Extension card · Fuzzy
Fuzzy BWM (Guo and Zhao, 2017)
Fuzzy BWM is the form of BWM used when the expert's comparisons against the best and worst criterion are expressed not as crisp numbers but as triangular fuzzy numbers. Its output is not a ranking but a weight vector that carries the fuzziness through and is defuzzified only in the final step.
Base method
BWM →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Fuzzy →
What this data type is, when to use it, how to write it in a cell: the family's full account is here.
What Changes from the Base Method?
Three things change; the decision logic does not.
Cells. In crisp BWM the best-to-others and others-to-worst comparisons are a single integer on a 1–9 scale. Here every comparison is a triangular fuzzy number (lower, middle, upper); the expert can say "roughly 2 to 2.5 times, most likely 2 times" instead of "exactly 3 times". The scale is converted into a triangle using pre-declared verbal terms; for example, "equally important" = (1;1;1), "fairly important" = (1.5; 2; 2.5). The best-to-itself and worst-to-itself comparisons remain fixed at (1;1;1).
Solving the weights. In crisp BWM the weights and the consistency indicator (ξ*) are solved from a single non-linear model. Here the model is built with triangular fuzzy arithmetic: the worst criterion's weight is taken as an anchor, the other criteria's weights are derived from their others-to-worst comparisons, and their sums are normalised to 1. The consistency indicator, now itself a triangular number (ξ̃*), is computed within the same framework.
Defuzzification. Crisp BWM already produces a single number. Here the weights are first found in triangular fuzzy form, then reduced to a single number by a graded mean (l+4m+u, divided by 6), and their sum is renormalised to 1. The uncertainty is consumed not in the first step but in the last.
DecisionMind fixes, for classical Fuzzy BWM, this defuzzification rule (the graded mean) and the solution path that anchors on the worst criterion.
How to Read the Output
As in base BWM, the weight shows a criterion's relative importance to the others within this comparison set; the weights always sum to 1, and the consistency indicator (ξ̃*) confirms only the internal coherence of the comparisons, not the criteria's "true" importance.
The difference lies here. The weight has emerged, through defuzzification, from a chain of triangular fuzzy comparisons, and the defuzzified single number conceals this approximation. If two criteria's weights are close, whether this closeness is sensitive to the width of the expert's fuzzy comparison, that is, to how wide the triangle is, must be tested separately.
Thus instead of writing:
"Since the Fuzzy BWM weight came out at this level, the criterion's importance is this precise"
the report should read:
"This weight is the defuzzified form of the expert's fuzzy comparisons; had the verbal term used in the comparison been different (for example 'very important' instead of 'fairly important'), the size of the weight would also have differed"
When to Prefer This over the Base Method
This extension is used when the expert says "approximately" or "around" rather than giving a crisp number while comparing the others against the best and worst criterion, and when reducing this approximation to a single number would create an artificial precision. Expanding a measured comparison into a triangle, when, for example, the objective ratio of two criteria's weight is already known, is not modelling uncertainty but manufacturing it.
BWM's exit condition applies here too. If the best and worst criterion cannot be chosen with confidence, or if the fuzzy comparisons contradict one another seriously, that is, if the defuzzified consistency indicator sits clearly above the widely accepted threshold of 0.30, the weights should be reviewed with the expert again before use.
Mistakes Specific to This Extension
Confusing the best and worst criterion. As in crisp BWM, if best and worst are labelled the wrong way round, the meaning of the two comparison vectors is reversed.
Entering the best-to-itself or worst-to-itself comparison as a triangle other than (1;1;1). This breaches the method's basic rule.
Not declaring the verbal scale, or letting it vary from expert to expert. The triangular number corresponding to the term "fairly important" is fixed before the analysis; using a different triangle for the same term makes the results incomparable.
Defuzzifying in the first step. Reducing the comparisons to a single number first and then running crisp BWM is not Fuzzy BWM; the fuzziness is erased in the first step, and the information carried in the width of the comparison is lost.
The governing principle is this:
Fuzzy BWM weights are a defuzzified summary of the expert's fuzzy comparisons against the best and worst criterion; which verbal term and which defuzzification rule were used must be stated explicitly in the report.
Cases
The first case is DecisionMind's validation example. In the manifest, this two-criterion table does not come from the literature but from the smallest hand-traceable construction (n=2), faithful to the formulas. The second case is an illustrative construction.
