Fuzzy WISP

Four things change. The decision logic stays the same.

Cells. In crisp WISP every cell is a single number. Here every cell carries three numbers: the lowest, most likely and highest plausible value. Criterion weights, too, may be entered as triangles; if a crisp weight is entered instead, the method treats its three components as identical. Weights are not defuzzified early here; the triangular structure is preserved right up to the weighting step.

Scale equalisation. In crisp WISP every column is divided by its own largest value. Here this division is carried out against the triangle's highest corner: the largest value among the highest corners of all alternatives in the column is found, and all three components of the triangle are divided by it. This applies in the same way to both benefit and cost criteria; direction information does not enter at this step but when the four measures are calculated.

Score and aggregation. The normalised triangles are multiplied by the criterion weight, component by component. After this step every cell is reduced to its central value, the average of its three components. The four measures from crisp WISP are then built from these central values: the difference between the benefit sum and the cost sum, the difference between the benefit product and the cost product, the ratio of the benefit sum to the cost sum, and the ratio of the benefit product to the cost product. All four are scaled against their own best alternative and then averaged.

Result. Defuzzification happens immediately after the weighted product, before the four measures are calculated. The output is a single crisp number. As in crisp WISP, larger is better.

In this fuzzy extension, DecisionMind keeps the weighting of the four measures equal, as in crisp WISP; all four are averaged with the same weight. This is not a user input.

The output is a score, as in crisp WISP. A value of 1 does not mean perfection; it means this is the alternative that all four logics find best within this set. The score is not a percentage and cannot be compared with the score of a different analysis.

The difference is this: here the four measures are computed from the triangles' central value. The width of the input triangles is not reflected in the final score at all; the width disappears immediately after the weighted product. Whether all four measures point to the same alternative should be checked separately here too, exactly as in crisp WISP.

Thus instead of writing:

"Fuzzy WISP found the best alternative"

the report should read:

"With these triangular inputs, the average of the four comparison logics places A1 ahead (1.0). All four measures agree on this alternative; the margin over A3 (0.914) stays robust unless A1's cost criterion worsens markedly"

This extension is suitable when criteria come from expert judgement or estimation, and reducing that approximation to a single number would create artificial precision. The Fuzzy data-type card describes which situations can be treated as fuzzy. DecisionMind requires a single data type; if the table is mixed, a measured criterion is also written as a triangle with all three components equal.

Crisp WISP's exit condition applies here too: this extension is unsuitable when no compromise is acceptable on one criterion. No cell should be very close to zero, otherwise the multiplicative measures break down.

Violating the value-space constraint. In every triangle the lowest value must be less than or equal to the most likely, and the most likely less than or equal to the highest; none may be negative.

Changing the defuzzification method without saying so. The canonical choice is the central value. A different defuzzification can change the four measures, and hence the ranking.

Never checking the four measures separately. Reporting only the final average, without checking whether the additive difference, multiplicative difference, additive ratio and multiplicative ratio point to the same alternative; when they disagree, the average conceals this disagreement.

Leaving a triangle corner close to zero. If any corner of a triangle is very close to zero, the multiplicative measures give an unreliable result; the four-logic consensus effectively drops to two.

The governing principle is this:

The Fuzzy WISP result is the average of the four comparison logics over the triangles' central value. If the logics disagree, or if the input triangles are wide, the report must show this uncertainty separately.