Z-fuzzy CRADIS (Puška et al., 2022)

Three things change; the CRADIS logic of keeping closeness to the ideal and distance from the anti-ideal as two separate ratios and then averaging them does not.

Cells and the prior conversion. In crisp CRADIS every cell is a single number. In this extension a cell is, conceptually, a Z-pair: the value an expert gives (A, converted from a verbal scale into a triangular fuzzy number) and the confidence placed in that value (B, converted from a separate verbal scale into a triangular fuzzy number). Puška and colleagues' method first reduces every expert's every Z-pair to a single triangular fuzzy number through Kang and colleagues' (2012) conversion; in this reduction, A's three corners are scaled by the square root of a coefficient drawn from B's centroid. Where more than one expert is involved, the triangles reduced in this way are combined by arithmetic mean into a single collective fuzzy decision matrix.

The step DecisionMind runs in this extension. A lasting and important limitation must be reported here: the engine that computes this extension for DecisionMind does not itself carry out the reduction from a Z-pair to a triangular fuzzy number described above; it expects a decision matrix that is already reduced, that is, made up of ready-made triangular fuzzy numbers. This card's author verified it by running the engine: when a cell is given six figures (a three-figure value triangle plus a three-figure reliability triangle), the engine reads only the first three figures and disregards the remaining three (reliability) entirely; changing the reliability between (0.1; 0.1; 0.1) and (0.9; 0.9; 0.9) leaves the result identical to the decimal place. This belongs to the same family as the "reliability component is not read" situation reported on the Z-number data-type card and on DecisionMind's Z-TOPSIS, Z-VIKOR, Z-EDAS, Z-COPRAS and Z-WASPAS cards. The reason here, however, differs. On those cards reliability is read and then cancels out mathematically. Here reliability is never read by the engine at all; the step converting a Z-pair into a fuzzy number has not been implemented in the engine's code.

Normalisation, weighting, the ideal/anti-ideal, distance, defuzzification and the compromise score. Once a ready triangular fuzzy decision matrix is in hand, all of these steps are the fuzzy form of crisp CRADIS and run according to Puška and colleagues' (2022) definition: direction-sensitive normalisation (ratioing to the largest value in a benefit column, to the smallest in a cost column), weighted triangles, a SINGLE ideal and a SINGLE anti-ideal triangle built over the whole matrix, the d+ and d- deviations found by triangular subtraction, their sums, defuzzification with the GMIR centroid (1,4,1)/6, K+ and K- against the best ratio, and the compromise score Q as their average.

DecisionMind fixes, in this extension, the GMIR defuzzification and the single-global-ideal/anti-ideal logic. Weights are taken from outside, as crisp numbers; the method does not generate weights.

The compromise score Q is read as in crisp CRADIS (see the CRADIS card): it is not a percentage, and it is not compared with another analysis.

The difference lies here. The purpose of the Z-number data structure is to carry confidence in the source into the decision. But the engine that computes this extension for DecisionMind never reads reliability; the calculation runs directly on the triangular fuzzy values entered. For that reason the report must not state that "reliability has been taken into account."

Thus instead of writing:

"Because Z-fuzzy CRADIS has been used, the supplier's reliability history is reflected in the result"

the report should read:

"Criterion values have been entered as triangular fuzzy numbers; the current engine does not read or take into account any reliability component accompanying these values. If a difference in reliability matters, it must be reflected, at the data-collection stage, in the value itself (A), not in a separate reliability field"

In formal terms, if criterion values come from a judgement or an estimate, and reducing them to a single number would manufacture an artificial precision, this extension, in effect functioning as a fuzzy CRADIS, is the right choice. But if the decision-maker expects that "a difference in the source's reliability will affect the ranking," this expectation is not met by the current engine; this limitation must be stated before data collection. Fuzzifying an already measured value manufactures uncertainty rather than modelling it; crisp CRADIS should be kept instead.

Crisp CRADIS's exit condition also applies here: if no compromise is acceptable on a criterion, sub-threshold alternatives must first be screened out, and only the remainder ranked with this extension.

Assuming that reliability (B) will change the ranking. The current engine does not read B; two identical A values with different reliability are, to the engine, exactly the same. This is not a software bug but a limit of the engine's present scope, and it must be stated explicitly in the report.

Confusing GMIR defuzzification with a simple centroid. Puška's formula uses GMIR weights of the form (l+4m+u)/6; replacing this with the simple centroid (l+m+u)/3 produces a different number.

Reversing s0+ and s0-. s0+ is the smallest s+ value (the alternative departing least from the ideal); s0- is the largest s- value (the alternative departing most from the anti-ideal). Reversing these two definitions inverts the ranking.

Adding a "reliability" field to a measured value out of caution. Adding a reliability field to an actual measurement is both unnecessary and, in any case, not read by the engine; a measured value should stay as a triangle whose three components are identical.

The governing principle is this:

Z-fuzzy CRADIS is designed to record the criterion value and the confidence placed in it separately. But DecisionMind's present engine never reads this confidence component; the calculation runs only on the triangular fuzzy value, and this limit must be stated to the decision-maker before data collection.