Bipolar fuzzy MARCOS

Cells. In crisp MARCOS 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 the effect in the opposite direction, on a scale of 0 to −1. Weights remain crisp numbers; MARCOS does not generate weights.

Scale equalisation. Bipolar MARCOS first reduces every cell to a single number with a score function: it adds the positive pole to the negative pole, adds 1, and divides by 2. Once the score function has produced these numbers, the remaining steps of crisp MARCOS run on them unchanged. Ideal (AI) and anti-ideal (AAI) rows, observed according to criterion direction, are added to the table; every cell is ratioed against the ideal.

Utility ratios and the final utility function. Every alternative's weighted row sum is ratioed against the sums of the ideal and anti-ideal rows to give two utility ratios (K⁺ and K⁻). As in crisp MARCOS, these two ratios are combined into a single final utility function. The result is a single number between 0 and 1, and the higher one is preferred.

DecisionMind fixes, for bipolar MARCOS, the score function and the crisp MARCOS steps that follow it (the ideal/anti-ideal extension, the utility ratios). Weights are taken from outside as crisp numbers.

No independent founding paper specific to bipolar MARCOS could be confirmed in the literature review. DecisionMind builds this extension by applying the crisp MARCOS algorithm to bipolar fuzzy data that has been defuzzified with a score function. It is not, like Fuzzy TOPSIS (Chen 2000), an independent fuzzy algorithm in its own right; it is the crisp method's direct adaptation to bipolar data.

The output is a final utility degree, as in crisp MARCOS, and it produces a ranking. It cannot be compared with a different set, because the ideal and anti-ideal references are built, in every analysis, from that analysis's own alternatives.

The difference is here: every cell beneath the utility degree now comes from a bipolar input that has already been reduced to a single number by the score function. Two alternatives can reach the same utility degree through different combinations of poles; the degree does not show this.

Thus instead of writing:

"In bipolar MARCOS, the alternative reaching the highest utility degree is definitively the best"

the report should read:

"The utility degree is a position relative to this set's ideal and anti-ideal references; because the positive and negative poles have already dissolved into a single number in the first step through the score function, the degree 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 piece of equipment's performance contribution alongside its implementation risk, or a supplier's price advantage alongside its delivery risk.

If a criterion is one-directional and already measured, there is no second pole, and crisp MARCOS is sufficient. The exit condition is the same as for crisp MARCOS: this extension is likewise suitable where a compromise value relative to ideal and anti-ideal references is being sought.

Deriving the negative pole from the positive pole. The negative pole comes from its own body of evidence; it is not computed from the positive pole.

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.

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. Where two independent viewpoints genuinely exist, a member of the m-polar family is used instead.

Trying to build the ideal and anti-ideal rows from the raw bipolar pairs rather than from the scored table. The ideal and anti-ideal references are built from the crisp table that follows the score function, according to criterion direction; searching for "the best pair" directly among bipolar pairs is undefined.

Changing the score function without declaring it. A different defuzzification form changes the utility ratios and, with them, the final utility degree.

The governing principle is this:

Every pole must rest on its own body of evidence, and the ideal and anti-ideal references must be built only from the crisp table that follows the score function.

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