Methods · Defuzzification
Bisector Defuzzification (Bisector Method)
The bisector finds the vertical line that splits the area under a fuzzy number exactly in half, and reports the point where that line falls as the single crisp number.
Base method's data type: Fuzzy
What Is the Method?
Defuzzification is the final step that reduces a fuzzy result to a single reportable crisp number. A method such as fuzzy SAW produces a three-cornered fuzzy score for every alternative; before this score can be fed into a ranking, it must first be reduced to a single number. The bisector performs this reduction on the idea of "splitting the area fairly": it finds the point with half the fuzzy number's area to its left and half to its right. Its output is a single crisp score; it produces no weight and performs no ranking, it only converts a fuzzy number into a number. The bisector is one of the three most frequently cited classical defuzzification methods, alongside the centroid and the mean of maxima (MOM); its origin lies in fuzzy control systems, and it is also used at the final step of fuzzy multi-criteria decision methods.
The Philosophy Behind It
The idea behind the bisector is a kind of areal median. If a fuzzy number is thought of as a mass represented by the area beneath it, the bisector searches for the point that splits this mass exactly in two. This differs from the centroid, which finds the centre of mass; the centroid takes the average of the area, whereas the bisector finds the boundary that leaves half the area on one side and half on the other. If the fuzzy number is symmetric, the two approaches meet at the same point. If the number is skewed in one direction, meaning one tail is longer than the other, the two methods diverge, and which one is chosen can change the result.
This philosophy carries a consequence. In an unbalanced triangle, the bisector shifts less towards the side where most of the area lies than the centroid does, because the only thing that concerns it is whether the area is split equally between the two sides, not where the area is concentrated. If an application calls for "a fair middle of the possible values", the bisector is a suitable choice; if what is wanted is "the most likely value itself" or "the average of all possibilities", the method's philosophy answers a different question.
How It Works
The method proceeds through three steps.
First, finding the total area. For a triangular fuzzy number, this area is tied to the triangle's base length; for a trapezoidal fuzzy number, the area consists of two triangles plus a rectangle. This total area is the reference quantity for the "equal split" being sought.
Second, searching for the point that splits the area in two. The method searches for the point x whose area to the left equals half the total area. In this search, the area of the triangle's left half (between the base and the apex) is computed first and compared with half the total. If the left half's area is smaller than this half, the point being sought lies to the right of the apex; if larger, it lies to the left.
Third, applying the closed-form formula. If the fuzzy number is symmetric, the bisector is equal directly to the apex and also coincides with the centre of mass. If it is not symmetric, the bisector is computed from whichever half was identified in the second step, and it generally comes out different from the centre of mass.
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 number the bisector gives is the point that splits the area beneath the fuzzy number into two equal halves. This number is not the fuzzy number's "most likely value"; the most likely value is the apex of the triangle, and the bisector can differ from it. Nor is this number "the average of all values"; that is the separate job of the centroid method. If the fuzzy number is symmetric, all three numbers (apex, centre of mass, bisector) coincide, and it makes no difference which method is chosen. If the number is skewed in one direction, the three diverge, and the method chosen can produce a different crisp score, and therefore sometimes a different ranking, from the same fuzzy data.
Thus instead of writing:
"The defuzzified score is the fuzzy number's most likely value"
the report should read:
"The defuzzified score is the point that splits the area beneath the fuzzy number equally in two; if the number is symmetric it coincides with the most likely value and the average, if not it may diverge from both"
Data Type and Inputs
The bisector works with fuzzy data: a triangular or trapezoidal fuzzy number. In DecisionMind, this building block has no independent extension; the bisector is not a family on its own but a component used at the final step of other fuzzy methods (such as fuzzy SAW, fuzzy TOPSIS, fuzzy VIKOR). You need a valid triangular fuzzy number (three corners, ordered from smallest to largest) or trapezoidal fuzzy number (four corners, ordered from smallest to largest). This number is usually the aggregated fuzzy score coming out of a host method's weighted-summation or distance step; the bisector clears up this score and prepares it for ranking. There is no limit on the number of alternatives or criteria; the method is applied separately to every alternative's own fuzzy score.
