Fuzzy SAW (Bonissone, 1982)

Three things change; the decision logic does not.

Cells. In crisp SAW every cell is a single number. Here every cell is a trapezoidal fuzzy number: left foot, left shoulder, right shoulder, right foot. DecisionMind's engine holds these four values in Bonissone's own notation: a central interval (a, b) with a left and a right spread (α, β) opening out to either side. The left foot is a−α, the right foot b+β; this is simply another way of writing the same number. Criterion weights are trapezoidal in the same way. This extension does not itself combine several experts' scores; if a group decision is not supported, scores must already be reduced to a single trapezoidal number before the analysis.

Scale equalisation. Crisp SAW ratios every column against its own best value. Bonissone's form does not do this. Inputs are assumed to already sit on a common, bounded band, typically 0 to 1, derived from a verbal scale; cells are multiplied directly by the criterion weight and summed. This means the "ratio against the best" step found in crisp SAW is absent here, so if the data is used without first being brought onto a common band, the result becomes meaningless.

Weighted sum and defuzzification. Every cell is multiplied by the criterion weight in fuzzy form; this multiplication is order-sensitive, and weight and score cannot swap places. The results are then summed per alternative in fuzzy form. The resulting trapezoidal number is the alternative's fuzzy utility; for ranking, this trapezoid is reduced to a single number by the centre-of-area method. In crisp SAW the sum is directly a single number. Here the sum stays trapezoidal throughout the calculation and is only defuzzified at the last step; uncertainty is consumed at the last step, not the first.

DecisionMind fixes, for classical Fuzzy SAW, Bonissone's definition of multiplication and addition and the centre-of-area defuzzification. Weights are taken from outside in trapezoidal form; the method does not generate weights.

The total score is a weighted average, exactly as in crisp SAW, and is read the same way: it is not a percentage, it cannot be compared with a different analysis, and full compensation still holds.

The difference is this. Beneath the score, both SAW's full-compensation logic and the verbal approximation in the input lie stacked on top of one another, and the single number conceals both. This extension also lacks crisp SAW's "ratio against the best" step; the size of the score depends directly on which band the input was given in. The report should therefore show not only the defuzzified score but also whether the ranking changes if any verbal score shifts by one step.

Thus instead of writing:

"Because the Fuzzy SAW score is highest, this alternative is definitely the best"

the report should read:

"This score is the fuzzy weighted sum reduced to a single point; full compensation still applies, and the gap may be sensitive to a one-step change in one of the verbal scores"

Use this when criteria are scored verbally or approximately and reducing these scores to a single number would create an artificial precision. Converting a measured criterion, such as price or time, into a trapezoid is not modelling uncertainty but manufacturing it; DecisionMind requires a single data type, so if the table is mixed, a measured value is also written as a trapezoid with zero spread.

SAW's exit condition applies unchanged: if no compromise is acceptable on one criterion, this extension too carries a full-compensation logic and will not screen out anything below a threshold.

Confusing the order of multiplication between weight and score. Bonissone's definition of multiplication is not symmetric; which of the weight and the score is the "first operand" changes how the left and right spreads are calculated. The order must be preserved.

Using data that has not been brought onto a common band. This extension does not carry crisp SAW's "ratio against the best" step; if the input does not already sit on a common band such as 0 to 1 (for instance, if one column is a raw currency amount and another a 0-to-1 verbal score), the sum becomes meaningless. The scale must be unified before the analysis.

Leaving a cost criterion in its raw form. This version of the engine does not process criterion direction itself; if a "lower is better" criterion exists, such as cost, its cells must be converted before input, otherwise the method treats it as "higher is better" too and reverses the ranking.

Defuzzifying first, then running crisp SAW. Reducing the trapezoids to a single number at the outset and applying the crisp method is not Fuzzy SAW; uncertainty is erased at the first step.

The governing principle is this:

Fuzzy SAW exists to carry the approximation in verbal scores through to the last step. Any application that uses the input without bringing it onto a common band, or that renders it precise from the outset, destroys the extension's one contribution.