Fuzzy WPM (Kahraman, Birgün and Yenen, 2008)

Three things change; the multiplicative logic does not.

Cells. In crisp WPM every cell is a single, strictly positive number. Here every cell is a triangular fuzzy number (lowest, most likely, highest), and it too must be strictly positive: a zero or negative corner leaves the product undefined. Criterion weights are also entered as triangles; DecisionMind never reduces the weight to a crisp value in this extension, and each corner (l, m, u) is paired with its own weight corner (w_l, w_m, w_u). This is where it departs from Fuzzy CoCoSo in the same family: there, the weight triple is reduced to its central value at the start of the calculation, whereas here the three corners are carried separately throughout.

Scale equalisation. Crisp WPM scales every cell against its own column's best value: it divides by the column's largest value for a benefit criterion, and divides the column's smallest value by the cell for a cost criterion. This scaling is the source of WPM's unit independence. The source paper (Kahraman, Birgün and Yenen, 2008) takes a different path here: no column scaling is applied at all; the raw TFN magnitudes are raised directly to the power of their own weight corners. The book's own worked example confirms this. The four criteria's raw values range from 3 to 52 and are of very different magnitudes; they are used directly, without any scaling, and match the book's reported result exactly (Case 1 of this card).

Handling direction (cost criteria). In crisp WPM a cost criterion is handled with the rule "divide the smallest by the cell." Here, for a cost criterion, both the value triple and the weight triple are inverted: (l, m, u) becomes (1/u, 1/m, 1/l), and (w_l, w_m, w_u) becomes (w_u, w_m, w_l). Inverting only the value and not the weight, or the reverse, produces an invalid triple that breaks the corner order (lowest ≤ most likely ≤ highest). This is why both are inverted together.

Result and defuzzification. The output is first a TFN: V(A_i) = (l-product, m-product, u-product). Defuzzification happens only at the final step, using the central value ((l+m+u)/3). This is where Fuzzy WPM differs from Fuzzy CoCoSo, which defuzzifies immediately after normalisation (before S/P), and from IF-CoCoSo, which defuzzifies right at the start, as soon as the cell is read. In Fuzzy WPM, fuzziness is carried all the way to the end of the multiplication.

DecisionMind keeps two things fixed in this extension. It uses the central value (centroid) as the defuzzification method at the final step, and preserves the source's own choice of "no column scaling, use the raw value directly." Weights are taken from outside; the method does not generate weights.

As in crisp WPM, the central-value (centroid) score shows a ranking within this alternative set. It is not a percentage or a probability. Nor can it be compared with a score from a different Fuzzy WPM analysis, because no column scaling is applied. The score's absolute magnitude therefore depends directly on the criteria's raw scale; for instance, the magnitude of net present value in local currency directly affects the score.

The score also cannot be compared with the score of an additive fuzzy method such as Fuzzy SAW; the two sit on different scales. This warning also appears in K.common_pitfalls. The triangle's three corners, that is, V's l, m and u components, can be read separately before they collapse to the central value. The gap between the corners shows how much of that alternative's input uncertainty has carried through to the final product. If two alternatives' central values are close but their corner spreads are very different, the alternative with the narrower corners represents a more reliable advantage.

Thus instead of writing:

"FMS_1's central score (15.18) is higher than FMS_2's (12.00), so FMS_1 is clearly better"

the report should read:

"FMS_1's central score is higher than FMS_2's; however, this figure shows only the relative ranking between these two alternatives, it is not an absolute percentage, and the score itself changes scale if the raw magnitude of one of the criteria, such as net present value, changes"

Criteria must be strictly positive and must come from an expert estimate or an approximate judgement. The "genuine ratio scale" requirement from the crisp WPM card still applies here, only now implemented with a triangular approximation. Opening a measured value into a triangle is the source's own choice. But turning a magnitude that is already expressed numerically, such as net present value or engineering effort, into a triangle can amount to manufacturing uncertainty. In the source's own example this is legitimate, because the corners come from expert estimates, that is, the most pessimistic, most likely and most optimistic values.

The crisp method's exit condition applies unchanged. If a zero or negative value is present, that is, if any corner of the fuzzy value is below or equal to zero, the method becomes undefined. Without a genuine ratio scale, that is, a meaningful zero point, the result stays sensitive to an arbitrary origin.

Adding column scaling. The source itself does not do this. If ratio-to-best normalisation, that is, the crisp WPM step, is added to the book's example, the ranking does not change but the central scores fall from 15.18 / 12.00 to 0.734 / 0.609. The result then sits on a scale that can no longer be compared with the figures the book reports.

Inverting only the value or only the weight on a cost criterion. The kernel's own documentation demonstrates this. In the book's X1 column, with value (3, 4, 6) and weight (0.15; 0.18; 0.24), inverting only the value while leaving the weight uninverted gives (0.7643; 0.7792; 0.7682). This triple's order is broken, because it violates the lowest ≤ most likely ≤ highest condition, and is invalid. The correct result, obtained by inverting both together, is the triple (0.6505; 0.7792; 0.8481), whose order is correct.

Comparing a Fuzzy WPM score with a Fuzzy SAW score. The two use different aggregation logics (multiplicative / additive) and different scales. Asking "which one is correct" is not a mistake but reflects the nature of the method. The same warning on the crisp WPM card applies here too.

Missing a corner that is very close to zero. If a criterion's lowest corner (l) is very close to zero, the product is pulled down sharply even if that criterion's weight is small. This shows that crisp WPM's "does not forgive a weak criterion" property also holds, at the corner level, in the fuzzy form.

The governing principle is this:

In Fuzzy WPM, fuzziness is carried all the way to the end of the multiplication, and defuzzification happens only at the final step; adding an extra scaling step without knowing that the source performs no normalisation produces a result that cannot be compared with the book's own figures.