MARCOS (Measurement of Alternatives and Ranking according to Compromise Solution)

MARCOS is a ranking method for when you already hold a decision table filled with numbers and want the alternatives placed in a single order. Its output is a final utility degree for every alternative and the rank that degree produces. Like TOPSIS, it uses both an ideal and an anti-ideal reference, but rather than the geometric distance between them, it looks at the alternative's ratio to these two references. It takes weights from outside, it does not generate them. Stević, Pamučar, Puška and Chatterjee proposed the method in 2020, for the problem of sustainable supplier selection in the healthcare sector; it has since spread rapidly into fields such as supply chains, transport and energy.

MARCOS's underlying idea is to find an alternative's value not by asking "how far or near is it to the ideal," but by asking two questions at once: "what is its ratio to the ideal" and "what is its ratio to the anti-ideal." The method builds the alternative's value by balancing these two ratios. The table is first extended, alongside the real alternatives, with two hypothetical rows built from every criterion's observed best (ideal) and worst (anti-ideal) value. Every alternative's weighted total score is divided by the ideal's total score to obtain the "utility ratio relative to the ideal"; the same score is divided by the anti-ideal's total score to obtain the "utility ratio relative to the anti-ideal." These two ratios are each passed through a utility function and combined into a single final utility degree. This is what separates TOPSIS from MARCOS. TOPSIS measures a metric distance (Euclidean). MARCOS measures a ratio instead, and turns this ratio into a utility function expressing the alternative's position between the ideal and the anti-ideal.

This idea carries a philosophical consequence. MARCOS is also compensatory, but it speaks the language of "utility ratio" rather than "distance." An alternative's final degree carries, at the same time, both "how much of the ideal it has reached" and "how far it has moved away from the anti-ideal." Using these two references together rewards a compromise point that is "both far from bad and close to good" more explicitly than methods that look only at closeness to the ideal.

The method proceeds through seven steps.

First step, the extended table. In addition to the real alternatives, two hypothetical rows, built from every criterion's observed best (ideal) and worst (anti-ideal) value in the table, are added.

Second step, normalisation relative to the ideal. Every cell is expressed as a ratio to that criterion's ideal value: for a benefit criterion, the cell is divided by the ideal; for a cost criterion, the ideal is divided by the cell. Every column is thereby converted to a scale of "how much of the ideal has been reached."

Third step, weighting. The normalised table is multiplied by the criterion weights.

Fourth step, row totals. For every real alternative, and also for the ideal and anti-ideal rows, the weighted values are summed horizontally; this gives a single "total score."

Fifth step, utility ratios. Every alternative's total score is divided by the ideal's total score to find the "ratio relative to the ideal"; the same total is divided by the anti-ideal's total score to find the "ratio relative to the anti-ideal."

Sixth step, utility functions. These two ratios are weighed against one another; how "close to the ideal" and how "far from the anti-ideal" the alternative is are each converted into individual functions.

Seventh step, the final utility degree. The two utility functions are combined into a single number; alternatives are ranked by this number from highest to lowest.

The formulas behind these steps, the intermediate tables and the citation formats are given on the DecisionMind method page; this card carries no formulas.

The final utility degree states where an alternative stands relative to the ideal and anti-ideal references within this set. It says nothing more. A value of 0.71 does not mean "71 per cent good" or "71 per cent likely to be the best." It cannot be compared with a MARCOS degree from a different analysis, because the ideal and anti-ideal points are built, in every analysis, from that analysis's own alternatives. When the alternative set changes, so do these references, and so do the degrees. A high degree does not mean "perfect," but "well positioned on this set's ideal-anti-ideal axis."

For this reason, instead of writing:

"MARCOS found the best supplier"

the report should read:

"With these weights and this alternative set, the alternative with the most balanced utility ratio relative to the ideal and anti-ideal references is this one; the ranking is sensitive to the weight on these criteria"

Classical MARCOS works with crisp data: one number per cell. If your data is uncertain from expert judgement, given as a range, or contradictory across experts, you change the data type rather than the method. MARCOS has fuzzy, grey, intuitionistic and other extensions; DecisionMind carries twenty MARCOS members alongside the base method.