When to Use It, When Not To
If the aggregated fuzzy score is markedly skewed (one tail longer than the other) and the "fair middle of the area" logic suits the report's needs, the bisector is a suitable choice. If the fuzzy number is symmetric, the bisector gives the same result as the centroid; in that case the more widely known and less computationally demanding centroid can be preferred, and choosing the bisector brings no extra gain. If the application needs "the most likely/typical value", meaning the peak of the uncertainty rather than its central tendency matters, MOM (the mean of maxima) is the more accurate answer.
A skewed fuzzy number, area fairness is the priority → Bisector
A symmetric fuzzy number → the bisector and the centroid give the same result; the centroid suffices
"The most likely/typical value" is needed → MOM
The width of the resulting interval must also be reported → Type Reduction for Type-2 fuzzy numbers
Data is intuitionistic, not fuzzy (a membership/non-membership pair) → Score Function
Strengths
The bisector's clearest strength is its guarantee of splitting the area "fairly": the area to its left and to its right are always equal, a property that holds by definition. In skewed fuzzy numbers it carries more information than looking only at the apex (MOM), because it also accounts for the size of the tails. The concept is relatively easy to teach: "the line that splits the area equally in two" is an intuitive definition and can be shown on a graph.
Weaknesses
The bisector's limitations also come from its definition. Runkler (1997), comparing which application properties different defuzzification methods suit, showed that the bisector's computational load is somewhat higher than the centroid's, because it first requires finding which half the point falls in. Talon and Curt (2017), in a real engineering application (dam-performance assessment), showed that different defuzzification methods produce visibly different scores from the same fuzzy input, and that the choice must be justified. The bisector itself uses a closed-form formula that depends on "which half" for asymmetric numbers; this formula grows more complex for trapezoidal numbers and carries a risk of error in manual computation. Finally, because the bisector is not as widely known as the centroid, an expert reading the report may assume it is the centroid without asking which defuzzification rule was used.
Common Mistakes
The most common mistake is confusing the bisector with the centroid and assuming the two always give the same result; this is true only for symmetric fuzzy numbers. A second mistake is applying the closed-form formula valid for a triangular fuzzy number to a trapezoidal fuzzy number without checking; the area calculation for a trapezoidal number requires an additional term. A third mistake is applying a formula directly without checking which half (left or right) the point falls in; skipping this step means the wrong segment's formula is used and the result comes out wrong.
The governing principle is this:
The bisector gives the point that fairly splits the area beneath a fuzzy number; this point coincides with the most likely value and the average only for symmetric numbers, and the report must state clearly which defuzzification rule was used.
Cases
Each case starts from an aggregated fuzzy score produced by a host method (usually fuzzy SAW), describes in words what different defuzzification rules do to that score, and shows how to read the result.
1. Software Procurement: the total score of two supplier bids in fuzzy SAW (illustrative example)
A company used fuzzy SAW in choosing an enterprise software supplier. Experts gave each of two suppliers a triangular fuzzy score on every criterion, and SAW multiplied these scores by the weights and summed them. The resulting aggregated fuzzy scores are as follows:
| Supplier | Aggregated fuzzy score (a, b, c) |
|---|---|
| T1 | (2.5; 3; 10) |
| T2 | (0; 7; 7.5) |
Both scores are markedly skewed: T1 has a long right tail (from 3 to 10), T2 a long left tail (from 0 to 7). Defuzzification reduces these two scores to a rankable single number; which rule is chosen makes a difference here.
