Interval-valued intuitionistic fuzzy COPRAS (Davoudabadi, Mousavi, Mohagheghi & Vahdani, 2019)

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

Cells. In crisp COPRAS every cell is a single number. Here every cell consists of four numbers: the lower and upper bound of the degree of support, and the lower and upper bound of the degree of rejection; the sum of the upper support and the upper rejection cannot exceed 1. Criterion weights are supplied from outside as crisp, single numbers.

Scale equalisation. Crisp COPRAS turns every column into a proportion by dividing it by its own column sum. Here the division is carried out without first reducing the cell to a single number, unlike in IV-COPRAS: each bound (lower support, upper support, lower rejection, upper rejection) is divided by the square root of the sum of the squared values of that same bound across all rows in its column (Xu vector normalisation). At the end of this step the four numbers are still four numbers, not yet a single score.

Weighting and summation. The normalised four-number cells are weighted by the criterion weight, using interval-valued intuitionistic fuzzy algebra's own rule (exponential reinforcement). Every alternative's cells on the benefit criteria are then combined with one another, and its cells on the cost criteria separately, through a genuine interval-valued intuitionistic fuzzy summation operation (⊕). This summation resembles the "either one or the other" logic of probability theory: the support components accumulate without consuming one another. The result is one benefit cell and one cost cell per alternative, each still four numbers.

Result and defuzzification. Only at this final step do the benefit and cost cells fall to a single number, through a score function (Garg's GIS function). This function carries a coefficient λ that weights the midpoint of the support interval: the larger λ is, the more the score leans on the support average; the smaller it is, the more it leans on the rejection complement. DecisionMind takes λ = 0.6 as the default for this family. The relative significance value Q is built from the benefit cell's score, with a correction term added that reflects the balance between the cost cells' scores.

DecisionMind fixes, for this classical form, Xu vector normalisation, the interval-valued intuitionistic fuzzy summation operation (⊕) and the GIS score function; λ can be adjusted by the user and must be stated in the report.

The utility degree is read as in crisp COPRAS: the best alternative is set to 100, and every other alternative receives a percentage relative to it. The difference lies beneath this percentage: a support–rejection interval has been carried through the steps as four numbers, and a coefficient λ reduces it to a single number. The choice of λ affects the result directly; λ = 0.6 and λ = 0.4 can give the same table a different ranking.

Thus instead of writing:

"Interval-valued intuitionistic fuzzy COPRAS finds A3 to be the best supplier"

the report should read:

"With a λ = 0.6 balance, A3 has the highest utility degree (D = 100); if λ is shifted to a different balance this result may change, so the λ value used must be stated in the report"

This extension suits situations where only an interval is known about a judgement's degree of support and rejection, and the width of that interval needs to be carried through to the end of the calculation, into the benefit and cost sums, rather than collapsed to a midpoint straight away. A typical example is a multi-criteria supplier selection where the lower and upper bounds of several assessors' support–rejection scores are known. If support and rejection are given as a single point each, that is, if the interval width is zero, the extension reduces to intuitionistic fuzzy COPRAS (IF-COPRAS). Crisp COPRAS's exit condition applies here too: if no compromise is acceptable on one criterion, this extension is equally fully compensatory. At least one benefit criterion must be present; if every criterion is a cost criterion, the Q formula this engine uses remains undefined.

Constraint violation. In every cell the lower bound must be less than or equal to the upper bound, and the sum of the upper support and the upper rejection must not exceed 1.

Swapping support and rejection on a cost criterion. In this extension the benefit–cost distinction is applied not at the normalisation step but at the step where the benefit and cost sums are built separately; swapping support and rejection before normalisation adds an incorrect conversion step.

Treating the λ parameter as a probability. λ is a balancing coefficient between the support average and the rejection complement, not a probability. λ = 1 counts only the support average; λ = 0 counts only the rejection complement.

Confusing this card with IV-COPRAS. Both use the same four-number cell format, but IVIF-COPRAS carries the four numbers through to the very last step and defuzzifies them there with a λ-weighted score, whereas IV-COPRAS reduces every cell to a single midpoint score as early as the first step. The two engines produce different numbers.

The governing principle is this:

In interval-valued intuitionistic fuzzy COPRAS, the support–rejection interval is carried as four numbers until the benefit and cost sums are built. Defuzzification happens only at the very last step, through a λ-weighted score, and this coefficient must be stated in the report.