Extension card · m-Polar
Bipolar fuzzy COPRAS
This is the form of COPRAS for situations where each cell records, separately, a judgement's effect in the desired direction and its effect in the opposite direction. The two effects are reduced to a single score in the very first step, and the rest of the calculation follows the same path as crisp COPRAS.
Base method
COPRAS →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
m-Polar →
What this data type is, when to use it, how to write it in a cell: the family's full account is here.
What Changes from the Base Method?
Cells. In crisp COPRAS 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; COPRAS does not generate weights.
Scale equalisation. Bipolar COPRAS 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. 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 COPRAS run on them unchanged: every column is equalised proportionally by dividing by its own column sum, then multiplied by the weight.
Benefit and cost sums, combination. Weighted values are added into either the benefit sum or the cost sum according to the criterion's original direction. This split is made using the direction label from before the score function, because cost criteria have already been complemented by the time the score function runs. These two sums are combined into a relative-significance value exactly as in crisp COPRAS, then divided by the highest value to convert to a percentage.
DecisionMind fixes, for bipolar COPRAS, the score function and the crisp COPRAS steps that follow it (proportional normalisation, the relative-significance formula). Weights are taken from outside as crisp numbers.
No independent founding paper specific to bipolar COPRAS could be confirmed in the literature review. DecisionMind builds this extension by applying the crisp COPRAS 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.
How to Read the Output
The output is a utility degree, as in crisp COPRAS: the highest value is set equal to 1 (100 per cent), and the others are scaled against it. It cannot be compared with a different set, because the highest value is determined, 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 COPRAS, the alternative reaching the highest utility degree is definitively the best"
the report should read:
"The utility degree is a percentage position relative to this set's best alternative; 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"
When to Prefer This over the Base Method
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 vehicle's fuel saving alongside its maintenance risk, or a quotation's price advantage alongside its hidden-cost risk.
If a criterion is one-directional and already measured, there is no second pole, and crisp COPRAS is sufficient. The exit condition is the same as for crisp COPRAS: this extension is likewise suitable only where the logic of keeping the benefit and cost sums separate is acceptable.
Mistakes Specific to This Extension
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.
Splitting the benefit and cost sums by the magnitude of the scored value rather than by the direction label from before the score function. Whether a criterion is a benefit or a cost is fixed by its original label before the score function; writing a criterion into the benefit side just because its scored number looks large is wrong.
Changing the score function without declaring it. A different defuzzification form changes the proportional equalisation and, with it, the utility degree.
The governing principle is this:
Every pole must rest on its own body of evidence, a criterion's benefit or cost direction must be fixed before the score function, and this order must not be changed during the calculation.
Cases
The first case is DecisionMind's engine validation fixture. The source field in the manifest shows an author and year for this table. But this field does not point to a real paper; it points to the engine's own test fixture, which is why the case is presented as an illustrative example. The second case is entirely fictional.
1. Illustrative example: Comparing three machine quotations on two criteria
A business is comparing three machine quotations on two criteria: efficiency contribution (higher is better) and failure risk (lower is better). Every cell is given as a bipolar fuzzy pair.
| Quotation | Efficiency contribution | Failure risk |
|---|---|---|
| A1 | (0.7; −0.2) | (0.6; −0.3) |
| A2 | (0.5; −0.4) | (0.8; −0.1) |
| A3 | (0.6; −0.3) | (0.4; −0.5) |
| Weight | 0.6 | 0.4 |
After the score function, the crisp scores come out at 0.75 and 0.35 for A1; 0.55 and 0.15 for A2; and 0.65 and 0.55 for A3. The benefit and cost sums are accumulated separately and combined into the relative-significance value.
| Quotation | Utility degree | Rank |
|---|---|---|
| A2 | 1.000 | 1 |
| A1 | 0.820 | 2 |
| A3 | 0.653 | 3 |
A2 comes first in the utility degree because it has the lowest failure risk (the lowest crisp failure value). Although it has the lowest efficiency contribution, this does not offset its low-risk advantage.
