Fuzzy PSI

Three things change. The logic of deriving the weight itself does not.

Cells. In crisp PSI every cell is a single number. Here every cell carries three numbers: lowest, most likely, highest. If an expert gives a verbal score ("high", "medium-high"), a pre-declared scale converts that word into a triangle. PSI does not require a weight; neither does this form — even if an external triangular weight is entered, DecisionMind's engine does not use it. It still derives the weight from the data itself, at the third step below. The base PSI rule, "no weight required, it produces its own", carries over unchanged into the fuzzy version; only the input's three-component structure changes.

Scale equalisation. Crisp PSI divides every column by its own best value. Here, the three components of every triangle are first scaled together by the same rule. On a benefit criterion, each is divided by the column's largest upper value. On a cost criterion, the column's smallest lower value is divided into the triangle's components with the order reversed, so that the scaled triangle also preserves the lowest-most likely-highest order. Then the three components of this scaled triangle are averaged and reduced to a single crisp number. This is a different order from fuzzy TOPSIS. There, defuzzification comes right at the end, after the closeness score has been computed. Here, defuzzification happens very early, finishing before the weight has even been derived. PSI's own weight-derivation step is already built to work on a crisp number. This is why DecisionMind consumes the fuzziness immediately after scale equalisation.

Preference variation value and weight. Everything after this step is identical to crisp PSI. In the defuzzified column, the sum of squared deviations from the mean is taken. This spread is subtracted from 1 and turned into a weight. The preference selection index is computed with this weight. The fuzziness no longer appears as a triangle; it leaves a trace only in the form of "how the fuzziness of the input affected the normalisation".

DecisionMind holds fixed, in this Fuzzy PSI, the scale equalisation and the centroid defuzzification that follows immediately after it. The TFN's three components are scaled together, by the same rule.

The preference selection score, as in crisp PSI, is only a ranking tool among the alternatives in this table. It is not a percentage or a probability. The difference lies in the score's magnitude range. In crisp PSI, weights are normalised to sum to 1 and multiplied into the score, so the score stays roughly between 0 and 1. In the fuzzy version, the score can visibly exceed 1, because the raw value produced by the weight-derivation step enters the score without being normalised. This does not disturb the ranking among alternatives. But it makes reading the score as "86 per cent good" even more meaningless.

So instead of:

"The Fuzzy PSI score is 2.53, this alternative is nearly perfect"

the report should read:

"A2's score is 2.53, the highest among these three alternatives. What should be read is not the magnitude of the score, but its gap to the other alternatives and whether that gap closes with a small change"

If you do not want to give the criteria a justified weight, that is PSI's basic condition. If the criterion scores you have are also not measured, but come from expert judgement or approximate estimation, Fuzzy PSI is appropriate. Opening a measured value into a triangle "to look more cautious" is wrong here too; the rule on the fuzzy data-type card applies just the same. The matrix must be a single type. If one criterion is triangular fuzzy and another is crisp, the crisp one's three components are written as the same number and entered into the matrix with a width of zero.

If the decision-maker's known priority is to be reflected in the criteria, that is, if a subjective weight is wanted, the exit condition on the base PSI card applies just the same. The same condition applies if no compromise is acceptable on a criterion. In that case you should prefer a subjective weighting method such as AHP, BWM or SWARA, or a screen-first-then-rank approach.

Breaking the triangle's l ≤ m ≤ u order. Scale equalisation on a cost criterion is done by reversing the order. If this step is skipped or applied incorrectly, the triangle falls into an undefined shape with l greater than u. The engine is expected to stop fail-closed.

Changing the defuzzification method and comparing the result. The centroid, that is, the average of the three components, is not the only standard. Placing a score produced with a different defuzzification rule (for example, taking only the most-likely value) side by side with this card's score is wrong.

Assuming the score is a ratio between 0 and 1. As explained above, this score can exceed 1. Using the language of "percentage" or "ratio" is misleading.

Entering an external weight and assuming the method uses it. PSI derives the weight from the data. Even if a weight is entered, it does not affect the ranking. If the weight needs to reflect the decision-maker's priority, PSI is already the wrong tool.

The governing principle is this:

In Fuzzy PSI, uncertainty is carried only up to the scale-equalisation step. Weight derivation and scoring then work, exactly as in crisp PSI, on a single defuzzified number.