Fuzzy COPRAS

Four things change; the benefit/cost ratio logic does not.

Cells. In crisp COPRAS every cell is a single number. Here every cell is three numbers: lowest, most likely, highest (a triangular fuzzy number). Criterion weights can also be entered as triangles; but in the weighting step DecisionMind uses not the weight's three corners but its average (its centroid), and applies this single number identically to all three corners of the triangle. So the cells' uncertainty is preserved through the calculation, while the weight's uncertainty ends at that step.

Scale equalisation. Crisp COPRAS divides every column by its own sum. Here the same is done, but the three corners (lowest, most likely, highest) are each divided by their own column sum separately. The triangle's shape is not distorted in this step, only its magnitude is scaled. The column sum is computed separately for each corner.

Distance/score/aggregation. COPRAS's logic of "sum of benefits, sum of costs, and their ratio" is run separately on each of the three corners. The method computes a separate benefit sum, a separate cost sum and a separate relative-significance value (Q) for each corner. In crisp COPRAS there was a single Q; here there are three separate Q's, one for each corner.

Result and defuzzification. The method takes the average of the three corners' Q values, that is the centre of gravity, to descend to a single number; this is the defuzzification step that ends the uncertainty. It divides the defuzzified value by the highest value and converts it to a percentage. The output is, in the same form as crisp COPRAS, a single degree of utility (the best alternative scores 100).

DecisionMind applies column-sum normalisation and centre-of-gravity defuzzification separately and consistently for each corner across this family. Even when a criterion weight is entered as a triangle, weighting proceeds through a single number (the centroid); the three corners are not weighted separately.

The output is, in the same form as crisp COPRAS, a degree of utility and a rank, and is read the same way. The best alternative scores 100, and the others receive a percentage relative to it; this percentage cannot be compared with the result of a different analysis.

Beneath the score lies an uncertainty coming from the three corners, which the centre-of-gravity defuzzification conceals. The score gap between two alternatives may be robust or fragile depending on the width of the triangles and the distribution of weights. The report should therefore show not only the degrees of utility but also at which weight distribution the ranking changes.

Thus instead of writing:

"Because Fuzzy COPRAS is fuzzy, its result is more reliable"

the report should read:

"Because criteria were assessed approximately, the uncertainty has been carried through the calculation and reduced to a single number by the centre of gravity; A2 leads with 100.00, and this lead holds until the weight distribution is shifted markedly in favour of C2"

Use this when criteria are assessed through a verbal or approximate judgement and reducing that judgement to a single number would create an artificial precision. If a measured criterion exists (price, duration, quantity), there is no need to expand it into a triangle; since DecisionMind requires a single data type, a measured criterion in the same matrix is written as a triangle with all three components equal, so its width is zero and it adds no information.

Crisp COPRAS's exit condition applies exactly here too. If no compromise is acceptable on one criterion, this extension is also compensatory and does not eliminate anything below a threshold. If weights rather than a ranking are needed, look to weighting methods such as AHP/BWM/SWARA/Entropy/CRITIC rather than the COPRAS family.

Violating the value range. Every cell must follow the order lowest ≤ most likely ≤ highest, with no component negative; if this order breaks down the calculation becomes meaningless.

Forgetting that the defuzzification method is a hidden assumption. DecisionMind uses the centre of gravity (the average of the three corners); this is the canonical choice but not the only one. A different defuzzification rule (such as taking only the most likely value) can give a different ranking; the rule used must be stated in the report.

Not noticing that the engine reduces a triangular weight to its centroid. Even if a criterion weight is entered with three components, such as (0.30; 0.35; 0.40), DecisionMind reduces it to 0.35 and uses it that way; writing in the report that "weight uncertainty is also carried" is wrong, because the weight's three corners do not enter the calculation separately.

Defuzzifying first and then running crisp COPRAS. Reducing the triangles to their centroid from the outset and running the crisp method skips Fuzzy COPRAS's own steps. In the illustrative example below, the two paths give nearly the same numbers (A2 100.00, A3 89.49 and 89.44), because COPRAS's summation-based structure happens to be insensitive to the order of defuzzification in this data set. This near-coincidence is not a general guarantee. Because the method carries the three corners separately, the two paths can diverge under a different weight distribution or matrix. Moreover, once a table has been defuzzified first, the width information in the corners can never re-enter the calculation.

The governing principle is this:

Fuzzy COPRAS exists to carry the approximation of criteria through until the benefit/cost ratio is computed; any application that crystallises the input from the outset or violates the value range removes this carrying-through's sole contribution.