Bipolar fuzzy VIKOR (Alghamdi, Alshehri & Akram, 2018)

Cells. In crisp VIKOR every cell is a single number. Here every cell is two numbers: a positive pole and a negative pole. The positive pole states, on a scale of 0 to 1, how strongly the alternative performs in the desired direction on this criterion. The negative pole states how strongly the same alternative performs in the opposite direction, on a scale of 0 to −1. The two poles do not bound one another. Bipolar fuzzy VIKOR is the special case of the m-polar family at m = 2 where the second pole's sign is reversed. Weights remain crisp numbers; VIKOR does not generate weights.

Scale equalisation. Bipolar VIKOR first reduces every cell to a single number. DecisionMind uses a score function for this: it adds the positive pole to the negative pole, adds 1, and divides by 2. For a cost criterion, the poles are complemented before this step: the positive pole is subtracted from 1, and the negative pole is rewritten as its absolute value minus 1. Once the score function has produced these numbers, the remaining steps of crisp VIKOR run on them unchanged. Every criterion's best and worst value is found and processed as in crisp VIKOR.

Group utility, individual regret and the compromise index. Once the score function is finished, the calculation follows exactly the same path as crisp VIKOR. Every alternative's weighted average distance (the group utility, S) and its single worst-criterion distance (the individual regret, R) are computed. The two are combined with the compromise coefficient to give the Q index. The conditions of acceptable advantage and acceptable stability are tested exactly as in crisp VIKOR.

DecisionMind fixes, for bipolar VIKOR, the score function and the crisp VIKOR steps that follow it (the compromise coefficient, the DQ threshold). Weights and the compromise coefficient are taken from outside.

The output is a Q index and an ascending ranking based on it, as in crisp VIKOR; the lowest Q is the best compromise. A low Q alone does not mean "the best": the compromise proposal reduces to a single alternative only when the conditions of acceptable advantage and stability are met; if they are not, a set is proposed instead.

The difference is here: the S and R values beneath the Q now come from a two-poled input that has already been reduced to a single number by the score function. The VIKOR family is already a weight-sensitive method. In its bipolar form, this sensitivity persists in exactly the same way once the poles have dissolved through the score function. Because S and R are computed separately, this sensitivity can be more pronounced than in TOPSIS.

Thus instead of writing:

"Bipolar VIKOR declares the alternative with the lowest Q the definitive winner"

the report should read:

"The alternative with the lowest Q is proposed alone only when the conditions of acceptable advantage and stability are met; if they are not, the compromise set contains more than one alternative, and this set must be shown in the report"

Use this extension when an alternative has a measurable effect in both the desired and the opposite direction on a criterion, and these two effects need to be preserved rather than reduced to a single net figure. Typical situations: a feed that both raises yield and carries a side effect, or an investment's return alongside its risk.

If a criterion is one-directional and already measured, there is no second pole, and crisp VIKOR is sufficient. The exit condition is the same as for crisp VIKOR. Where the decision seeks a compromise solution and no single criterion carries absolute priority, the VIKOR family, including its bipolar form, is suitable. If no compromise is acceptable on one criterion, pre-screening with a threshold on that criterion should be done first.

Deriving the negative pole from the positive pole. Saying "if the positive pole is 0.7, the negative pole is −0.3" collapses the structure into a single number. The negative pole comes from its own body of evidence.

Reading the negative pole as a degree of rejection. In the bipolar structure there is no constraint that the two poles must sum to no more than 1; the negative pole is the opposite-direction effect, not a rejection.

Confusing the bipolar fuzzy set with the m-polar fuzzy set at m = 2. A bipolar set measures a single criterion's two-directional effect. An m-polar set at m = 2 measures two independent viewpoints, both between 0 and 1. Where two independent viewpoints genuinely exist, a member of the m-polar family is used, not bipolar VIKOR.

Ignoring the compromise set. Declaring the alternative with the lowest Q the sole winner without the acceptable-advantage condition being met; in that case the set contains more than one alternative, and the report must state this.

Changing the score function without declaring it. A different defuzzification form changes the S and R values, and with them the Q.

The governing principle is this:

Every pole must rest on its own body of evidence, the score function must stay fixed, and when the compromise set cannot be reduced to a single alternative, this must be explicitly reported.

---