Neutrosophic WPM (Ye, 2014)

Three things change; the multiplicative logic and defuzzification happening at the very end do not.

Cells. In crisp WPM every cell is a single, strictly positive number. Here every cell consists of three independent components: truth (T), indeterminacy (I) and falsity (F). All three lie within 0-1, and their sum can take any value between 0 and 3. Criterion weights are taken from outside as crisp numbers.

Handling direction (cost criteria). On a cost criterion, every triple is converted to its own complement; (F, 1−I, T) is written in place of (T, I, F). Here it is not only T and F that swap; I also turns into its own complement (1−I). This differs from the intuitionistic fuzzy extension's simple (μ, ν) → (ν, μ) complement; the presence of a third component extends the complementation rule to all three.

Aggregation and defuzzification. Criteria are combined with the neutrosophic weighted geometric aggregation operator (SVNWG, Ye 2014). The combined truth is computed as Π T_j^{w_j}; this is multiplicative, meaning a weak T on one criterion pulls the whole down. The combined indeterminacy is computed as 1 − Π(1 − I_j)^{w_j}, and the combined falsity as 1 − Π(1 − F_j)^{w_j}; both take the form of a probabilistic sum, meaning a high I or F on one criterion pulls the whole up in proportion to its weight. T behaving multiplicatively while I and F behave in a sum-like way means that WPM's unforgiving character towards a weak criterion is preserved in the truth component, while indeterminacy and falsity accumulate.

Defuzzification happens at the very last step. DecisionMind uses a score here that weights indeterminacy twice: s = (1 + T − 2I − F) / 2. This differs from simply reading truth minus falsity (T − F). Of two alternatives with the same T and F, the one with the higher indeterminacy (I) is penalised more heavily in the score. In the event of a tie, a second criterion takes over: the accuracy function a = T − F breaks the ranking.

For this extension, DecisionMind fixes three things: the SVNWG operator (Ye 2014) as the aggregation rule, the score function s = (1 + T − 2I − F) / 2 as the defuzzification rule, and the accuracy function a = T − F as the tie-breaker. Weights are taken from outside; the method does not generate weights and does not support group decision-making.

As in crisp WPM, the score is only a ranking within this particular alternative set. The unit-independence advantage does not apply here, because the criteria are no longer measured quantities but degrees of truth, indeterminacy and falsity.

The difference lies here. A high T alone does not guarantee a high score. If an alternative's truth is high but the information behind it is incomplete or contradictory, that is, I is high, the score does not ignore this; it pulls the score down with double weight. The report should therefore show not only the combined score but also what the T, I and F components that make it up say separately.

Thus instead of writing:

"A3 is the best alternative because its truth is the highest"

the report should read:

"A3's combined truth is the highest; the score reduces this truth by also taking into account the accompanying indeterminacy and falsity (with indeterminacy at double weight), so A3's advantage should be read from s as a whole, not from T alone"

This extension is used when the information about a criterion is incomplete, inconsistent or contradictory, and this matters for the decision itself. Examples: alternatives with no track record, conflicting expert opinions, partially observable systems. Forcing T + I + F to equal 1, or computing I as 1 − T − F, zeroes out this extension's contribution; the general warning on the neutrosophic data-type card applies here too.

The crisp method's exit condition applies here too. A situation where every alternative gets the same score on a criterion can render scale equalisation and aggregation meaningless. T, I and F must each lie within 0-1, and the sum T + I + F must not exceed 3.

Not knowing that indeterminacy (I) enters the score with double weight. In the illustrative example below, A1's truth (0.604) is lower than A3's (0.668), but the gap is small. What actually decides it is that A1's indeterminacy (0.295) is higher than A3's (0.261); this gap is reflected in the score twice over.

Forcing T + I + F to sum to 1. The neutrosophic components are independent; forcing their sum to equal 1 removes the structure's one contribution, carrying contradictory or incomplete evidence separately.

Mistaking the accuracy function (a = T − F) for the main score. In DecisionMind, a is only a tie-breaker that comes into play when s ties; the main ranking criterion is s.

Swapping only T and F on a cost criterion and leaving I unchanged. The correct complement, (T, I, F) → (F, 1−I, T), changes all three at once; leaving I unchanged leads to an invalid aggregation.

The governing principle is this:

In N-WPM, the score is not simply truth minus falsity; the indeterminacy share enters the score with double weight, so an alternative with high truth but also high indeterminacy can come out weaker than it appears.