m-Polar and Bipolar Fuzzy

This family covers two closely related structures.

The bipolar fuzzy structure expresses an alternative's standing on a criterion with two figures: the positive pole measures how strongly the judgement holds in the desired direction, between 0 and 1; the negative pole measures how strongly it holds in the opposite direction, between 0 and −1. For the judgement "this supplier lowers cost", the positive pole might be 0.6 (it does lower cost) and the negative pole −0.3 (it also raises cost on some line items). The two poles do not constrain each other; there is no rule governing their sum.

The m-polar fuzzy structure instead grades the same judgement separately from m distinct points of view: three stakeholder groups, four regional offices, five periods. Each point of view carries its own membership degree between 0 and 1, and these degrees are independent of one another. The bipolar structure can be thought of as the special case of the m-polar structure where m = 2 and the sign of the second pole is reversed.

The key difference from the intuitionistic fuzzy structure is this: there, the second figure is a degree of rejection, and its sum with the first must not exceed 1; here, the second figure is the judgement's effect in the opposite direction, not rejection, and there is no constraint on the sum. Something can carry a strong positive effect and a strong negative one at the same time.

The bipolar structure is used where an alternative produces both benefit and harm at once on a criterion, and the two do not cancel out: a drug's effect alongside its side effect, a policy's winners alongside its losers, an investment's return alongside its risk. Reducing the two to a single "net" figure erases who bears the harm.

The m-polar structure is used where the same judgement is graded independently from several sources or points of view, and those grades should not be collapsed to an average: stakeholder groups, regions, periods, measurement channels.

Conversely, if a criterion is one-directional and single-source ("is the price low"), there is no second pole or second point of view, and the structure should not be forced.

Yes, but each pole or point of view must have its own source of information. Two steps are needed.

The first is converting the criterion into a judgement: "the alternative affects this criterion in the desired direction." A measurement alone carries no pole; a judgement is needed as to whether the measurement works in the desired direction or the opposite one.

The second is, for a bipolar structure, deriving the positive pole from evidence in the desired direction and the negative pole from evidence in the opposite direction, separately; for an m-polar structure, taking each point of view's degree from that point of view's own data. Deriving the negative pole from the positive one (saying −0.4 negative because the positive is 0.6) empties the structure of its contribution, and writing the same figure m times in an m-polar structure does the same.

What must not be done is splitting a single figure into two poles by calling it "somewhat positive, somewhat negative", or copying one expert's score across three stakeholder groups.

The negative pole in a bipolar structure does not mean "this judgement is false"; it means "this alternative also affects the criterion in the opposite direction". A drug both lowers blood pressure (positive pole) and increases kidney load (negative pole); the second is not a rejection of the first but a separate effect of the same drug.

For this reason:

"If the positive pole is 0.6, the negative pole is −0.4, because the sum must equal 1"

should give way to:

"The positive pole comes from evidence in the desired direction, the negative pole from evidence in the opposite direction; neither constrains the other"

If a degree of rejection is needed, use an intuitionistic fuzzy structure; if both rejection and an unknown component are needed, use a neutrosophic structure.

The chief advantage of the bipolar structure is that it does not dissolve benefit and harm into a single "net effect". Two alternatives with the same net effect may differ, one with small benefit and small harm, the other with large benefit and large harm; the second is riskier, and the bipolar structure keeps that difference visible.

The advantage of the m-polar structure is that it makes agreement or conflict between points of view visible. If three stakeholder groups' degrees are (0.8, 0.7, 0.9), there is agreement; if they are (0.9, 0.2, 0.8), one group is clearly opposed, and the average, 0.63, would conceal this.

In the bipolar structure, the two poles must be reduced to a single value for ranking, and this reduction is a choice; how heavily the negative pole is weighted changes the result. In the m-polar structure, how the points of view are combined (equal weight, worst case, majority) is likewise a choice, and it must be justified in the report.

The burden of data collection rises: every cell needs two or m figures, and each must have its own source. That the points of view are genuinely independent is also an assumption; two different treatments of the same data are not "two points of view". Nor does this family carry an unknown component; where missing information must be represented separately, a neutrosophic structure is needed.

m is the actual number of points of view for which data were collected; it is not a parameter the analyst chooses. If views were obtained from three stakeholder groups, m = 3; if there is no data for a fourth group, m is not written as 4 with a fourth degree guessed in. The same principle holds for the bipolar structure: if no effect in the negative direction was measured, the negative pole is written as 0, not guessed.

The most frequent mistake is deriving the negative pole from the positive one, which collapses the structure to a single figure. The second is treating the negative pole as rejection and applying a "must sum to 1" constraint that does not exist in this structure.

In the m-polar structure, a common mistake is copying a single-source score across m points of view, or averaging the points of view down and treating the result as though the structure had been used properly. Inflating the number of poles "to look richer", and writing guesses into poles for which there is no data, are equally methodologically unsound.

The governing principle is this:

Every pole and every point of view must rest on its own data; the negative pole must not be derived from the positive one, nor one point of view from another.