You need alternatives in rows, criteria in columns, one number per cell with no empty cells; "higher is better or lower is better" information for every criterion; and criterion weights summing to 1. MARCOS does not produce weights, it asks for them. A minimum of two alternatives and two criteria is required; three to twelve criteria work comfortably. If all alternatives are identical to one another on a criterion, the ideal and anti-ideal fall on the same point. In that case, that criterion's contribution to the calculation becomes indeterminate, and the engine treats it as an edge case.

MARCOS is a suitable choice if your criteria can be measured numerically, your table is fully populated, and you want to see, together, both "how much of the ideal has been reached" and "how far it has moved from the anti-ideal." Its typical territory includes supplier and vendor evaluation, digital-transformation and strategy selection, and prioritising transport and energy investment.

The case where it should not be used follows from its own philosophy: if you will not compromise on one criterion, MARCOS will not prevent this, because it is compensatory. If your alternative set is very small, say two or three alternatives, the ideal and anti-ideal points are built from the alternatives themselves, so the references can be overly sensitive to the number of alternatives. Larger alternative sets give a more stable ideal/anti-ideal reference. Where criteria are strongly linked, this link needs handling first.

A numerical table, a utility ratio relative to ideal and anti-ideal is wanted, the goal is ranking → MARCOS

Same goal, but the data is fuzzy / grey / intuitionistic → the relevant MARCOS extension

Direct geometric distance to ideal and anti-ideal is wanted → TOPSIS

No compromise allowed on one criterion, sub-threshold alternatives must be screened out → screening first, then ranking, or elimination-based methods

Not a ranking but weights are needed → AHP, BWM, SWARA (subjective); Entropy, CRITIC (objective)

MARCOS's principal strength is that it shows, explicitly and in ratio terms, both closeness to the ideal and distance from the anti-ideal at once. This gives the decision-maker two separately readable pieces of information: "how much of the ideal this alternative has reached" and "how far it has moved from the worst." Thanks to the extended-table approach, the ideal and anti-ideal references stay visible at every step of the calculation. A broad body of applied literature has built up in a short time; results have been reported as consistent in fields such as airlines, supply chains and sustainability (Stević et al., 2020; Büyüközkan et al., 2021).

Its limitations stem from the same structure. First, the assumption of full compensation: a weakness on one criterion can be papered over by strength on others. Second, MARCOS, like other ratio- or distance-based ranking methods, is open to rank reversal. Because the ideal and anti-ideal points are built from the alternatives themselves, an alternative added to or removed from the set shifts these references (Aires and Ferreira, 2018). Third, the method is comparatively new (2020); it has not yet built up as long a body of critique and comparison literature as TOPSIS and VIKOR. Fourth, criteria are treated as independent. Fifth, the quality of the weights lies outside the method itself.

The most common mistake is marking criterion direction wrongly. Because the ideal and anti-ideal points are built directly from criterion direction, a directional error swaps the ideal and anti-ideal and reverses the ranking. A second mistake is reading the final utility degree as a percentage or a probability. A third mistake is assuming that, in a very small alternative set, say two or three alternatives, the resulting ranking is as stable as in larger sets. When the ideal and anti-ideal are built from a small number of alternatives, the references are more sensitive to the alternatives themselves. A fourth mistake is adding an alternative once the analysis is finished and failing to notice that the ideal/anti-ideal points, and therefore all the degrees, have changed. A fifth is choosing a compensatory method for a situation where one criterion can never be compromised.

The governing principle is this:

A MARCOS result is a summary of the alternatives' ratio position relative to this set's own ideal and anti-ideal points; these points change together with the alternative set, and the report must show this.