1. Illustrative example: Weighting two criteria (DecisionMind validation example)
An e-commerce company will weight only two criteria before choosing new call-centre software: response speed and interface simplicity. This is deliberately set up with two criteria to show the method's smallest and plainest form; the team has designated response speed as the most important (Best) criterion and interface simplicity as the least important (Worst).
| Comparison | Verbal term | Triangle (lower; middle; upper) |
|---|---|---|
| Response speed (Best) → Response speed | Equally important | (1; 1; 1) |
| Response speed (Best) → Interface simplicity | Fairly important | (1.5; 2; 2.5) |
| Response speed → Interface simplicity (Worst) | Fairly important | (1.5; 2; 2.5) |
| Interface simplicity (Worst) → Interface simplicity | Equally important | (1; 1; 1) |
With only two criteria, these four judgements collapse into a single ratio; the method is solved directly in closed form, and the consistency indicator comes out at zero, because with two criteria there is no third comparison to conflict with.
| Criterion | Weight |
|---|---|
| Response speed | 0.6667 |
| Interface simplicity | 0.3333 |
The result can be read as follows. Response speed carries exactly twice the weight of interface simplicity; this follows directly from the middle value (2) of the "fairly important" term's triangle.
The team has one hesitation. What would happen if a term one notch stronger than "fairly important", namely "very important" (2.5; 3; 3.5), had been chosen instead? In that case the weights shift to 0.75 and 0.25; this has been confirmed with an independent Python calculation. In this simplest two-criterion case, which criterion carries more weight never changes, because Best is always heavier than Worst, but the size of the margin depends directly on the verbal term chosen.
In the report: "In this simplest two-criterion case, response speed comes out ahead with twice the weight of interface simplicity (0.6667 to 0.3333); this ratio depends directly on the triangle of the chosen term 'fairly important', and a stronger term would have pushed the ratio towards 3 to 1."
Source: DecisionMind Fuzzy BWM manifest, validation example; the two-criterion closed-form solution rests on Guo and Zhao's (2017) fuzzy BWM framework, and the step order on the formulation in Özdağoğlu and colleagues' (2025) book chapter. The weights have been confirmed with an independent Python calculation.
2. Local government: Weighting the criteria for an electric bus fleet tender
A municipality's transport department will weight three criteria before proceeding to an electric bus fleet tender: operating cost, charging time and domestic-production share. The department has designated operating cost as the most important (Best) criterion and domestic-production share as the least important (Worst), and has had an expert team complete the verbal comparisons.
The method solves the weights and the defuzzified consistency indicator from these fuzzy comparisons. Suppose operating cost receives the highest weight, domestic-production share the lowest, and the consistency indicator comes out below the acceptable limit.
The department has a hesitation. Some members of the council committee had argued that domestic-production share should not be treated as least important, given its political and strategic significance. The method works only from the given best/worst choice as supplied. Had this choice been different, that is, had domestic-production share been ranked in the middle rather than as the worst, the entire weight distribution would have had to be recalculated from scratch; the method does not say which of these two choices is "correct".
In the report: "The weights rest on the transport department's choice of operating cost as the most important and domestic-production share as the least important criterion; this initial choice is contested in the council committee, and the report must state this explicitly."
3. What Not to Do
The first error is entering the "fairly important" comparison as (1.5; 2; 2.5) in one place and as a different triangle, such as (1.4; 2; 2.6), elsewhere in the illustrative example. The same verbal term must correspond to the same numerical triangle.
The second error is entering the best-to-itself comparison as a triangle other than (1;1;1), such as (0.9;1;1.1). This breaches the method's basic rule.
The third error is reducing the triangles to their middle values from the outset and running crisp BWM. The result usually gives a similar weight ordering, but the information about how wide, that is how uncertain, the comparison was is lost, and the weight's sensitivity to the chosen term becomes invisible.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/fuzzy-bwm
Guo, S., & Zhao, H. (2017). Fuzzy best-worst multi-criteria decision-making method and its applications. Knowledge-Based Systems, 121, 23–31. DOI: 10.1016/j.knosys.2017.01.010
Rezaei, J. (2015). Best-worst multi-criteria decision-making method. Omega, 53, 49–57. DOI: 10.1016/j.omega.2014.11.009
Özdağoğlu, A., et al. (2025). Bulanık En İyi-En Kötü Yöntemi (Bölüm 10). In G. Demir (Ed.), Sosyal Bilimlerde Stratejik Karar Verme: Çok Kriterli Karar Verme Yöntemleri ile Uygulamalar. Özgür Yayınları. DOI: 10.58830/ozgur.pub768
Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353. DOI: 10.1016/S0019-9958(65)90241-X