Intuitionistic fuzzy MARCOS (Ecer & Pamučar, 2021)

Four things change; the ratio-to-ideal/anti-ideal logic does not.

Cells. In crisp MARCOS every cell is a single number. In IF-MARCOS every cell is a pair: a degree of support (μ) and a degree of rejection (ν), whose sum does not exceed 1. Criterion weights stay crisp. The founding article (Ecer & Pamučar, 2021) proposed the method to rank health-service performance among insurance companies during the COVID-19 period. In DecisionMind, IF-MARCOS can combine more than one decision-maker's pairs with Xu's (2007) intuitionistic fuzzy weighted average (IFWA); the validation example is single-decision-maker.

Scale equalisation. Crisp MARCOS builds the ideal and anti-ideal references from the column's observed best and worst crisp value. IF-MARCOS first looks at each criterion's direction. On a cost criterion, the cell's degrees of support and rejection are swapped, so that a "little-rejected" cost cell can be read as a "much-supported" cell. Then a score is computed for every cell (Chen-Tan 1994: support minus rejection), and this score is ratioed to the ideal's score. This is the intuitionistic fuzzy counterpart of crisp MARCOS's division-by-the-ideal step.

Distance / score / aggregation. After normalisation, every row (the actual alternatives plus the ideal plus the anti-ideal) is aggregated across the criteria into a SINGLE intuitionistic fuzzy pair with IFWA; this is the intuitionistic fuzzy counterpart of crisp MARCOS's weighted sum, but it is a multiplicative aggregation (the IFWA operator), not an additive one. The utility ratios K+ and K− are then built by division, over the Chen-Tan scores of these aggregated pairs.

Result and defuzzification. The intuitionistic fuzzy pair is reduced to a single number by the Chen-Tan score function (μ − ν) immediately after the IFWA aggregation; the remaining utility function and final score run on exactly the same formula as the steps of crisp MARCOS.

The choice DecisionMind fixes in IF-MARCOS: the Chen-Tan (1994) score function (μ − ν) and Xu's (2007) IFWA aggregation operator (F2, F3).

The output is a final utility degree and a ranking, in the same form as in crisp MARCOS, and is read the same way.

The difference lies here: the final degree dissolves the support and rejection information into a single difference (μ − ν). The gap between two alternatives does not answer, at the same time, the question "was one more supported" or "was one less rejected"; the score function merges the two into a single number.

Thus instead of writing:

"IF-MARCOS gives a more reliable result because it accounts for indecision"

the report should read:

"Degrees of support and rejection were aggregated separately, but because the final degree is built on their difference, the share of indecision stays invisible in the result; μ and ν should also be shown separately in the report"

Use this method when support for a judgement and the opposing view can be measured separately, and the gap between the two matters for the decision itself (the criterion given on the intuitionistic fuzzy data-type card). There is no need to move to this structure by writing the degree of rejection directly as 1 − μ; this carries the same information as crisp data.

The exit condition is the same as crisp MARCOS: if no compromise is acceptable on one criterion, this extension is also compensatory.

Breaking the μ + ν ≤ 1 rule. If support and rejection sum to more than 1, the pair is invalid; every cell must be checked before the calculation.

Changing the score function without noticing. The Chen-Tan score (μ − ν) is the canonical choice but not the only one; a different score function can produce a different ranking.

Defuzzifying with the score first and then running crisp MARCOS. Reducing every cell to a μ − ν score and then applying crisp MARCOS's cell-by-cell ideal normalisation and direct weighted sum is not IF-MARCOS. In the correct method, rows are aggregated first with IFWA, and only after this aggregation is the score taken. In the illustrative example below, this wrong path completely reverses the ranking. The correct method gives A2 first (0.754), A3 second (0.606), A1 third (0.574), while the score-first-then-crisp-MARCOS path gives A1 first (0.971), A2 second (0.493), A3 third (0.366).

Forgetting the complement on a cost criterion. A direction error swaps the ideal and the anti-ideal and reverses the ranking; in the intuitionistic fuzzy setting this complement (swapping μ and ν) must be applied separately.

The governing principle is this:

In IF-MARCOS, degrees of support and rejection are aggregated row by row with IFWA, and the score is taken only after this aggregation; any shortcut that starts the ranking from the score breaks the method's ideal/anti-ideal logic.

---