Z-number VIKOR (Shen et al., 2018)

Three alternatives are scored as Z-numbers on three criteria. Every cell gives the A (value) triangle; reliability B is the same across all three criteria and all three alternatives: (0.70; 0.80; 0.90). The compromise coefficient is taken as v = 0.50.

The method reduces every cell's Z-number to a crisp number by the Kang transformation. It then applies classical VIKOR's linear normalisation, the group-utility (S) and individual-regret (R) calculation, the compromise index (Q), and the two conditions.

The result reads as follows. A2 is best both overall (S) and on the single worst criterion (R); neither the majority nor the minority side can object to it. The acceptable-advantage condition is met, because A2's Q gap to A3 (0.780) exceeds the threshold of 0.50 that applies for three alternatives. The acceptable-stability condition is met too, because A2 ranks first on both S and R. A2 is the single compromise solution.

The decision's hesitation is this. If the weights are changed to C1=0.20, C2=0.55, C3=0.25, A3 comes first (Q=0.040), and A2 stays second (Q=0.107). But the Q gap between these two (0.067) is below the threshold (0.50). The acceptable-advantage condition is not met, and VIKOR proposes A3 and A2 together as a compromise set rather than a single winner.

Reliability B has no effect at all on these results. Even if B is changed, S, R and Q come out exactly the same under both weighting scenarios. This has been verified by running the kernel.

In the report: "With weights C1=0.40, C2=0.35, C3=0.25, A2 is the single compromise solution (Q=0.000); both conditions are met. When the weight of C1 is lowered below that of C2, the acceptable-advantage condition is not met, and the compromise set consists of A3 and A2. The degree of reliability is not reflected in this result by the current engine."

Source: DecisionMind's Z-VIKOR validation example; a synthetic 3x3 fixed dataset, built following the Z-number framework of Zadeh (2011) and Kang et al. (2012). The S, R, Q values and the compromise scenario have been independently recomputed by this card's author using the same algorithm, and match the manifest's expected values exactly (tolerance 1e-9).

A publisher will choose one of three translated-book proposals for its new season's list. The criteria are expected sales potential, translation-and-rights cost, and a literary-value score. All but cost are "higher is better," cost is "lower is better." Sales potential is entered as a Z-number because it rests on a forecast. A is the estimate given by the editor; B is whether that estimate rests on the author's previous-book sales data or only on the editor's personal opinion.

The method reduces the three proposals' Z-numbers to crisp numbers by the Kang transformation and applies classical VIKOR. Suppose the result gives, as the single compromise solution, the proposal with the highest sales-potential forecast, where that forecast rests only on the editor's personal opinion.

The publisher's hesitation is this. If this forecast has low reliability, it should be known that the current engine does not reflect this gap in the compromise proposal at all. The publisher should discuss this reliability gap separately at the editorial-board meeting.

In the report: "With the weights given, the proposal with the highest sales-potential forecast is the single compromise solution. This forecast rests only on the editor's personal opinion, and the current calculation does not reflect this gap; it should be considered separately at the editorial board."

In the illustrative example, sharply lowering A2's reliability on the C1 criterion and expecting "the compromise solution should now change" is wrong: S, R and Q do not change at all, because the current engine does not read B. The second error is writing only "A3 is first" under the weight scenario C1=0.20/C2=0.55 and concealing the compromise set; when the acceptable-advantage condition is not met, A2 is also part of the proposal. The third error is reading a Q of 0.000 as "a flawless alternative"; Q only positions these three alternatives relative to one another.