Intuitionistic fuzzy COPRAS

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

Cells. In crisp COPRAS every cell is a single number. Here every cell is a pair: a degree of support (μ) and a degree of rejection (ν), whose sum cannot exceed 1. Criterion weights are entered as crisp (single) numbers; only the decision matrix is intuitionistic fuzzy.

Scale equalisation. DecisionMind first reduces every pair to Chen and Tan's (1994) score difference: support minus rejection (μ − ν). Since this difference can range from −1 to 1, it is shifted by adding 1. The column-sum-based normalisation used in crisp COPRAS is then applied to the scores obtained after this shift. The shift prevents negative values from disrupting the column sum.

Distance / score / aggregation. The normalised, weighted scores are split into a benefit sum and a cost sum, as in crisp COPRAS, and the relative significance value (Q) is built from the same ratio.

Result and defuzzification. Defuzzification happens at the very start: the support-rejection pair is reduced to a single score (μ − ν) in the first step, and the share of indecision (π = 1 − μ − ν) does not enter any subsequent step of the calculation on its own. The output is directly a single benefit degree, in exactly the same form as in crisp COPRAS, with the best alternative set at 100.

DecisionMind fixes the Chen-Tan score difference and the +1 shift throughout this family; it does not use the share of indecision as a separate penalty or bonus, nor does it normalise support and rejection separately.

The output is a benefit degree and a ranking, in the same form as in crisp COPRAS, and is read the same way. The best alternative receives 100, and the others receive a percentage relative to it.

The difference lies here: this score rests only on the difference between support and rejection, and never shows the share of indecision. The share of indecision shows how much information is missing or contradictory. The same μ − ν difference can come from very different evidentiary situations. For example, at μ=0.60/ν=0.30 the indecision is 0.10; at μ=0.90/ν=0.60 the indecision is not 0.10, yet the difference is still 0.30. The benefit degree does not distinguish between these two.

Thus instead of writing:

"IF-COPRAS found this alternative more reliable because it takes indecision into account"

the report should read:

"The benefit degree rests only on the difference between support and rejection; A3 leads with 100.00, and this difference does not by itself mean 'low indecision' unless the support-rejection pairs behind each criterion are separately examined"

Use this method when an assessment is given as support for and rejection of a judgement (for example "this alternative satisfies this criterion"), and the opposing view can be measured from a separate source, that is, a separate body of evidence. If the degree of rejection is computed as 1 − μ, that is, if there is no separate source for the opposing view, there is no reason to move to an intuitionistic fuzzy structure; if a criterion is measured, or expressed by a single score, the crisp or fuzzy structure is sufficient.

The exit condition of crisp COPRAS applies here in exactly the same way. If no compromise is acceptable on one criterion, this extension is also compensatory and will not eliminate anything below a threshold. The matrix must be of a single data type; a measured criterion is embedded in the same matrix as a "full support, zero rejection" pair such as (1, 0). This is an honest embedding that adds no uncertainty at all.

Range violation. In every cell, μ and ν must lie in [0,1] and their sum must not exceed 1. The share of indecision (π = 1 − μ − ν) must not come out negative.

Forgetting that the score function is a hidden assumption. DecisionMind uses the Chen-Tan (1994) score difference (μ − ν). This is a canonical choice but not the only one. A different score or accuracy function, for example one that penalises indecision, can give a different ranking. The function used must be stated in the report.

Writing ν as 1 − μ. This effectively reduces intuitionistic fuzzy data to crisp data: the share of indecision is set to zero, and ν carries no new information because it is derived from μ. In the illustrative example below, this mistake lowers A1's benefit degree from 91.45 to 89.43, and A2's from 96.67 to 95.06. The ranking does not change, but the true share of counter-evidence is left out of the calculation.

Marking the criterion's direction incorrectly. If a cost criterion is marked "more is better," the high support on that criterion is included in the benefit sum, and the result can be reversed completely.

The governing principle is this:

IF-COPRAS exists to keep support and rejection from separate sources; deriving ν from μ, or reducing the support-rejection pair to a single crisp number from the start, removes intuitionistic fuzzy data's only contribution.