Complex Fuzzy VIKOR

Four things change; the two conditions, acceptable advantage and acceptable stability, are computed here too, exactly as in crisp VIKOR.

Cells. In crisp VIKOR every cell is a single number. Here every cell consists of four numbers: a support amplitude r_μ and support phase ω_μ, and a rejection amplitude r_ν and rejection phase ω_ν. Amplitudes lie between 0 and 1; phases are angles between 0 and 2π. Amplitude measures the same thing as the support and rejection degree in intuitionistic fuzzy. Phase is a separate dimension, carrying where the judgement sits on a cyclical or second axis. Criterion weights are taken from outside as crisp numbers.

Scale equalisation (direction reversal). On a cost criterion, crisp VIKOR swaps the best and worst value. Here, on a cost criterion, the support pair (amplitude and phase together) and the rejection pair (amplitude and phase together) swap places wholesale; not only the amplitudes but the phases move together with them.

Best and worst value. On every criterion, the best value is built by bringing together that column's highest support amplitude and phase with its lowest rejection amplitude and phase; the worst is built the opposite way round. This is not the value of an alternative that actually exists, but a reference point built from the separate best of four components; it is crisp VIKOR's "the best and worst of every column" logic carried into four dimensions.

Distance, score and aggregation. Each alternative's distance to these two reference points is measured with a Euclidean distance over all four components (support amplitude, rejection amplitude, support phase, rejection phase; phases are brought into the 0–1 range by dividing by 2π). This distance's ratio to the distance between the best and worst is the counterpart of the normalised difference in crisp VIKOR. The method weights and sums this ratio (S) and separates out its largest value (R); it then computes Q with v = 0.5. From this point on the calculation proceeds entirely through crisp numbers.

DecisionMind fixes the compromise coefficient v at 0.5 for this extension; v, which can be chosen by the user in base VIKOR, cannot be changed here. DecisionMind genuinely computes and produces the two conditions (acceptable advantage: DQ = 1/(number of alternatives − 1); acceptable stability: the candidate also ranking first by S or by R) in this extension. Weights are taken from outside; the method does not generate weights.

A smaller Q is better, as in base VIKOR. Whether the two conditions are met, that is, whether the outcome is a single solution or a set, is computed here too, because DecisionMind genuinely runs this test. The difference is this: beneath Q sits a four-component (amplitude-amplitude-phase-phase) source of uncertainty, and this uncertainty is carried only inside the distance calculation, becoming invisible in the comparison. Where phase's meaning has not been clearly defined, half of S and R look to the reader like a repeated score of unclear origin.

Thus instead of writing:

"Complex fuzzy VIKOR is more reliable because it carries uncertainty in two dimensions"

the report should read:

"This result was computed over a distance carrying the difference in both the support/rejection amplitude and the phase angle; phase here represents (X), and without that definition half of S and R remain unreadable"

This extension is worth considering when the support or rejection given to a judgement genuinely comes from two independent dimensions: for instance, how strong an assessment is (amplitude) might be known separately from which cyclical phase it holds under (phase). If these two dimensions are not genuinely independent, that is, if phase is derived from amplitude, the complex structure adds nothing to intuitionistic fuzzy VIKOR and only makes the model harder to explain. Turning a measured criterion into an amplitude-phase pair does not model uncertainty, it manufactures it. Crisp VIKOR's exit condition applies here too: where no compromise is acceptable on a criterion, this extension limits regret but does not eliminate it either.

Giving phase proportionally to amplitude, or arbitrarily. If phase is derived from amplitude (a rule such as ω = 2π·r), the distance calculation produces a result indistinguishable from intuitionistic fuzzy VIKOR; the complex structure adds no extra information at all. Case 1 below uses exactly this situation and states it openly.

Mistaking the phase angle for degrees and entering it as such rather than radians. The engine expects ω in radians between 0 and 2π; entering a value on a 0–360 degree scale directly violates the value range.

Swapping only the amplitudes on a cost criterion and forgetting the phases. In this extension the cost transformation swaps the support pair (amplitude + phase) and the rejection pair (amplitude + phase) as a whole. Swapping only the amplitudes and leaving the phases as they are leaves the direction reversal half-done.

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

Ignoring the two conditions' outcome and declaring the alternative with the smallest Q the "winner." Because DecisionMind genuinely computes the conditions in this extension, ignoring them means the most common mistake in base VIKOR applies here just as much.

The governing principle is this:

Complex fuzzy VIKOR's phase adds something only when it genuinely comes from a source independent of amplitude; the two compromise conditions are genuinely tested here by the engine and must be given in the report together with the S and R columns.