Bipolar Fuzzy WASPAS

Four things change; the logic of blending the sum and the product does not.

Cells. In crisp WASPAS every cell is a single number. Here every cell is two numbers: a positive pole (between 0 and 1, the desired-direction effect) and a negative pole (between −1 and 0, the opposite-direction effect). The two poles do not bound one another; there is no rule for their sum. Weights remain crisp numbers; WASPAS does not generate weights, it takes them from outside.

Scale equalisation. Crisp WASPAS brings every column into the 0-to-1 range by dividing by the best value. Here a benefit criterion is left unchanged; for a cost criterion the poles are complemented, the positive pole is subtracted from 1, and the negative pole's absolute value is subtracted from 1 and its sign rewritten as negative. This complementing is the bipolar counterpart of crisp WASPAS turning a "lower is better" criterion into "higher is better."

Sum and product components. Crisp WASPAS builds the weighted sum (WSM) and the weighted product (WPM) with plain numbers. Here these two components are built with bipolar addition and multiplication: the WSM component is computed with a bipolar weighted-addition operator, and the WPM component with a bipolar weighted-geometric operator. Both still yield, in the end, a bipolar number (one positive pole, one negative pole); at no step of the calculation is it reduced early to a single number.

Combination and defuzzification. Crisp WASPAS weights the WSM and WPM scores with λ and adds them. Here these two bipolar components are likewise combined with bipolar addition; λ again determines the share carried by the sum logic. The single bipolar number obtained at the end is then defuzzified into one number with a score function (adding the positive pole to the negative pole, adding one, and dividing by two). Defuzzification happens only at this final step; just as crisp WASPAS performs no defuzzification in its first step, here too the two poles are carried separately until the very last moment.

DecisionMind fixes, for bipolar WASPAS, these addition and multiplication operators, the score function, and the combination with λ. Weights are taken from outside as crisp numbers; the method does not generate weights.

The output is a combined score and a ranking, as in crisp WASPAS, and is read the same way. The size of the score depends on the chosen λ; a different λ can give a different ranking, and the report should state which λ was used.

The difference is here. Beneath the score lies a bipolar calculation carried through to the last step, but this calculation descends to a single number only at that final step, through the score function. Two alternatives can reach the same score through different combinations of poles; an alternative with a strong positive pole but also a noticeable negative pole can give the same score as one where both poles are moderate.

Thus instead of writing:

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

the report should read:

"The two poles are carried through with their own arithmetic all the way to the last step; the combined score reflects the combination of these two poles only once they are reduced to a single number at that final step, and does not directly 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 carried separately up to a score that blends the sum and the product logic. Typical situations: an investment's return alongside its risk, or a policy's gains for one group alongside its losses for another.

The situation calling for a return to crisp WASPAS is when the criterion is one-directional and already measured. Forcibly splitting a single measured value into two poles is not modelling uncertainty; each pole must have its own body of evidence.

The exit condition is the same as for crisp WASPAS: if the data contains a zero or negative positive pole, the product component becomes undefined, and this extension, like the base method, is compensatory if no compromise is acceptable on one criterion.

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.

Leaving λ at its default of 0.5 without testing it. The WSM and WPM components, even in bipolar arithmetic, can suggest a different ranking; in this extension's validation example, raising λ from 0.5 to 0.7 swaps the ranking, and the difference is very small. Leaving λ untested keeps this fragility hidden.

Skipping pole complementing on a cost criterion. Using the negative pole directly without complementing it leaves the "lower is better" criterion's direction unreversed and makes the result meaningless.

The governing principle is this:

Bipolar WASPAS carries the two poles through with their own arithmetic to the last step, and only there reduces them to a single number with the score function; deriving the negative pole from the positive one, leaving λ untested, or skipping the pole complementing invalidates this carrying-through.