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

Three things change; the decision logic does not.

Cells. In crisp TOPSIS 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; there is no rule for their sum. Bipolar fuzzy TOPSIS is the special case of the m-polar family at m = 2 where the second pole's sign is reversed. Here it is not different viewpoints being measured through m, but the two directions of a single judgement. Weights remain crisp numbers; TOPSIS does not generate weights, it takes them from outside.

Scale equalisation. Bipolar TOPSIS 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. This operation gives a cell with a positive pole of 1 and a negative pole of 0 a score of exactly 1; a cell with both poles at 0 gets a score of 0.5. 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. This complementing is the bipolar counterpart of turning crisp TOPSIS's "lower is better" criterion into "higher is better." Once the score function has produced these numbers, the remaining steps of crisp TOPSIS run on them unchanged. Every column is divided by its own magnitude, multiplied by the weight, and the ideal and anti-ideal alternatives are built.

Distance and closeness score. Once the score function is finished, the calculation follows exactly the same path as crisp TOPSIS. The method measures every alternative's Euclidean distance to the ideal and the anti-ideal, and computes the closeness score as the ratio of the distance to the anti-ideal over the sum of the two distances. The result is again a single number between 0 and 1.

The difference from Fuzzy TOPSIS (Chen 2000) lies here. There, uncertainty is carried in triangular form all the way to the vertex-distance calculation, and only descends to a single number at the very last step. Here, the positive and negative poles merge into a single number as early as the first step, through the score function; the remaining four steps operate on this single number.

DecisionMind fixes, for bipolar TOPSIS, the score function and the crisp TOPSIS steps that follow it (vector normalisation, Euclidean distance). Weights are taken from outside as crisp numbers; the method does not generate weights.

The output is a closeness score and a ranking, as in crisp TOPSIS; it is read the same way. It is not a percentage, it cannot be compared with a different analysis, and the ideal point and the scores change when the alternative set changes.

The difference is here: beneath the score lies a two-poled input, but this input has already dissolved into a single number as early as the first step, through the score function. The score itself no longer shows how strong either pole was; two alternatives can reach the same score through different combinations of poles. A cell with a positive pole of 0.9 and a negative pole of −0.1 produces a score of (1 + 0.9 − 0.1)/2 = 0.9. A cell with a positive pole of 0.6 and a negative pole of −0.4 also produces the same score, (1 + 0.6 − 0.4)/2 = 0.9. The first case is a strong, one-directional effect; the second is a weaker judgement whose opposing effect is also relatively strong.

Thus instead of writing:

"Because bipolar TOPSIS evaluates the two poles together, the result is richer"

the report should read:

"The positive and negative poles have already been reduced to a single number by the score function in the first step; the closeness score carries only the difference between these single numbers, and does not show which pole contributed what to the result"

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 supplier that both lowers one cost item and raises another, an investment's return alongside its risk, a drug's effect alongside its side effect.

If a criterion is one-directional and already measured (price alone, time alone), there is no second pole, and crisp TOPSIS is sufficient. Forcibly splitting a single measured value into two poles is not modelling uncertainty; as explained on the data-type card, each pole must have its own body of evidence.

The exit condition is the same as for crisp TOPSIS: if no compromise is acceptable on one criterion, the TOPSIS family, including its bipolar form, is compensatory and will not eliminate anything below a threshold.

Deriving the negative pole from the positive pole. Saying "if the positive pole is 0.7, the negative pole is −0.3, because the total must equal 1" collapses the structure into a single number. The negative pole comes from its own body of evidence; it is not computed from the positive one.

Reading the negative pole as a degree of rejection. In an intuitionistic fuzzy structure, the second number is a degree of rejecting the judgement, and its sum with the first number cannot exceed 1. In the bipolar structure there is no such constraint; 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 fuzzy set measures a single criterion's two-directional effect; the positive and negative poles are an opposite-property pair for the same judgement. An m-polar set at m = 2, by contrast, measures two independent viewpoints, both between 0 and 1, with no opposite-property relationship. Where two independent viewpoints genuinely exist (such as two stakeholder groups), a member of the m-polar family (mpf-*) is used, not bipolar TOPSIS.

Changing the score function without declaring it. Other defuzzification forms besides (1 + positive + negative)/2 have also been defined, and can give a different ranking. DecisionMind fixes this function; if a comparison is being made against another piece of software, this difference should be checked first.

Forcibly splitting a measured criterion into poles. A measured value should remain crisp; poles are meaningful only when there are genuinely two separate bodies of evidence.

The governing principle is this:

Every pole must rest on its own body of evidence, and the score function must stay fixed throughout the analysis. Deriving the negative pole from the positive one, or changing the defuzzification form, erases the extension's own contribution.

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