical/most likely scenario," MOM answers the right question; where "a balanced summary of all possible scenarios" is wanted, centroid or the bisector answer a different question.
How It Works
The method proceeds through two steps.
First, find the point (or interval) at which the degree of membership is highest. For a triangular fuzzy number, membership reaches its highest value at a single point only, the peak. For a trapezoidal fuzzy number, membership holds its highest value over a flat interval between two vertices; the whole of this interval counts as the "peak."
Second, take the average of this point (or interval). For a triangular fuzzy number, since the peak is a single point, the result is that point directly and no further operation is needed. For a trapezoidal fuzzy number, since the peak is an interval, the result is the average of the interval's two ends.
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 MOM returns is the fuzzy number's "most likely" region. This figure is not "the average of the whole area"; if the fuzzy number is skewed, centroid moves away from this figure and shifts towards the longer tail, while MOM never shifts at all. If two alternatives' MOM scores differ, this difference reflects only the difference in their peak positions; the width of the tails, that is, the size of the uncertainty, is never reflected in this score. For this reason, two alternatives that look equal under MOM may in fact carry very different levels of uncertainty.
Thus instead of writing:
"The defuzzified score reflects the fuzzy number's general tendency (its average)"
the report should read:
"The defuzzified score reflects only the fuzzy number's most likely (peak) region; the width of the tails, that is, the size of the uncertainty, is not reflected in this score and must be reported separately"
Data Type and Inputs
MOM works with fuzzy data: a triangular or trapezoidal fuzzy number. DecisionMind holds no independent extension for this building block on its own; MOM 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; MOM 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
Where the application needs the "typical/most likely scenario," meaning the position of the peak matters and not the width of the tails, MOM is the correct choice. Being the lightest of the defuzzification rules to compute, it can also be preferred when a quick first reading is needed. Where the fuzzy number is markedly skewed and the width of the tails, that is, the size of the uncertainty, would affect the decision, MOM alone is not enough; centroid or the bisector then offer a summary that takes the tails into account too. Where the fuzzy number has a very wide flat peak (many values equally "most likely"), the single figure MOM returns can conceal this wide uncertainty; in that case reporting the peak interval itself is the more honest option.
"Typical/most likely scenario" is needed, and speed of computation matters → MOM
The width of the tails (uncertainty) affects the decision → Centroid
"The point that divides the area fairly" is needed → Bisector
The peak is a very wide flat interval → The peak interval itself should also be reported
Data that is intuitionistic (membership/non-membership pair) rather than fuzzy → Score function
Strengths
MOM's clearest strength is the ease of its computation: for a triangular fuzzy number, reading off the peak is all that is required, with no area calculation of any kind. It answers the question "what is the most likely scenario" directly, which in some applications, finding a device's most likely point of failure, say, is exactly the information sought. Because it is unaffected by extreme values in the tails, it offers an advantage in applications where a rarely observed extreme scenario should not be allowed to distort the result.
Weaknesses
MOM's limitations arise from this same narrow focus. Van Leekwijck and Kerre (2001) examined MOM's continuity problem in detail: even a small change in the fuzzy number (a slight widening of the peak interval, say) can cause the figure MOM returns to jump. This is because the method looks only at the peak and disregards entirely the information lying just outside it. Runkler (1997), comparing defuzzification rules, showed that MOM's disregard of the tails leads to a loss of information in applications where the tails carry meaningful content. Where a fuzzy number has a flat peak, MOM treats the whole of that interval as "equally likely," and cannot detect even a subtle gradient within the interval. Finally, MOM cannot distinguish between two alternatives carrying different levels of uncertainty (one with a narrow tail, one with a wide tail) if their peaks coincide.
Common Mistakes
The most common mistake is reporting the MOM score as "the average of the fuzzy number"; MOM reads the peak, not the average, and the two concepts coincide only for symmetric fuzzy numbers. A second mistake is applying MOM to a markedly skewed fuzzy number (with a very long tail) and never reporting the uncertainty carried by that tail; MOM does not see this uncertainty at all, so the report must state it separately. A third is comparing only the MOM scores of two alternatives without considering that they might be differently shaped fuzzy numbers, and concluding they are "equal" or "different"; very different tail widths can accompany the same MOM score.
The governing principle is this:
MOM gives a fuzzy number's most likely region; it does not see the width of the tails, so where the size of the uncertainty matters, this must be shown separately in the report.
