Cubic fuzzy VIKOR (Fahmi, Amin, Abdullah, Aslam & Ul Amin, 2019)

Four things change.

Cells. In crisp VIKOR every cell is a single number. Here every cell consists of three numbers: an interval [μᴸ, μᵁ] ⊆ [0,1] AND a point λ ∈ [0,1]. The interval part carries the information "this assessment is plausible within this band"; the point part carries "the single most likely value"; the two are held together, neither reduced to the other. Criterion weights are taken from outside as crisp numbers.

Scale equalisation (direction flipping). For a cost criterion, crisp VIKOR swaps the best and worst values. Here, for a cost criterion, a cell is complemented to ⟨[1−μᵁ, 1−μᴸ], 1−λ⟩: the interval's two ends are complemented together and in reverse order, and the point is likewise subtracted from 1.

Score and best/worst values. Every cell is first reduced to a single score: s = (μᴸ+μᵁ)/2 + (μᵁ−μᴸ)/4 − (1−λ)/2. The first term of this score is the interval's midpoint, the second is a correction from the interval's width, and the third is the contribution of the point assessment. On every criterion, the best and worst values are set as the largest and smallest of these scores, that is, they are the score of an alternative that genuinely exists. This is identical to crisp VIKOR's own logic of "the best and worst of the column." The difference is that the score itself comes from a combination of interval and point.

Distance, score and combination. The difference between every cell's score and the best score is set against the difference between the best and worst scores; this is the counterpart of crisp VIKOR's normalised difference. The method weights and sums this ratio (S) and separates out its largest value (R); it computes Q with v = 0.5. Because the score can come out negative (the third term −(1−λ)/2 is negative), the gap between the best and worst scores can come out small; in that case the ratio grows and the difference between alternatives can look exaggerated.

DecisionMind fixes, for this extension, the score function and the compromise coefficient v at 0.5; v, which is user-selectable in base VIKOR, cannot be changed here. DecisionMind does not compute crisp VIKOR's two conditions (acceptable advantage, acceptable stability) in this extension; it produces only a ranking by Q. The user must check this by looking at the S and R columns. Weights come from outside; the method does not generate weights.

A small Q is good. However, the two conditions described on the base VIKOR card (acceptable advantage, acceptable stability) are not tested by DecisionMind in this extension; the user must check this by looking at the S and R columns. Two alternatives' Q values can come out close; this is a normal and honest result of VIKOR, not a computational error. There is also an interval width lying beneath Q: two alternatives can share the same point value while differing in interval width, and this difference merges into the score, not appearing separately in Q.

Thus instead of writing:

"Cubic fuzzy VIKOR finds A2 to be the best alternative"

the report should read:

"A2 has the smallest Q with these weights and at v = 0.5; because DecisionMind does not test the compromise conditions in this extension, whether this is a single compromise solution or a compromise set should be separately assessed by looking at the S and R columns"

This extension is worth considering when experts give a criterion both as a band (at least this, at most that) and as a single most-likely value together; the aim is to carry this pair of pieces of information without reducing one to the other. Converting a measured criterion into an interval-point pair models uncertainty rather than manufacturing it. Crisp VIKOR's exit condition applies here too: where one criterion cannot be traded away, this extension also limits regret rather than eliminating it.

Value-domain violation. In every cell, μᴸ ≤ μᵁ must hold and both must lie within [0,1]; λ must also lie within [0,1]. If the order is broken, the score calculation produces a meaningless number.

Trying to change v. In base VIKOR, v is a user decision; in this extension the engine keeps v fixed at 0.5 in the code, and changing it has no effect on the result whatsoever.

Assuming the two conditions have been tested. DecisionMind does not compute the acceptable-advantage and acceptable-stability conditions in this extension; it gives only a Q ranking. Before using the phrase "single compromise solution," S and R must be read separately.

Ignoring the difference when the score comes out negative. Because the score's third term can be negative, the gap between the best and worst scores can come out small; in that case the normalised ratio grows and the difference between alternatives can look sharper than it really is.

The governing principle is this:

In this extension the result is only a Q ranking; crisp VIKOR's two compromise conditions are not tested by the engine, and the distinction between "single solution" and "compromise set" must be made by the user by looking at the S and R columns.