Methods · Defuzzification
Type Reduction Defuzzification
Type reduction first collapses a Type-2 fuzzy number, defined by an upper and a lower bound, into an interval, then reduces that interval's midpoint to a single crisp figure; the width of the interval also shows the size of the uncertainty the number carries.
Base method's data type: Fuzzy
What Is the Method?
In classical fuzzy numbers, every value's degree of membership is given by a single figure. In Type-2 fuzzy numbers, membership itself is uncertain: an upper bound (the most optimistic membership curve) and a lower bound (the most cautious membership curve) are defined, and the region lying between them is called the footprint of uncertainty. This is one way of modelling a situation where "the experts cannot even fully agree on the degree of membership itself." This card describes the defuzzification process that reduces such a Type-2 fuzzy number to a single crisp figure. The process has two stages: first type reduction (obtaining an interval from the upper and lower bounds), then taking that interval's midpoint. Its output is both a single crisp score and the interval of uncertainty surrounding that score.
The Philosophy Behind It
The idea is to preserve the uncertainty in an intermediate step first, rather than reducing it immediately to a single figure. The larger the gap between the upper and lower bounds, the greater the disagreement among experts about the degree of membership. The method developed by Karnik and Mendel (2001) derives two separate classical (Type-1) fuzzy numbers from these two bounds: one the "smallest centroid that could fall to the left," the other the "largest centroid that could fall to the right." These two values form an interval, and the interval itself directly shows the size of the uncertainty the Type-2 fuzzy number carries. In the final step, this interval's midpoint gives the single crisp figure the decision requires.
This philosophy carries a consequence: even if two alternatives' single crisp figures after type reduction come out close together, the width of their intervals can differ considerably. An alternative with a narrow interval carries an assessment on which the experts largely agree; an alternative with a wide interval, even if it reaches the same crisp figure, carries far more disagreement.
How It Works
The method proceeds through three steps.
First, define the Type-2 fuzzy number by its upper and lower bounds. The upper bound shows the most optimistic membership curve; the lower bound shows the most cautious membership curve. The region between them is the footprint of uncertainty, and it is processed by dividing it into discrete points along the x-axis.
Second, calculate the left and right bound values (the two ends of the interval). The Karnik-Mendel iterative algorithm first estimates a switch point. It determines which combination of the upper and lower bounds is used to the left and to the right of this point, and then calculates a new centroid using that combination. This operation is repeated until the result stops changing; it is carried out once for the "left" bound and once, separately, for the "right" bound.
Third, take the midpoint of the interval. Once the left and right bound values have been found, the single crisp figure the decision requires is the average of these two values. The width of the interval (right minus left) can additionally be reported as the size of the uncertainty.
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 single figure type reduction returns is a balanced summary of the uncertainty jointly represented by the upper and lower bounds. This figure must not be confused with the centroid the upper bound (the most optimistic curve) alone would give; defuzzifying the upper bound on its own means disregarding the lower bound (the cautious view) entirely, which amounts to ignoring the footprint of uncertainty. The width of the interval is itself informative: a narrow interval shows a strong expert consensus, a wide interval a serious difference of opinion.
Thus instead of writing:
"The defuzzified score is the centroid of the upper bound (the most optimistic curve)"
the report should read:
"The defuzzified score is the midpoint of the interval jointly formed by the upper and lower bounds; the width of the interval shows the size of the disagreement among experts and should be reported separately"
Data Type and Inputs
This rule works with Type-2 fuzzy data: an upper-bound and a lower-bound curve in every cell, on condition that the lower bound never exceeds the upper bound at any point. DecisionMind holds no independent extension for this building block on its own; it is used at the final step of extensions such as Type-2 fuzzy AHP and Type-2 fuzzy TOPSIS. You need, for every alternative, both an upper-bound and a lower-bound fuzzy number (usually triangular, defined by three vertices); the lower bound must sit entirely within the upper bound. The frequency of discretisation (the number of points into which the x-axis is divided) affects the stability of the result; using fewer than twenty points can disrupt the convergence of the iterative algorithm.
