Neutrosophic PSI

Three things change. The logic of deriving weights from the data itself does not change.

Cells. In crisp PSI every cell is a single number. Here every cell carries three separate degrees: truth, indeterminacy and falsity. PSI still does not ask for a weight here; it derives the weight from the data itself, in the third step.

Scale equalisation. Crisp PSI divides every column by its own best value. Here every cell's T-I-F triple is first reduced to a single score. The score subtracts twice the indeterminacy and the falsity from the truth, then halves the result; high truth and low indeterminacy increase the score. These scores are then normalised within the column by dividing by the column sum. For a benefit criterion, the score itself is divided by the column sum. For a cost criterion, the score's reciprocal is divided by the column sum. This differs from crisp PSI's "divide by the column's best value" rule. Because the neutrosophic score can come out negative, division by the sum is used instead of PSI's rule of dividing by the largest value.

Preference variation value and weight. There is a further departure from crisp PSI here. In base PSI, the spread is subtracted from 1 and turned into a weight. In this neutrosophic version, DecisionMind instead derives the weight by dividing the spread itself directly by the total spread; there is no "subtract from 1" step. As a result, a criterion with a large spread receives a relatively high weight, the opposite of crisp PSI. A criterion with a low spread receives a relatively low weight. This is the exact opposite of crisp PSI's logic of "rewarding the criterion that differentiates least." This reversal comes not from the data type but from this neutrosophic adaptation's own design choice. The report must state this.

DecisionMind holds this score function and this sum-based normalisation fixed.

The preference selection index, as in crisp PSI, is only a ranking device among the alternatives in this table. It is not a percentage or a probability. Here the direction of the weight is also reversed. As explained above, a large spread here means a high weight. A criterion receiving a high weight does not mean "this criterion discriminates poorly"; it means "the alternatives differ a great deal on this criterion." This is the exact opposite of the reading given on the crisp PSI card. The report must state clearly which direction applies. Otherwise the reader will carry over the interpretation they are used to from crisp PSI.

Thus instead of writing:

"In N-PSI this criterion's weight came out low, so the alternatives are close together here"

the report should read:

"In this neutrosophic PSI adaptation, weight is directly proportional to the criterion's own internal spread; here a low weight shows that the alternatives differ little on this criterion, and a high weight shows that they differ a great deal"

N-PSI is suitable if you do not want to assign a reasoned weight to the criteria and the criterion evaluation at hand rests on incomplete or contradictory information. Favourable evidence, adverse evidence and an unknown share must be kept separate. The rule on the neutrosophic data-type card applies here too: opening a measured value into a T-I-F triple without justification adds no information. Only if "no indeterminacy" is honestly embedded, that is, written as (t, 0, 1−t), is this a valid path.

If the decision maker's known priorities are to be reflected in the criteria, or if no compromise is acceptable on one criterion, the exit condition on the base PSI card applies in exactly the same way.

Forcing the sum of T+I+F to equal 1. The general mistake on the neutrosophic data-type card applies here too. The three components must come from independent sources.

Reading the direction of the weight as in crisp PSI. As explained above, in this adaptation a high weight means "much spread," not "little spread." Carrying crisp PSI's reading directly across produces a reversed error.

Applying sum-based division on a cost criterion without inverting the score first. The score can come out negative or close to zero. If the inversion step is skipped, the normalisation of the cost criterion becomes meaningless.

Deriving indeterminacy from truth and falsity. I must rest on an independent source of information. If it is computed as 1−T−F, the rationale for the neutrosophic data type disappears.

The governing principle is this:

N-PSI's weight direction is the reverse of crisp PSI's. Here the data-derived weight grows the more a criterion discriminates, not the less. If this reversal is not reported, the reader will carry crisp PSI's logic to the wrong place.