Neutrosophic MARCOS (Martin et al., 2023)

Four things change; the ideal/anti-ideal ratio logic does not.

Cells. In crisp MARCOS every cell is a single number. In N-MARCOS every cell is three independent degrees: truth (T), indeterminacy (I), falsity (F). Criterion weights remain crisp. The founding paper (Martin, Broumi, Sudha & Priya, 2023) proposed the method for software selection in a smart manufacturing system.

Scale equalisation. Crisp MARCOS divides a cost cell by the ideal's (small) value. N-MARCOS instead first swaps the T and F components of a cost cell and complements I (I → 1 − I). This complementing step turns a low-cost cell that is "low in truth, high in falsity" into one readable as "high in truth, low in falsity." A single score function (Ye 2014: (1 + T − 2I − F) / 2) is then applied to every cell; this score is the input to what corresponds to normalisation in crisp MARCOS.

Distance / score / combination. After the score function, the weighted sum (S_i for each alternative) is built directly as the weighted sum of the scores. The ideal row (the highest observed T, lowest I, lowest F at each criterion) and the anti-ideal row (the reverse) are also passed through the same score function, giving their own weighted sums (S* and S⁻). The utility ratios K+ = S_i/S*, K− = S_i/S⁻ are built as ratios of these sums. Crisp MARCOS divides by the ideal cell by cell first and sums afterwards; N-MARCOS reverses this order, summing first and only then taking the ratio of the total to the ideal total.

Result and defuzzification. The three components (T, I, F) collapse to a single number by the score function at the second step; the remaining utility function and final score proceed exactly as in crisp MARCOS's utility-function step.

DecisionMind's fixed choices for N-MARCOS are: the Ye (2014) score function, (1 + T − 2I − F)/2; the T↔F swap plus I-complementing rule for a cost criterion; and the order of ratio-taking against the ideal at the total level (F1, F2, F5).

The output is a final degree of utility and a rank, in the same form as crisp MARCOS, read the same way.

The difference is here: the final degree does not show T, I and F separately. An alternative can reach a high final degree either through "strong, consistent evidence" or through "weak evidence but also a low unknown share"; both can compress into the same score.

Thus instead of writing:

"The N-MARCOS result rests on strong evidence"

the report should read:

"The final degree is T, I and F combined into a single score; which alternative carries a high unknown share must be read separately from the T-I-F triple, not from the single degree"

When information is incomplete, inconsistent or contradictory, and this matters for the decision itself (the criterion given on the neutrosophic data-type card). There is no need to compute I as 1 − T − F and move to this structure; doing so reduces the triple to a dependent structure and removes N-MARCOS's one contribution, making contradiction visible.

The exit condition is the same as for crisp MARCOS: if no compromise is acceptable on one criterion, this extension is also compensatory.

Breaking the T + I + F > 3 rule. Even though each component lies in [0,1], their sum must not exceed this bound; this must be checked when the cell is entered.

Deriving I from T and F. I must come from an independent source; computing it as 1 − T − F reduces N-MARCOS to intuitionistic fuzzy (IF) and erases the neutrosophic structure's one contribution, keeping contradiction separate.

Defuzzifying with the score first, then applying crisp MARCOS's cell-level normalisation. Reducing every cell to a single number with the Ye score and then applying crisp MARCOS's "divide by the ideal first, then sum" order is not N-MARCOS; the correct method sums first, and only then takes the ratio of the total to the ideal total. In the illustrative example below, this wrong path does not change the order (A3 still comes first), but it narrows the gap between A1 and A2 from 0.0759 to 0.0165. This shrinks the gap to less than a quarter of its size; the ranking staying the same does not mean the two methods are equivalent.

Forgetting to apply the T↔F swap and I-complementing rule on a cost criterion. In this case, the ideal and anti-ideal swap places, and the low-cost alternative is mistakenly penalised.

The governing principle is this:

In N-MARCOS, T, I and F are first gathered from independent sources, only reduced to a score at the cell level, and totals are only then ratioed against the ideal; any shortcut that changes this order distorts the size of the gap even if the ranking stays the same.

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