Fuzzy SMART

There is an important difference in scope here, not merely a calculation difference. The base SMART card explains the method from beginning to end. First, the worst and best end of every criterion is fixed. Then alternatives are placed between 0 and 1 relative to this end. Next, importance ratings are converted into weights, and finally the alternatives' utility scores are calculated and ranked.

This DecisionMind extension makes only the third step fuzzy. This is the step that converts importance ratings into weights.

Input. No alternative table is entered into this extension. The input is only the list of criteria and the importance rating given to each. Every importance rating consists of three numbers: minimum, most likely, maximum.

Weight derivation. In crisp SMART, the decision-maker gives each criterion an importance rating, and these ratings are normalised so that they sum to 1. Fuzzy SMART carries out this same normalisation separately on all three components. One weight vector emerges from the minimum ratings, one from the most-likely ratings, one from the maximum ratings. These three vectors are then averaged criterion by criterion and renormalised so that the total is again 1.

Result. The output is a weight vector, not a ranking. Scoring and ranking alternatives falls outside this extension's scope; it corresponds to the fourth step on the base SMART card and is not run here.

DecisionMind, in this extension, also calculates the difference between the averaged weight and the weight derived from the most-likely component alone. This difference shows how much the uncertainty margin the expert gave changes the weight. If all three components produce identical weights, this is flagged as a warning.

The output is a weight vector showing the relative importance of the criteria. The criterion with the highest weight is the one the decision-maker found most important in the ratio scoring. This weight does not say which alternative comes out ahead, because this extension never sees any alternatives.

The difference is here: the weight is now derived from three separate rating bands, minimum, most likely and maximum. If all three bands give the same weight, the expert's uncertainty has not affected the weight. If the three bands give different weights, the report must show which band is driving the weight.

Thus instead of writing:

"The Fuzzy SMART analysis showed K1 to be the most important criterion"

the report should read:

"K1 obtained the highest weight in all three of the minimum, most-likely and maximum rating bands. Because these three bands come out close together, the weight is not sensitive to the width of the range the decision-maker gave"

The decision-maker may think of an importance rating as a range rather than a single number when scoring criteria. If they say, for instance, "I would give this criterion a minimum of 50, most likely 60, maximum 70," Fuzzy SMART is preferred. If the importance rating is already a single crisp number, fuzzification adds nothing, and base SMART suffices.

This extension produces weights only. If alternatives are to be scored and ranked, these weights should be used as the input to base SMART or another ranking method. If no compromise is acceptable on one criterion, screening should be applied before weighting.

Trying to give this extension an alternative table. The input is only a triple of importance ratings per criterion. An alternative table does not belong here; it belongs in base SMART itself.

Failing to notice proportional fuzzification. If the same proportional width has been added to every criterion, for instance if all ratings have been scaled up and down by the same factor, the three bands give exactly the same weight. This is not an error, but if it goes unnoticed it can create the misleading impression that "the three bands were each calculated separately."

Presenting the weight as an objective result. The weight comes from the decision-maker's own subjective ratio scoring. DecisionMind does not generate these ratings; it only converts them into weights in fuzzy form.

Widening the rating band without justification. Opening up a single crisp rating into three numbers afterwards adds no information; it only manufactures an apparent uncertainty.

The governing principle is this:

Fuzzy SMART exists to carry the approximation in the decision-maker's importance ratings into the weight vector. Any application that presents this weight as though it were a ranking outcome, or that conceals its subjective source, misrepresents the method's scope.