Interval-valued intuitionistic fuzzy MARCOS (aggregated, a DecisionMind derivation)

Three things change; the extended-matrix and ideal/anti-ideal ratio logic does not.

Cells. In crisp MARCOS every cell is a single number. Here every cell consists of four numbers: the lower and upper bound of the degree of support ([μL, μU]) and the lower and upper bound of the degree of rejection ([νL, νU]); the sum of the two upper bounds (μU + νU) cannot exceed 1. Criterion weights are supplied from outside as crisp (single) numbers.

Ideal and anti-ideal. In crisp MARCOS the ideal and the anti-ideal are built from the best and worst observed numbers. Here, on a benefit column, the ideal is taken as the component-wise highest of the support interval and the lowest of the rejection interval; on a cost column, the reverse is taken as ideal, and the anti-ideal is reversed accordingly.

Aggregation and defuzzification. Crisp MARCOS multiplies each row by the weights and sums it. Here each row is first combined into a single support-rejection interval pair with the interval-valued intuitionistic fuzzy weighted averaging operator (IIFWA), and this pair then descends to a single number through a score function ((μL + μU − νL − νU)/2). What remains is crisp MARCOS itself: the ratio to the ideal and the anti-ideal (K+, K−), the two utility functions and the final utility degree that combines them are built through the same steps.

DecisionMind fixes, for this classical form, the IIFWA aggregation operator and the mid-point score function; no path is followed that additionally treats interval width as a penalty or a reward.

The final utility degree is read exactly as in crisp MARCOS: a position relative to this set's own ideal and anti-ideal references (see the MARCOS card). The difference is here: this degree rests only on the mid-point of the support-rejection interval, and does not show how wide that interval is. A degree arising from a narrow interval and the same degree arising from a wide interval are indistinguishable in the final utility degree.

Thus instead of writing:

"According to interval-valued intuitionistic fuzzy MARCOS, A1 is the best alternative"

the report should read:

"The final utility degree rests only on the mid-point of the support-rejection interval; A1 comes out ahead, but this lead should not be taken as certain without separately examining how narrow or wide A1's intervals are"

This extension is appropriate when only an interval is known for a judgement's degree of support and degree of rejection, that is, when an expert says "support is not exactly 0.6, but somewhere between 0.5 and 0.7." Examples include supplier audits where several assessors' support-rejection scores are summarised as an interval, and committee evaluations that are deliberately left unreduced to a single crisp ratio. If support and rejection are each given as a single number (interval width zero), this extra width contributes nothing, and intuitionistic fuzzy MARCOS (IF-MARCOS) suffices. Crisp MARCOS's exit condition applies here too: if no compromise is acceptable on one criterion, this extension is likewise fully compensatory.

Allowing the lower bound to exceed the upper bound. In every interval 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.

Forgetting that the score function conceals the interval. DecisionMind uses the mid-point score. This is a canonical choice, but it does not report interval width; a narrow interval and a wide one can give the same score, and this loss of information must be stated in the report.

Domain violation. Every μU + νU ≤ 1 must hold; entering the calculation without checking this invalidates the method.

Confusing this card with IVIF-MARCOS. Both use the same four-number cell format (the support-rejection interval), but this card first aggregates with IIFWA into a single interval pair and only then descends to a score; a different aggregation-and-score order can produce a different number. Which engine was used must be stated in the report.

The governing principle is this:

In interval-valued intuitionistic fuzzy MARCOS, the support-rejection interval descends to a single mid-point score only at the very last step. The interval's width never returns at any step after that, and must be stated separately in the report.