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. Parks and Recreation: the combined score of two renewal projects under fuzzy SAW (illustrative example)
A municipality has used fuzzy SAW to prioritise between two park renewal projects. Experts gave each project a triangular fuzzy score on the criteria of usage intensity, maintenance cost and community demand, and SAW multiplied these scores by weights and summed them. The resulting aggregated fuzzy scores are as follows.
| Project | Aggregated fuzzy score (a, b, c) |
|---|---|
| Project A | (1; 4; 6) |
| Project B | (2; 3; 10) |
Project A's tails are relatively narrow, whereas Project B's right tail is very wide (3 to 10). Applying MOM gives Project A a peak of 4 and Project B a peak of 3; Project A appears ahead. Applying centroid gives Project A (1+4+6)/3 = 3.67 and Project B (2+3+10)/3 = 5.0; this time Project B appears ahead.
| Project | MOM score | Rank (MOM) | Centroid score | Rank (centroid) |
|---|---|---|---|---|
| Project A | 4 | 1 | 3.67 | 2 |
| Project B | 3 | 2 | 5.0 | 1 |
The result reads as follows. Project B's most likely value (3) is lower than Project A's (4); a majority of experts found a lower score most likely for Project B. But Project B's score carries a wide probability tail extending to a much higher value (10), meaning some experts assigned Project B a far higher probability of strong community demand. MOM never sees this wide tail and brings Project A ahead purely on the "most likely" scenario; centroid takes this wide tail into account and brings Project B ahead instead.
The municipal team hesitates here: if the budget is tight and the decision should follow the "most likely" scenario, Project A is the sensible choice. But if Project B's possibility of high demand is too large to ignore, looking only at MOM may miss this opportunity.
In the report: "Under the MOM rule Project A (most likely score 4) comes ahead, and under the centroid rule Project B (5.0) comes ahead; because Project B carries a wide tail of uncertainty, the results of the two rules should be considered together."
Source: The project scores in this case are constructed for illustrative purposes; the defuzzification formulas are anchored to Lee (1990) 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 MOM and centroid building blocks.
2. Fire Service: defuzzifying response priority
A fire service is combining building age, occupancy rate and rate of fire spread using fuzzy SAW to decide which of two simultaneous call-outs should receive priority. The aggregated fuzzy urgency score of two call-outs has been calculated; one has a low peak but a very wide tail, the other a high peak but a narrow tail.
When MOM is applied, only the peaks are considered, and the call-out with the narrow tail and high peak may come ahead. But the wide-tailed call-out's possibility of reaching a much higher urgency in the "worst case" never appears in the figure MOM reports.
The team hesitates here: in urgent response, should the "most likely scenario" or "the probability of the worst scenario" take priority. With life safety at stake, relying on MOM alone can be risky, and the width of the tail should be assessed separately.
In the report: "Under MOM, priority falls to the first call-out; however, the second call-out's wide urgency tail shows that this priority could reverse under a worst-case scenario."
3. Telecommunications: defuzzifying base-station fault priority
A telecommunications operator is combining number of users affected, outage duration and the criticality of the region using fuzzy SAW to decide which base-station fault should be repaired first. The aggregated fuzzy scores of two faults have been calculated, each skewed in a different way.
When MOM is applied, only the most likely impact is considered; a fault's possibility of a rare but very wide-reaching outage is not reflected in this score. The operations team must decide whether this possibility can be disregarded.
The team hesitates here: MOM's simplicity makes for a quick decision, but a wide-tailed fault's true cost can remain invisible under MOM. Where critical infrastructure is concerned, the width of the tail should also be reported.
In the report: "The repair order was set according to the MOM score; because the second fault carries a wide impact tail, this order will be reconsidered should additional resources become available."
4. What Not to Do
If Project B's score (2; 3; 10) from the parks and recreation case were treated as a trapezoidal fuzzy number and an interval average computed in place of the peak, the result would be incorrect; this score is triangular and has a single peak (3). A second error is reporting the MOM score as "Project B's general tendency is 3"; MOM gives only the most likely value, and Project B's total area and wide tail are not reflected in this figure. A third error is comparing Project A's and Project B's MOM scores and declaring the 1-point gap between them "certain and beyond dispute"; this gap reflects only the peak positions and can reverse under another rule such as centroid.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/defuzz-mom
Lee, C. C. (1990). Fuzzy logic in control systems: fuzzy logic controller, Part I. IEEE Transactions on Systems, Man, and Cybernetics, 20(2), 404–418. DOI: 10.1109/21.52551
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
van Leekwijck, W., & Kerre, E. E. (2001). Continuity focused choice of maxima: yet another defuzzification method. Fuzzy Sets and Systems, 122(2), 303–314. DOI: 10.1016/S0165-0114(00)00025-7
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