When to Use It, When Not To
Where experts disagree not only over "how well this criterion is met" but also over "how uncertain the degree of membership itself is," Type-2 fuzzy data and type reduction are appropriate. This carries a full order of magnitude more uncertainty than classical (Type-1) fuzzy modelling, and correspondingly demands more data collection and computational effort. Where experts broadly agree on the degree of membership itself, that is, where the uncertainty lies only at the level of "how well is the criterion met," classical triangular fuzzy data together with centroid, the bisector or MOM is sufficient; Type-2 modelling then adds needless complexity.
The degree of membership itself is uncertain, experts disagree on this → Type reduction
Uncertainty lies only at the level of "how well is the criterion met" → Centroid, the bisector or MOM
Membership/non-membership pair (intuitionistic fuzzy) data → Score function
The width of the interval must also be reported → Type reduction's natural output already provides this
Strengths
Type reduction's clearest strength is its ability to preserve uncertainty in an interval first, rather than reducing it immediately to a single figure; this interval offers information about the size of the disagreement among experts that classical fuzzy defuzzification rules cannot provide. The enhanced Karnik-Mendel iterations developed by Wu and Mendel (2009) have significantly reduced the computational burden, making the method usable in large-scale applications too. Compared with simplified approaches that rely on the upper bound or the lower bound alone, it accounts for both views, optimistic and cautious, in a balanced way.
Weaknesses
Type reduction's limitations arise from its computational complexity. As Liu (2008) showed, the classical Karnik-Mendel algorithm, being iterative, can remain slow on large datasets and requires more efficient strategies. Mendel, John and Liu (2006) proposed simplified forms of this iterative process to ease the application of Type-2 fuzzy systems, but every simplification carries the risk of overlooking part of the footprint of uncertainty. If the frequency of discretisation is not set high enough, the iterative algorithm can converge on the wrong point. Finally, collecting Type-2 fuzzy data, asking experts for both an upper and a lower bound for every cell, demands far more time and attention from them than classical fuzzy data does; whether this additional cost is worthwhile must be questioned separately in every application.
Common Mistakes
The most common mistake is applying the classical triangular centroid formula (the mean of the three vertices) directly to Type-2 fuzzy data and disregarding the footprint of uncertainty entirely; this loses all the information carried by the lower bound. A second mistake is setting the number of discretisation points too low (fewer than twenty); this can cause the iterative algorithm to stop at the wrong point. A third is confusing which bound, upper or lower, is used on which side for the left- and right-bound calculations (one assignment for the left-bound calculation, the reverse for the right); these two calculations require separate iterations and cannot be substituted for one another.
The governing principle is this:
Type reduction gives the midpoint of the interval jointly formed by the upper and lower bounds; the width of the interval is separate information, and a shortcut that looks only at the upper bound loses entirely the disagreement carried by the lower bound.
Cases
Each case begins with the Type-2 fuzzy assessment produced by a host method, describes in words what different readings do to it, and shows how to read the result.
1. Water Management: the combined score of two dam operation plans under fuzzy SAW (illustrative example)
A water management authority has used fuzzy SAW to choose between two dam operation plans. There is disagreement among expert teams over the plans' performance in reducing flood risk. This disagreement lies not only at the "how good" level but at the level of "how uncertain is the degree of membership itself"; the assessment was therefore made with Type-2 fuzzy numbers, and each plan was given an upper-bound (optimistic view) and a lower-bound (cautious view) triangle.
| Plan | Upper bound (a, b, c) | Lower bound (a, b, c) |
|---|---|---|
| Plan A | (0; 2; 10) | (6; 7; 8) |
| Plan B | (2; 6; 8) | (2.5; 3; 4) |
Looking only at the upper bound's own centroid (using the classical centroid formula), Plan A gives (0+2+10)/3 = 4.0 and Plan B gives (2+6+8)/3 = 5.33; Plan B appears ahead. Applying full type reduction (the Karnik-Mendel algorithm, upper and lower bounds together) gives Plan A 5.29 and Plan B 4.49; this time Plan A appears ahead.
| Plan | Upper bound alone | Rank (upper bound alone) | After type reduction | Rank (type reduction) |
|---|---|---|---|---|
| Plan A | 4.00 | 2 | 5.29 | 1 |
| Plan B | 5.33 | 1 | 4.49 | 2 |
The result reads as follows. Plan A's lower bound (the cautious view, narrow and high, at 6-7-8) differs considerably from its upper bound; this shows a wide gap between the optimistic and cautious views of Plan A among the experts, but also that the cautious view still points to a high value. When type reduction weighs these two views together, it lifts Plan A upward. For Plan B, by contrast, the lower bound (2.5-3-4) sits in a much lower region than the upper bound; the cautious view finds this plan poor, and type reduction takes this into account and pulls Plan B downward. A reading that looks only at the upper bound (the optimistic view) misses this difference entirely.
