MULTIMOORA (Multi-Objective Optimization by Ratio Analysis plus Full Multiplicative Form)

MULTIMOORA is a ranking method that arranges alternatives into a single order once you hold a numerical decision table. Its output is one final order, arising from the combination of the orders given by three separate sub-methods; it does not produce criterion weights, it takes them from outside. Brauers and Zavadskas proposed the method in 2010 by adding a third viewpoint, the full multiplicative form, to MOORA (Multi-Objective Optimization by Ratio Analysis), which they had developed themselves. It is used in fields such as project management, materials selection and supplier evaluation.

The idea behind MULTIMOORA is not to trust a single ranking logic. There is more than one reasonable way to rank a set of alternatives. The first asks "which is best on average," and this is called the ratio system. The second asks "which is furthest from the worst position," and this is called the reference point. The third asks "which is highest in the product of the criteria," and this is called the full multiplicative form. These three viewpoints use different mathematical logics and usually, though not always, give the same order. MULTIMOORA computes all three separately and then combines them with dominance theory: an alternative that stands out in most of the three sub-rankings also stands out in the final order.

This idea carries a philosophical consequence. MULTIMOORA does not reduce the question "which ranking logic is correct" to a single method. Instead it seeks the consensus of several viewpoints; this is a majority or dominance logic, not a compromise. Where the three sub-methods agree, the result is strong. Where they disagree, MULTIMOORA does not paper over the disagreement; it leaves visible which viewpoint supports which alternative. At heart the ratio system and the full multiplicative form are compensatory; the reference-point component, by focusing on the worst criterion gap, carries a more cautious, risk-averse logic.

The method proceeds through five steps.

First, scale equalisation (normalisation). Every column is divided by the square root of the sum of the squared values in that column, making it unit-free and comparable. This is vector normalisation, the same normalisation TOPSIS uses.

Second, the ratio system. In the equalised table, a ratio-system score is built for every alternative by subtracting the sum of the "lower is better" criteria from the sum of the "higher is better" criteria. This is an average performance measure.

Third, the reference point. A reference point is built from the best value on each criterion. Every alternative's largest, that is worst, weighted deviation from this reference point is computed. This evaluates the alternative according to its weakest criterion, looking not at the average but at the worst case.

Fourth, the full multiplicative form. A single ratio is built for every alternative by dividing the product of the "higher is better" criteria by the product of the "lower is better" criteria. Unlike the ratio system, this rests on multiplication rather than addition and measures the balance across all criteria more severely.

Fifth, combination by dominance theory. Each of the three sub-methods (ratio system, reference point, full multiplicative form) ranks the alternatives on its own. An alternative's positions across the three rankings are summed; whichever has the smallest such sum, that is, whichever ranks highest across all three viewpoints, becomes first in the final order. Where the three sub-rankings diverge sharply, dominance theory produces a consensus by finding which alternative is in a poor position in none of the sub-methods.

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

The final order is a combined summary of three different viewpoints: average performance, distance from the worst case, and full multiplicative balance. It is not a single "closeness score" or "percentage" arising from one formula. An alternative ranking first in the final order does not mean it ranks first in ALL three sub-methods; it means its total position across the three rankings is smallest, that is, best.

A reading point specific to MULTIMOORA is this: the three sub-rankings must be compared. If the ratio system and the full multiplicative form favour the same alternative while the reference point favours a different one, this means two different alternatives diverge: one is "best on average performance," the other is "least weak on the worst criterion." The decision-maker should interpret the final order according to which of these two philosophies matters more to them. Where the three sub-rankings agree fully, the result is strong; where they disagree, the "winner" given by dominance theory may be a fragile winner, supported by only two of the three viewpoints.

