Neutrosophic COPRAS (Şahin, 2019)

Four things change; the benefit/cost ratio logic does not.

Cells. In crisp COPRAS every cell is a single number. Here every cell is six numbers: the lower and upper bound of truth, of indeterminacy, and of falsity (an interval-valued neutrosophic number). Criterion weights are entered as crisp numbers. Şahin (2019) generates these weights by a separate method, blending both expert judgement (subjective) and the dataset itself (objective) into a combinative weight determination (CWD). But DecisionMind's COPRAS-IN step does not generate these weights itself; like the rest of the family, it takes them ready-made. The weights in Case 1 were therefore produced by a separate calculation in Şahin's paper and carried here as an input.

Scale equalisation. In DecisionMind, cells are not converted for direction. A benefit cell is entered as a "degree of being good," a cost cell as a "degree of the cost or effect being high," and both are kept as they are; cost cells go to the divided side of the calculation and are penalised there directly. Şahin's paper swaps truth and falsity and complements indeterminacy in cost cells, because the paper codes every cell as a "degree of satisfaction." DecisionMind's input takes cost directly as a magnitude and needs no such conversion; the calculation is identical, only the coding differs. Weighting is done not by multiplication, as in crisp COPRAS, but by an exponentiation operation specific to neutrosophic numbers.

Distance / score / combination. In crisp COPRAS, the benefit and cost sums are simple addition. Here, both are combined by a Maclaurin symmetric mean operator; this operator does not simply add criteria one by one but carries a k parameter that also accounts for the interaction between them. The combined benefit and cost values are then reduced to a single number by a risk indicator (λ): as λ rises, the score gives more weight to the truth component; as it falls, more weight to the scarcity of indeterminacy and falsity.

Result and defuzzification. The benefit and cost scores, defuzzified by the risk indicator, are combined into a relative significance value (Q) exactly as in crisp COPRAS, divided by the highest value and converted to a percentage.

DecisionMind runs this family with the Maclaurin operator's k parameter and the risk indicator λ fixed at values stated in the manifest (in Case 1, k=2, λ=0.5); both must be stated alongside the report, because, as shown below, the ranking can change when λ changes.

The output is a degree of utility and a rank, in the same form as crisp COPRAS, and read the same way. The best alternative scores 100, and the others receive a percentage relative to it.

The difference is here: this score depends on a risk indicator (λ). If λ is not stated in the report, the reader cannot tell whether priority was given to truth or to the scarcity of indeterminacy and falsity. Two different λ values can produce different rankings from the same data. For this reason, it is not a single ranking that should be reported, but the range of λ over which the ranking stays fixed.

Thus instead of writing:

"N-COPRAS found this alternative the most trustworthy"

the report should read:

"With the risk indicator λ=0.5, A4 has the highest degree of utility; this ranking holds up to λ=0.64, A1 moves ahead from λ=0.65, and A3 from λ=0.85. Which risk attitude is adopted affects the decision"

When information about a criterion is incomplete, inconsistent or contradictory, and this needs to be preserved through three separate components (truth, indeterminacy, falsity); and, further, when experts can give each of these three components only as a range rather than a single number. If the assessment rests on a reliable measurement, or experts do not express any share of indeterminacy, there is no need to carry the data into the neutrosophic structure merely to use a more elaborate model.

The exit condition is the same as for crisp COPRAS. If no compromise is acceptable on one criterion, this extension is also compensatory and will not eliminate anything below a threshold. The matrix must be of a single data type; a measured criterion is written into the same matrix with an honest embedding, in which truth equals the measured value, falsity equals 1 minus it, and indeterminacy is entered as equal, narrow intervals around zero.

Sensitivity to the risk indicator λ. Different λ values can reverse the ranking (Şahin 2019, Table 7). The λ used must always be stated in the report.

Value-range violation. In every cell, the sum of the upper bounds of truth, indeterminacy and falsity must not exceed 3. If this bound is exceeded, the cell is invalid.

Ignoring the Maclaurin parameter k. k=1 reduces the calculation to a simple weighted average; when k equals the number of criteria, the behaviour approaches a product-like form; k=2 is the most common choice in the literature, but the choice must be stated in the report.

Collapsing the three components to their midpoints and treating them as single-valued neutrosophic. In the literature example, this changes the winner: A1 (0.2255) moves ahead of A4 (0.2369), which drops to second (0.2143). Once the "how much uncertainty remains" information carried by the interval width is dropped from the calculation, A4's wide indeterminacy interval on the environmental-impact cell ([0.5;0.9]) disappears from view.

Marking a criterion's direction wrongly. Marking the cost criterion "higher is better" moves that criterion from the divided side to the summed side: a high cost is then rewarded. In the literature example, if environmental impact is marked "higher is better," the relative significance values are compressed (between 0.4253 and 0.4645) and A2 and A3 swap places.

The governing principle is this:

N-COPRAS exists to preserve the incompleteness and contradiction of information through interval width; collapsing intervals to their midpoints at the outset, declaring a single ranking without stating the risk indicator, or marking a criterion's direction wrongly, invalidates all of this protection at once.

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