The authority hesitates here: choosing Plan B on the strength of the optimistic view alone would mean disregarding the cautious experts' serious reservations. In a decision with an irreversible consequence such as flood risk, the cautious view must be given its full weight too.
In the report: "According to the optimistic view alone, Plan B (4.00 against 5.33) appears ahead; however, type reduction, which also accounts for the cautious view, shows Plan A ahead instead (5.29 against 4.49), because for Plan A the gap between the two views closes in Plan A's favour."
Source: The plan scores in this case are constructed for illustrative purposes; the type-reduction algorithm is anchored to Karnik and Mendel (2001), and the figures were calculated and verified in Python (with 2000 discretisation points) while this card was prepared. This serves as the validation example for DecisionMind's type-reduction building block.
2. Aviation: defuzzifying runway maintenance priority
An airport operator is assessing surface wear data with Type-2 fuzzy numbers to decide which runway section should be maintained first. Because engineers disagree over the scale of wear itself, each section has been assessed with both an optimistic and a cautious triangle.
A reading that looks only at the optimistic view (the upper bound) may show one section as less of a priority. Once type reduction also accounts for the cautious view, this section's true priority can change; this is a critical difference for flight safety.
The team hesitates here: because runway maintenance is directly bound up with safety, a priority order based on the optimistic view alone is unacceptable. The width of the interval (the gap between the two views) should be reported separately for each section, and sections with a wide interval should receive additional inspection.
In the report: "Maintenance priority has been set according to the type-reduction result; on-site inspection has additionally been planned for the section with the widest interval."
3. Publishing: defuzzifying content-moderation priority
A digital publishing platform is assessing the reliability of automated classification with Type-2 fuzzy numbers to decide which content category should be prioritised for human moderation. Different moderation teams disagree over the degree of reliability of the classification, so each category has been given an optimistic and a cautious triangle.
A ranking that looks only at the optimistic view can leave a seemingly low-risk category outside the priority list. Once type reduction also incorporates the cautious team's view, this category can return to the priority list.
The team hesitates here: with limited human moderation capacity, which category is genuinely the priority must rest not on the optimistic estimate alone but on the full interval between the two views. Categories with wide intervals should be kept on the list despite their seemingly low scores.
In the report: "Moderation priority has been set according to the type-reduction result; categories with a wide difference of opinion between the two teams have been kept on the priority list despite their low scores."
4. What Not to Do
If Plan A's lower bound (6; 7; 8) from the water management case were disregarded and the value from the classical triangular centroid formula applied to the upper bound alone (0; 2; 10), which is 4.00, reported as the final score, the cautious expert view would be lost entirely. A second error is setting the number of discretisation points too low (ten, say) and halting the iterative algorithm before it has fully converged; this causes the left- and right-bound values to come out wrong. A third error is reporting the single figure after type reduction (5.29 for Plan A) without ever stating the width of the interval (the gap between the left and right bounds); this width is separate information showing the size of the disagreement among experts, and the single figure does not reflect it.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/defuzz-type-reduction
Karnik, N. N., & Mendel, J. M. (2001). Centroid of a type-2 fuzzy set. Information Sciences, 132(1–4), 195–220. DOI: 10.1016/S0020-0255(01)00069-X
Liu, F. (2008). An efficient centroid type-reduction strategy for general type-2 fuzzy logic system. Information Sciences, 178(9), 2224–2236. DOI: 10.1016/j.ins.2007.11.014
Mendel, J. M., John, R. I., & Liu, F. (2006). Interval type-2 fuzzy logic systems made simple. IEEE Transactions on Fuzzy Systems, 14(6), 808–821. DOI: 10.1109/TFUZZ.2006.879986
Wu, D., & Mendel, J. M. (2009). Enhanced Karnik-Mendel algorithms. IEEE Transactions on Fuzzy Systems, 17(4), 923–934. DOI: 10.1109/TFUZZ.2008.924329