Therefore, instead of writing:

"MULTIMOORA found the best alternative"

the report should read:

"Two of the three viewpoints (ratio system, full multiplicative form) favour this alternative, while the third (reference point) favours a different one; the final order, combined through dominance theory, is as follows, and it is sensitive to which viewpoint is given weight"

MULTIMOORA works with crisp data: one number per cell, no empty cells. You need alternatives in rows, criteria in columns, direction information (higher or lower is better) for every criterion, and criterion weights that sum to 1. MULTIMOORA does not produce weights, it asks for them; you can derive them from expert opinion (AHP, BWM, SWARA) or from the data itself (Entropy, CRITIC). The full multiplicative component requires every cell to be positive; a zero or negative value renders this component undefined, and this must be checked beforehand. DecisionMind holds nine MULTIMOORA members alongside the base method, including fuzzy, intuitionistic fuzzy, interval-valued neutrosophic, two-dimensional uncertain linguistic and probabilistic linguistic uncertainty types; which one fits depends on your data type. A minimum of two alternatives and two criteria is required; three to twelve criteria work comfortably.

MULTIMOORA is a suitable choice if your criteria can be measured numerically, your table is fully populated, all values are positive, and you want to see the combination of several viewpoints rather than trust a single ranking logic. A single logic means either average performance alone or worst-case performance alone. Its typical territory includes supplier evaluation, materials and equipment selection, and project and investment prioritisation.

The case where it should not be used arises when your data contains a zero or negative value, since this renders the full multiplicative form undefined. Where there is not enough time or data to bear the computational load of three sub-methods, a single simple method (such as SAW) should be preferred. Where no compromise can ever be made on one criterion, meaning a strict below-threshold elimination is required, MULTIMOORA's compensatory components (ratio system, full product) do not provide this.

A numerical table, all values positive, the combination of several viewpoints is wanted → MULTIMOORA

A zero or negative value is present → not MULTIMOORA; transform the data first, or use another method

A single, simple ranking logic is enough → SAW or TOPSIS

A cautious ranking against the worst case alone is enough → VIKOR, which relies on reference-point logic alone

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

MULTIMOORA's greatest strength is that it does not commit to a single ranking logic, using three different mathematical viewpoints at once: average performance, distance from the worst case, full multiplicative balance. This catches a weakness that single-method approaches can miss, for example an alternative that is good on average but very poor on one criterion. Comparing the three sub-rankings gives additional information about how robust the result is, an insight not directly available in single-score methods such as TOPSIS or SAW. The computational load is moderate and requires no complex external parameters.

Its limitations arise largely from the joint operation of the three components. First, when the three sub-rankings give different results, how dominance theory produces a "winner" may not be easy to explain to a user; compared with TOPSIS's single distance measure, interpretation is more complex. Second, the full multiplicative form is undefined at zero or negative values, closing MULTIMOORA off from some data types. Third, the method was introduced in 2010 as an extension of MOORA, so it lacks as long a history of critique and rank-reversal examination as TOPSIS or VIKOR; its scope and applications are still under active investigation (Baležentis and Baležentis, 2014). Fourth, it remains a compensatory method: a serious weakness on one criterion can be papered over by others through the ratio system and the full multiplicative form. Fifth, the reference-point component looks only at the worst criterion, disregarding information from other criteria in this step; this is one reason for using it alongside the ratio system, but it can be misleading on its own.

The most common mistake is reporting only the final dominance order without ever comparing the three sub-rankings. Where the sub-methods disagree, this information is lost and the decision-maker remains unaware of how fragile the result is. A second mistake is computing the full multiplicative form without noticing a zero or negative value in the data; this step then produces an undefined or meaningless result. A third mistake is assigning equal weights without justification and not presenting this as a choice. A fourth mistake is presenting the "winner" found by dominance theory as the best alternative on every dimension, when in fact the winner is only best in the sum of the three rankings, and may not be best in each individually. A fifth mistake is comparing a MULTIMOORA result directly with another method's score, such as TOPSIS's; different aggregation logics can give different orders.

The governing principle is this:

A MULTIMOORA result is a summary of three different viewpoints, combined through dominance theory. If these three viewpoints do not agree, the final order is fragile, and the report must show which viewpoint supports which alternative.