Neutrosophic CoCoSo (Nabeeh & Sallam, 2024)

Three things change; the decision logic does not.

Cells. In crisp CoCoSo every cell is a single number. Here every cell consists of three independent degrees: truth, indeterminacy and falsity; all three lie between 0 and 1, and their sum can run as high as 3. Indeterminacy is not a residual share here; it is separate information drawn from its own source. Criterion weights are supplied from outside as crisp numbers.

Score and scale equalisation. Every cell is first reduced to a single number using Nabeeh and Sallam's (2024) score function: s = (1 + T − 2I − F) / 2. For a cost criterion, the triple is complemented before this reduction: (T, I, F) is rewritten as (F, 1−I, T). Crisp CoCoSo pulls each column into the 0–1 range against its own best and worst value at this point; the neutrosophic extension has no such pulling step. The additive measure (S) and the multiplicative measure (P) are computed directly from these scores, without any 0–1 scaling.

Additive and multiplicative measure. S_i is the weighted sum of the scores; it works exactly as in crisp CoCoSo. P_i is the weighted power product of the scores. Because a score can be negative (the expression T − 2I − F can fall below zero), a problem arises: the multiplicative measure is undefined for a negative base. DecisionMind clips the base to a very small positive number (ε) in this case. This keeps the calculation from collapsing, but it has a cost: as the illustrative example below shows, once a criterion's score falls below zero, P_i loses almost all of that criterion's real information.

DecisionMind fixes, for this extension, the Nabeeh–Sallam (2024) defuzzification score and the balance coefficient in the third compromise strategy. The balance coefficient λ is a user-adjustable parameter here, unlike the rest of the family (Pythagorean, spherical, q-rung fuzzy CoCoSo), where the engine fixes λ at 0.5.

As in crisp CoCoSo, the K score is only a ranking within this particular set of alternatives. Whether the three compromise strategies agree indicates robustness.

The difference is here: in crisp CoCoSo, K is always positive, because S and P are built from numbers already pulled between zero and one. In the neutrosophic extension, no such pulling takes place, so S_i can come out negative, and K can then fall negative too. Moreover, once a criterion's score drops below zero and is clipped to ε, P_i carries almost none of that criterion's real difference; the ranking is then effectively carried by S_i (the additive measure) alone.

Thus instead of writing:

"The result is reliable because the K score comes out positive and large"

the report should read:

"In this extension, the sign and magnitude of K cannot be compared with crisp CoCoSo's; if any criterion's score has fallen below zero, the multiplicative measure may be almost disabled, and this must be checked separately"

Use this extension when criteria come not from a measurement but from a judgement that can be expressed as a truth–indeterminacy–falsity triple, and when these three components are independent and drawn from separate sources. Indeterminacy here is not the expert's hesitation but a lack or contradiction of information; the distinction is drawn on the Neutrosophic data-type card. If support and rejection already sum to no more than 1 and there is no separate source of indeterminacy, intuitionistic fuzzy CoCoSo (if-cocoso) is sufficient; the neutrosophic structure adds one further assumption.

Converting a measured criterion into a T-I-F triple is not modelling uncertainty but manufacturing it. The base method's exit condition applies here too: the matrix must be of a single type, and if no compromise is acceptable on one criterion, this extension is also compensatory and will not eliminate anything below a threshold.

Failing to notice that a criterion's score has fallen below zero. In the illustrative example below, the cost criterion's complemented score is negative for all three alternatives (−0.25; −0.15; −0.45). In this case P_i is computed from a base clipped to ε, and falls to between a thousandth (0.22–0.35) of the magnitude of S_i, that is, to 0.00053–0.00055. Had the same three alternatives been assessed on benefit criteria only, P_i would have remained the same order of magnitude as S_i (roughly 0.38–0.47). The report must state that P_i has been all but zeroed out in this way; otherwise, saying "two separate measures, additive and multiplicative, were used" is misleading when only one (S_i) is in fact carrying the ranking.

Allowing T+I+F to exceed 3. This is the manifest's own warning. Even though each component lies in [0,1], if the sum of all three exceeds the validity bound (3), the cell is invalid.

Forgetting that K can come out negative and blindly applying "the highest K wins." If the weight loads heavily onto the cost criterion (0.70 on C3 in the illustrative example), S_i and hence K can fall negative. The ranking is still taken from highest to lowest, but an expectation such as "K should be positive" is wrong here.

Changing the defuzzification score. The Nabeeh–Sallam (2024) score, (1+T−2I−F)/2, is a fixed rule. A different neutrosophic score function (such as T−F alone) can produce a different ranking.

The governing principle is this:

In neutrosophic CoCoSo, scores are never pulled into 0–1 at any intermediate step; if a criterion's score falls below zero, the multiplicative measure loses almost all of that criterion's information, and the ranking is then effectively carried by the additive measure alone.