Fuzzy LMAW (Božanić, Pamučar, Milić, Marinković and Komazec, 2022)

Three things change; the logarithmic-compression logic does not.

Cells. In LMAW's ranking form, every cell is one option's crisp performance on one criterion. Here there is no cell; instead, for every criterion, each expert chooses a single verbal term from a pre-declared nine-point priority scale. The scale begins with "Absolutely Low" (1; 1; 1), passes through "Medium" (2; 2.5; 3) and "Equal" (2.5; 3; 3.5), and ends with "Absolutely High" (4.5; 5; 5). With more than one expert, each expert's priority triangles are combined per criterion with a Bonferroni mean; with a single expert this step leaves the triangle unchanged.

Solving for the weights. In crisp LMAW, direction-sensitive standardisation is followed by a logarithmic transform applied to option performances. Here, the counterpart of standardisation is ratioing each criterion's priority triangle against a fixed absolute anti-ideal point (0.5; 0.5; 0.5); this ratio is the fuzzy, one-directional (only "how high is the priority") counterpart of LMAW's direction-sensitive standardisation. The logarithm of the resulting ratio triangles is taken and divided by the product of the criteria's middle values; this follows the same logic as crisp LMAW's second step, except that it now operates on the criteria's priority rather than their performance.

Defuzzification. Crisp LMAW already produces a single number. Here the weights are first found as triangular fuzzy numbers, then reduced to a single number by a graded mean (lower + 4×middle + upper, divided by six) and renormalised to sum to 1 again. Uncertainty is consumed not at the first step but at the last.

DecisionMind fixes, in classical Fuzzy LMAW, the anti-ideal point (0.5; 0.5; 0.5), the reduction of the Bonferroni step to an identity transform with a single expert, and the graded-mean defuzzification.

The weight is read as in base LMAW: a criterion's relative importance to the others within this comparison set; the weights always sum to 1. It is not a ranking and says nothing about any alternative.

The difference is here. The weight has emerged, through defuzzification, from a triangular fuzzy priority judgement, and the defuzzified single number conceals this approximation. If two criteria's weights sit close to one another, it must be tested separately whether this closeness is sensitive to the expert's chosen verbal term shifting by one step.

Thus instead of writing:

"Since fuzzy LMAW produced this weight, the criterion's importance is fixed at this level with certainty"

the report should read:

"This weight is the defuzzified form of the fuzzy priority term chosen by the expert; had the term shifted by one step, the size of the weight would have changed too"

This extension is used when experts give the criteria's priority not as a crisp number but as a verbal term such as "approximately" or "absolutely high", and reducing this approximation to a single number would create an artificial precision. If a priority is already measured, for instance if a criterion's objective weight is already known from another method, opening it out into a triangle does not model uncertainty, it manufactures it.

The exit condition of LMAW's crisp form does not apply here, because the nature of the task has changed: here no alternative is ranked, a criterion is weighted. The scale must be declared before the analysis and applied identically to every expert; otherwise the same verbal term corresponds to different numerical values.

Choosing an anti-ideal point larger than the scale's lower bound. As the manifest record itself warns, the anti-ideal point's (0.5; 0.5; 0.5) upper bound must be smaller than the lower bound of the lowest term on the scale; otherwise the inside of the logarithm falls below zero and the calculation is undefined.

Trying to apply the Bonferroni mean as it stands with a single expert. The Bonferroni mean requires more than one expert; with a single expert this step reduces to an identity transform (leaving the triangle unchanged). This is not a shortcut but the method's own rule.

Not declaring the verbal scale, or changing it from expert to expert. The triangular number that corresponds to the term "Absolutely High" is fixed before the analysis; using a different triangle for the same term makes the results incomparable.

Defuzzifying at the first step. Reducing the priority triangles to a single number at the outset and then running a crisp weighting method is not Fuzzy LMAW; the uncertainty is erased at the first step and the information carried by the term's width is lost.

The governing principle is this:

Fuzzy LMAW's weights are a defuzzified summary of the fuzzy priority term the expert gave to the criteria; which verbal scale and which anti-ideal point were used must be stated explicitly in the report.