Z-number MARCOS (Yazdani, Pamucar, Chatterjee & Torkayesh, 2021)

Three things change; the logic of the utility ratio relative to the ideal and the anti-ideal does not.

Cells. In crisp MARCOS every cell is a single number. Here every cell has two parts: A, the value itself (as a triangular fuzzy approximation); B, how far that value is trusted (again a triangular fuzzy number). A performance score an expert calls "roughly 0.6" is written together with a separate triangle showing how far that estimate is trusted.

Conversion to a crisp number. Before entering crisp MARCOS's extended table (the real alternatives plus the ideal and anti-ideal rows), every Z-number cell is converted into a single crisp number. This conversion is carried out by scaling the value triangle with the square root of the reliability triangle's centroid, and then taking the centroid of this scaled triangle. In this centroid-based conversion that DecisionMind uses, the scaling constant enlarges or shrinks only the membership height, not the position of the value. Because of this, once the centroid enters the calculation, this constant cancels out; the crisp result comes out equal only to the centroid of the value (A) triangle. This is an important finding, addressed further in the card's "Mistakes Specific to This Extension" section.

The extended table, ratios and utility functions. On the matrix once it has been converted to crisp numbers, exactly the same steps run as in crisp MARCOS. The ideal and anti-ideal rows are added, normalisation and weighting relative to the ideal are carried out, and the row totals build the utility ratios and the final utility degree.

DecisionMind holds this conversion rule, and the crisp MARCOS steps that follow it, fixed in classical Z-MARCOS.

The final utility degree says the same thing as in crisp MARCOS: the alternative's proportional position, in this set, relative to the ideal and anti-ideal references.

An important caveat is needed here. The promise of the Z-number is not to treat sources giving the same value at different reliability levels as equal. But in DecisionMind's current Z-MARCOS engine, the reliability (B) component never changes the crisp result at all, because the centroid-based conversion described above cancels B out. This has been independently tested and verified by this card's author, cell by cell. In the current engine's output, a reliability difference produces no effect worth reporting; the final utility degree depends only on the centroid of the value (A) triangle.

Thus instead of writing:

"This alternative's utility degree has been found by also taking the reliability of the inputs into account"

the report should read:

"This utility degree has been calculated from the centroid of the cells' value (A) component. Under the current conversion rule, the reliability (B) component does not change the crisp result, and no interpretation should therefore be built on a reliability difference"

If the assessments come from sources of differing reliability, and this difference is expected to show up in the report, the Z-number data type is logically appropriate. But because of the finding above, choosing Z-number MARCOS in DecisionMind for this purpose at present does not meet the expectation that a reliability difference will show up in the result. If a reliability difference genuinely needs to show up in the result, that difference should for now be carried into the criterion weight or into the value (A) itself. Where this is not possible, Fuzzy MARCOS (A alone, without any reliability claim) should be preferred.

The exit condition of crisp MARCOS applies here too: this extension is not suitable if no compromise is acceptable on one criterion, because it is compensatory. If the alternative set is very small, the ideal and anti-ideal references can be overly sensitive to the alternatives themselves.

Assuming a difference in reliability (B) will change the result. As shown above, even if B is changed within the same cell from very low to very high, the final utility degree does not change. A sentence such as "the reliability difference has affected the result" should not be written in the report; it would be an unverified claim.

Writing the same B value into every cell and treating this as the justification for using a Z-number. If B is the same everywhere, it already carries no discriminating information. Even entering different B values shows no effect in the current engine.

Confusing the width of the value (A) triangle with reliability. The width of A is the estimate's own uncertainty; B is the reliability of the source. This distinction is set out on the Z-number data-type card and applies here as well.

Adding an alternative once the analysis has finished. As in crisp MARCOS, the ideal and anti-ideal references are built from the alternatives themselves. Adding a new alternative changes these references, and therefore every utility degree.

The governing principle is this:

In Z-number MARCOS, the final utility degree comes from the centroid of the cells' value component. It has been verified that the reliability component does not change the result under DecisionMind's current conversion rule; the report must state this limit plainly.