The business's hesitation: if the weights were swapped, that is, 0.6 to failure risk and 0.4 to efficiency, the ranking stays the same: A2 first (1.000), A1 second (0.655), A3 third (0.493). Even if A1's efficiency-contribution cell were pulled down one notch (0.5 instead of 0.6), the ranking would not change. In this example the ranking is robust to both a weight swap and a single-cell change.
In the report: "Quotations have been scored with bipolar fuzzy pairs, and the positive and negative poles have been reduced to a single number with the score function. A2 reaches the highest utility degree (1.000) thanks to its lowest failure risk; a weight swap and a single-notch score change do not alter the ranking."
Source: DecisionMind's validation fixture for the bipolar fuzzy COPRAS engine; utility degrees and sensitivity values were obtained by independently recomputing the kernel in Python.
2. Public transport: Choosing among three bus-fleet renewal quotations
A municipality will choose one of three procurement quotations to renew its bus fleet. Two criteria are set: contribution to passenger satisfaction (higher is better) and maintenance failure risk (lower is better). Each criterion has two separate bodies of evidence: a pilot-route satisfaction survey supplies the desired-direction effect, and a technical service record supplies the opposite-direction effect.
| Quotation | Contribution to passenger satisfaction | Maintenance failure risk |
|---|---|---|
| O1 | (0.66; −0.1) | (0.5; −0.34) |
| O2 | (0.44; −0.2) | (0.5; −0.82) |
| O3 | (0.5; −0.1) | (0.5; −0.58) |
| Weight | 0.55 | 0.45 |
O1 delivers the highest contribution to satisfaction but also has the highest failure risk. O2 is the opposite: the lowest contribution to satisfaction and the lowest failure risk. O3 sits in the middle on both criteria.
| Quotation | Utility degree | Rank |
|---|---|---|
| O1 | 1.000 | 1 |
| O3 | 0.840 | 2 |
| O2 | 0.719 | 3 |
The municipality's hesitation: if the weights were swapped, that is, 0.55 to failure risk and 0.45 to satisfaction, the ranking stays the same: O1 first (1.000), O3 second (0.828), O2 third (0.703). Even if O1's failure risk were worsened by one notch (from −0.34 to −0.44), the ranking would not change. The ranking is robust under both scenarios.
In the report: "When passenger satisfaction is given a weight of 0.55, O1 reaches the highest utility degree. When the weight is shifted to failure risk, and when O1's risk cell is worsened by one notch, the ranking still does not change; the result is robust to these two scenarios."
3. What Not to Do
In the public-transport example, deriving O2's failure-risk cell from its positive pole and writing "0.5; −0.4" is wrong. This collapses two separate bodies of evidence, the satisfaction survey and the technical service record, into a single source. The second error is adding the failure-risk criterion to the benefit sum because its scored value looks high; a criterion's direction is fixed by its original label before the score function, and cannot be changed afterwards. The third error is reporting the illustrative example's A2 utility degree of 1.000 as "perfect"; 1.000 only means it is the best among these three quotations.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/bf-copras
Zavadskas, E. K., & Kaklauskas, A. (1996). Determination of an efficient contractor by using the new method of multicriteria assessment. In International Symposium for the Organization and Management of Construction: Shaping Theory and Practice, Vol. 2: Managing the Construction Project and Managing Risk (CIB W65), 94–104. (no DOI)
Zhang, W.-R. (1994). Bipolar fuzzy sets and relations: A computational framework for cognitive modeling and multiagent decision analysis. Proceedings of NAFIPS/IFIS/NASA '94, 305–309. DOI: 10.1109/IJCF.1994.375115
Chen, J., Li, S., Ma, S., & Wang, X. (2014). m-Polar fuzzy sets: An extension of bipolar fuzzy sets. The Scientific World Journal, 2014, 416530. DOI: 10.1155/2014/416530
Alghamdi, M. A., Alshehri, N. O., & Akram, M. (2018). Multi-Criteria Decision-Making Methods in Bipolar Fuzzy Environment. International Journal of Fuzzy Systems, 20(6), 2057–2064. DOI: 10.1007/s40815-018-0499-y