Picture fuzzy VIKOR (Fan, Han and Wu, 2023)

Four things change. The decision logic stays the same.

Cells. In crisp VIKOR every cell holds a single number. Here every cell holds three degrees: support, abstention, rejection; their sum cannot exceed 1. This triple comes from a committee's or a panel's vote distribution on the same judgement. Weights come from outside, as single numbers. The compromise coefficient v is also supplied from outside, as in crisp VIKOR, and its default value is likewise 0.5.

Scale equalisation. Crisp VIKOR finds the best and worst value on each criterion, then measures where each alternative sits within that range. A similar logic applies here, but cost-oriented criteria are completed first. Every cell of a cost criterion is replaced by its own inverse: support and rejection swap places, abstention stays the same. This brings every criterion to read the same direction, "high support is good." Then, in each criterion column, the real row closest to each of two fixed corners (the full-support point and the full-rejection point) is found; these become that criterion's ideal and anti-ideal point respectively. This is not a hypothetical extreme as in crisp VIKOR, but a real row chosen from within the set itself.

Group utility and individual regret. Each alternative's distance to the ideal on each criterion is divided by the distance between that criterion's ideal and anti-ideal, bringing it into the 0-to-1 range. These ratios are multiplied by the weights and summed to give group utility (S); taking the largest of them gives individual regret (R). The logic up to this point is identical to crisp VIKOR; what differs is only that distance is measured between three-degree triples rather than between two points.

Result and defuzzification. The compromise index Q is built from S and R exactly as in crisp VIKOR, and tested against the two conditions (acceptable advantage, acceptable stability). The result is again either a ranking or a compromise set. Uncertainty is not defuzzified; all three degrees are used both when the ideal and anti-ideal points are chosen and when distances are computed.

DecisionMind fixes, for this member, that the ideal and anti-ideal points are chosen from the set's own real rows; no hypothetical corner is built.

The S, R and Q triple is read as in crisp VIKOR: S is total distance, R is the distance on the worst criterion, and Q balances the two. The actual result is not the Q ranking but the decision the two conditions produce; the distinction between a single compromise solution and a compromise set also holds here.

What differs is this: the ideal and anti-ideal points are now chosen from the set's own rows. These points also carry the abstention share of the vote distribution. Even if an alternative holds the highest support on a criterion, if that criterion's abstention share is large, the ideal point itself carries uncertainty, and this uncertainty is reflected in S and R.

Thus instead of writing:

"According to VIKOR, the best alternative is A1"

the report should read:

"With these weights and this vote distribution, A1 is the sole compromise solution; it comes first on both group utility and individual regret, and the gap to the second alternative exceeds the acceptance threshold"

If one of the conditions is not met, the compromise set is written out as it stands.

Use this extension when criterion evaluations come from a committee's or a survey's yes, abstain, no vote distribution, and the interests of more than one party are in conflict. VIKOR's philosophy of "the alternative that draws the least objection" still applies here; the only difference is that the input is gathered in the form of a vote distribution. As the picture fuzzy data-type card explains, the abstention share must be counted separately; it must not be derived from a single support percentage.

There is no need to expand a measured criterion into a picture fuzzy triple. DecisionMind requires a single data type; if some criteria in the matrix are based on votes and others on measurement, all of them must be written in the same type. Spherical fuzzy and Pythagorean fuzzy data types use similar triple or pair structures, but their constraint forms differ; those cards should be consulted.

The exit condition is the same as for crisp VIKOR. If no compromise is acceptable on one criterion, VIKOR, including this extension, limits regret but does not eliminate it; if strict elimination is required, dominance-based methods should be used instead.

Violating the value space. In every cell, the sum of support, abstention and rejection must not exceed 1. This check must be made before the ideal and anti-ideal points are chosen.

Changing the defuzzification method without justification. Using support minus rejection as the score function is the established choice, but other functions can also be defined. Whichever function is chosen, the same function must be used throughout every step of the same analysis; changing it partway through distorts the ranking.

Forgetting to complete a cost criterion. If support and rejection are left unswapped on a cost-oriented criterion, the alternative with high support on that criterion, that is, the expensive one, is treated as ideal, and the ranking is reversed.

Reporting only the alternative with the smallest Q. VIKOR's defining feature is the compromise set, not a single winner. Reporting only the Q ranking without checking the two conditions means the method has been applied incorrectly.

Defuzzifying first and then running crisp VIKOR. Reducing the three degrees to a single number from the outset and then applying the crisp method is not this extension. The abstention and opposing-vote information is erased at the first step.

The governing principle is this:

Picture fuzzy VIKOR exists to carry the abstention share of a committee's or survey's vote distribution through to the ideal and anti-ideal points. Any implementation that fabricates a degree or crisps the input from the outset destroys the method's sole contribution.

---