With the centroid (centre of mass) applied, T1 comes out at (2.5+3+10)/3 = 5.17 and T2 at (0+7+7.5)/3 = 4.83; T1 appears ahead. With the bisector applied, T1 comes out at 4.88 and T2 at 5.12; this time T2 appears ahead.
| Supplier | Centroid score | Rank (centroid) | Bisector score | Rank (bisector) |
|---|---|---|---|---|
| T1 | 5.17 | 1 | 4.88 | 2 |
| T2 | 4.83 | 2 | 5.12 | 1 |
The result reads as follows. T1's score carries a long tail stretching to the right; because the centroid folds this tail into the average, it pulls T1 upward. The bisector, searching instead for the point that leaves half the area on the left, shows that T1's wide right tail pushes the midpoint backward rather than forward. The opposite happens for T2: its left tail is wide, and the bisector weighs this width differently.
The procurement team hesitates here: the two suppliers' scores sit very close together, and the order reverses depending on which defuzzification rule is chosen. In this situation, rather than deciding with a single rule, both should be reported and the experts asked which logic, area average or area fairness, they find more appropriate.
In the report: "Under the centroid rule T1 (5.17) comes ahead, under the bisector rule T2 (5.12) comes ahead; because the two suppliers' scores sit close together, the order is sensitive to the defuzzification rule, and the decision should be presented together with this sensitivity."
Source: The supplier scores in this case were constructed for illustrative purposes; the defuzzification formulas are anchored to van Leekwijck and Kerre (1999), and the figures were computed and verified in Python while preparing this card. This is the validation example for DecisionMind's bisector and centroid building blocks.
2. Care Home: defuzzifying a care-priority score
A care home uses a three-criterion fuzzy scoring system to determine which resident its daily care team should attend to first: mobility restriction, medication-tracking risk, and the number of complaints in the last 24 hours. Expert nurses assessed each criterion with a triangular fuzzy number, and the weighted summation produced a fuzzy priority score for every resident.
Suppose one resident's score comes out markedly skewed to the right: the lowest and most likely values sit close together, while the highest value sits much further away. The centroid pulls the score upward because it folds this wide tail into the average; the bisector, searching for the point that leaves half the area on the left, leaves the score in a more cautious place. The two rules can change this resident's priority order.
The team hesitates here: for care priority, is it safer to "also account for the worst possible case" or to "split the area fairly"? Declaring a single priority order without showing the results of both defuzzification rules together risks conflicting with the nurses' own clinical judgement.
In the report: "The resident's priority score shifts by one rank depending on the defuzzification rule chosen; the results of both rules have been shown together and left to the clinical team's final approval."
3. Courier: defuzzifying a delivery-route priority
A courier company combines traffic density, package fragility and customer waiting time with fuzzy SAW to prioritise routes for same-day delivery. Two candidate routes' aggregated fuzzy scores come out, one skewed left, the other skewed right.
Assessing the right-skewed route with the centroid, the method puts it ahead, because the long tail pulls the average upward. Assessing the same route with the bisector, the area-fairness criterion weighs the tail differently and the order can change. The operations team wants to see this difference before deciding which route is genuinely the higher priority.
The team hesitates here: morning traffic forecasts can change during the day, which can also change the shape of the routes' fuzzy scores, and therefore which tail is long. Rather than committing to a fixed defuzzification rule, the team should monitor daily which rule the sensitive route's result depends on.
In the report: "The priority order between the two routes is sensitive to the defuzzification rule; both rules' results should be re-checked alongside the morning traffic update."
4. What Not to Do
If T1's score (2.5; 3; 10) in the software-procurement case were treated as if it were a trapezoidal fuzzy number and the "area" computed directly without applying the triangular formula, the total area comes out wrong and the bisector point is found wrong as well. A second mistake is choosing a formula without checking which half (left or right) the point falls in; because T2's score (0; 7; 7.5) is wide on the left, applying the right-segment formula renders the result meaningless. A third mistake is reporting the bisector score as "T1's most likely value is 4.88"; 4.88 is the point where the area is split fairly, while the most likely value (the apex) is a separate number, here 3.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/defuzz-bisector
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
Filev, D. P., & Yager, R. R. (1991). A generalized defuzzification method via BADD distributions. International Journal of Intelligent Systems, 6(7), 687–697. DOI: 10.1002/int.4550060702