Neutrosophic CODAS

Four things change; the two-distance comparison logic does not.

Cells. In crisp CODAS every cell is a single number. Here every cell has three components: truth (T), indeterminacy (I) and falsity (F). Each lies between 0 and 1 and is independent of the others. Their sum can exceed 1. Criterion weights remain crisp here too. Indeterminacy is carried only in the criterion values.

Scale equalisation. Crisp CODAS divides a benefit criterion's value by the column's largest value. For a cost criterion, it takes the ratio of the column's smallest value to the value itself. In the neutrosophic cell, the cost direction is handled not by division but by complementing: the triple (T, I, F) is reversed to (F, I, T). The reason is that in neutrosophic algebra, a judgement's "less is better" status is expressed by swapping truth and falsity. Numerical ratio-taking is not defined here. Beyond this complementing, there is no further normalisation step; the neutrosophic components already lie in the 0–1 range.

Weighting and the negative-ideal. Every neutrosophic cell is combined with its criterion weight by the weighted neutrosophic summation rule (SVNWA). The negative-ideal is built, component by component, from the worst neutrosophic value at each criterion (smallest T, largest I, largest F). As in crisp CODAS, a single "worst" reference is used here too.

Score function and distance. Before the Euclidean and taxicab distances are computed, every neutrosophic value is reduced to a single number by a score function, (T − F + 1)/2. The negative-ideal is reduced by the same function. The distances are then computed between these scores in the same way as in crisp CODAS.

The assessment score and the pairwise comparison rule (threshold τ=0.02, Euclidean first, taxicab only if the difference does not exceed it) are identical to crisp CODAS. DecisionMind keeps this score function and threshold fixed for classical N-CODAS.

The assessment score is read exactly as in crisp CODAS: a relative position measure, not a percentage, rebuilt whenever the alternative set changes. The difference is here: the score function uses T and F but does not feed the indeterminacy (I) component into the score directly. The score is affected by I only indirectly: whether T+I+F exceeds 1 can change the result. This means an alternative with high indeterminacy (large I) can receive the same score as another alternative with low T and F. The score alone does not answer the question "how much incomplete information underlies this assessment"; the I component must be examined separately for that.

Thus instead of writing:

"The N-CODAS score came out high, so there is strong evidence for this alternative"

the report should read:

"This score is derived from T and F; without separately checking the I component, one cannot know how incompletely this alternative was assessed"

When criterion assessment rests on incomplete, contradictory or inconsistent evidence, and this matters for the decision. Measured criteria should not be carried into this extension. If the matrix must be of a single type, a measured value with satisfaction degree t is written using the honest embedding (t, 0, 1−t). The base CODAS's exit condition applies here in exactly the same way: if no compromise is acceptable on one criterion, a compensatory method is not suitable.

Defuzzifying with the score function first, then running crisp CODAS. Reducing the T-I-F triple to the (T−F+1)/2 score at the outset and then running crisp CODAS can look the same as N-CODAS. But the difference lies here: whether neutrosophic weighting and cost-complementing are applied before or after the score is exactly what distinguishes N-CODAS from crisp CODAS. In the illustrative example below, taking the score first and then running crisp CODAS preserves the ranking (A3, A1, A2), but shifts the scores to 0.1456 / -0.3334 / 0.1878. Genuine N-CODAS gives 0.3632 / 0.1020 / -0.4652 instead. Both the magnitudes and the relative gaps in the ranking are distorted.

Deriving indeterminacy (I) from truth and falsity. Writing I = 1 − T − F reduces the neutrosophic structure to crisp data. Each component must come from its own body of evidence (see the neutrosophic data-type card).

Forcing the three components to sum to 1. Fixing the sum at 1 reduces the structure to a dependent triple and removes the entire rationale for using a neutrosophic structure.

Changing the score function without stating it in the report. Score functions other than (T−F+1)/2 also exist in the literature. Which one is used affects the magnitude of the score.

The governing principle is this:

N-CODAS exists to preserve, through the T-I-F triple, situations where evidence is incomplete or contradictory. Any shortcut that derives I from T and F, or applies the score function before normalisation and weighting, erases the structure's one contribution: not confusing the unknown with negative evidence.