CoCoSo (Combined Compromise Solution)

CoCoSo is a ranking method that orders alternatives into a single sequence once you hold a numerical decision table. Its output is a combined compromise score for every alternative, together with the rank that score produces. The method does not generate criterion weights; weights are supplied from outside. Yazdani, Zarate, Zavadskas and Turskis proposed the method in 2019. It is used in areas such as supplier selection, sustainability assessment and public-project selection.

The idea behind CoCoSo is not to rely on a single form of aggregation. There are two classical ways of reducing criteria to a single score. The first is the weighted sum, the SAW logic: simple and fully compensatory, but it can flatten out the proportional differences between alternatives. The second is the weighted product, the WPM logic: it punishes a weak criterion heavily, but is highly sensitive to small values. CoCoSo calculates both. It then combines these two measures through three different compromise strategies: the relative share within the total, the sum of ratios against the worst values, and a balanced ratio against the best values. It builds the final score as both a multiplicative and an additive combination of these three strategies. This is where the method's name comes from: not a single compromise, but a combination of compromises.

This idea carries a philosophical consequence. CoCoSo does not assume that a single aggregation logic is correct. It has a structure that can test its own robustness by checking how far the three viewpoints agree with one another. If all three strategies give the same order, the result is supported by all three views. If the strategies give differing orders, this is a sign that the result is sensitive to the choice of aggregation. At bottom, CoCoSo remains compensatory: a weakness on one criterion can be offset by strength on another, particularly through the additive component.

The method proceeds through five steps.

First, scale equalisation (normalisation). Depending on the criterion type (higher is better / lower is better), each column is scaled to between 0 and 1 against its own best and worst value, so that criteria measured in different units become comparable.

Second, the additive measure (S). Each equalised column is multiplied by the criterion's weight and summed; this is a weighted-sum performance measure, as in SAW.

Third, the multiplicative-like measure (P). Each equalised value is raised to the power of its own weight and summed. This is a measure that mimics multiplicative logic while remaining additive; it punishes a weakness in a small value more severely than the S measure does.

Fourth, three compromise strategies. The S and P measures are combined in three different ways. The first strategy takes each alternative's relative share within the total S+P; this is called the share strategy. The second strategy sums the ratios of S and P against the worst values across all alternatives; this is called the worst-relative strategy. The third strategy blends S and P with a balance coefficient and takes their ratio against the best values; this balance coefficient is a parameter that adjusts the decision-maker's additive-multiplicative preference. This is called the balanced-compromise strategy.

Fifth, the combined score. The method builds a single final score by taking both the geometric and the arithmetic mean of the three strategies' scores. Alternatives are ranked by this score from highest to lowest.

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

The combined score states an alternative's relative standing against the other alternatives in this particular analysis. It is not an absolute percentage or probability. It cannot be compared with a score from a different analysis either, because normalisation is built afresh, in every analysis, from that analysis's own alternative set. The magnitude of the score carries no meaning on its own. Unlike TOPSIS's closeness score, which sits between 0 and 1, a CoCoSo score can exceed 1. Only the ranking is meaningful here.

CoCoSo has its own indicator of robustness: the separate rankings produced by the three compromise strategies (share, worst-relative, balanced) can be compared. If all three give the same order, the result is supported by all three viewpoints and is more reliable. The strategies can also give diverging orders; one, for instance, might favour A2 while another favours A1. In that case the final ranking is fragile, and the report must state this.

For this reason:

"CoCoSo found A3 to be the best alternative"

should be written as:

"With these weights and this alternative set, A3 leads according to the combined result of the three compromise strategies; this result is robust if all three strategies give the same order"

CoCoSo works with crisp data: a single number in every cell is required, and empty cells are not accepted. You need alternatives in rows, criteria in columns, whether more is better or less is better for each criterion, and criterion weights that sum to 1. CoCoSo does not produce weights, it asks for them from outside. You can derive weights from expert opinion (AHP, BWM, SWARA) or from the data itself (Entropy, CRITIC). The decision-maker also sets, from outside, the balance coefficient (λ) that combines the S and P measures. DecisionMind uses 0.5 by default, giving S and P equal weight, and states this in the report. DecisionMind holds nine CoCoSo members alongside the base method; these cover uncertainty types such as fuzzy, intuitionistic fuzzy, neutrosophic, Pythagorean fuzzy and q-rung orthopair. Which one is suitable depends on your data type. If all the values in a criterion column are equal, that is, the largest equals the smallest, the normalisation denominator becomes zero. The user should check for this in advance. A minimum of two alternatives and two criteria is required; three to twelve criteria work comfortably.

CoCoSo is a suitable choice if your criteria can be measured numerically, your table has no gaps, and you accept compensation between criteria. You may also prefer CoCoSo if you want to see the combination of several compromise viewpoints rather than relying on a single form of aggregation, additive alone or multiplicative alone. Its typical territory is supplier and vendor evaluation, sustainability and environmental-project selection, and public-investment prioritisation.

The case where it should not be used is one where a criterion column carries the same value throughout, that is, a constant column. Here the normalisation denominator becomes zero and the calculation is undefined. If you will not compromise on one criterion at all, that is, if compensation is unacceptable, CoCoSo is not suitable, because both the additive and multiplicative components underlying the method involve some degree of compensation. In that case, switch to a method built on elimination logic.

A numerical table, compensation accepted, a combination of several compromise viewpoints is wanted → CoCoSo

A single, simple weighted sum is sufficient → SAW

A weak criterion should be punished heavily, additive smoothing is not wanted → WPM

A criterion's column is entirely constant (max = min) → check the data before running CoCoSo, or remove that criterion

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

CoCoSo's chief advantage is this: it is not tied to a single form of aggregation, and it offers a built-in robustness check through the combination of three different compromise strategies. If the strategies agree, the result looks more reliable. Using additive and multiplicative measures together partly offsets the weaknesses carried by purely additive methods (such as SAW) or purely multiplicative ones (such as WPM) on their own. The computational burden is small, and the method brings criteria measured in different units together into a single table.

Its limitations are largely inherited from its components. First, the user chooses the balance coefficient (λ) from outside, and the result is sensitive to that choice; a different λ can give a different order. Second, the assumption of full compensation still holds: through the additive component, a serious weakness on one criterion can be papered over by others. Third, the original authors themselves proposed an extension of CoCoSo to widen the method's limits, named CoCoFISo; this proposal shows that classical CoCoSo's additive-multiplicative combination does not always give the most discriminating result (Rasoanaivo, Yazdani, Zaraté and Fateh, 2024). Fourth, if a criterion column is constant, that is, every alternative takes the same value, min-max normalisation becomes undefined; the user must check for this by hand. Fifth, CoCoSo is a newer method compared with well-established ranking methods such as TOPSIS and VIKOR. Rank-reversal behaviour, meaning how the order shifts when the alternative set changes, has therefore not yet been studied in the literature to the same depth.

The first mistake is leaving the balance coefficient λ at its default value without ever discussing it, and failing to show whether the result is sensitive to this choice. The second mistake is feeding in a criterion without noticing that it takes the same value across all alternatives; its min-max normalisation then produces a division-by-zero error. The third mistake is reporting only the final combined score without ever checking the three compromise strategies' separate rankings (share, worst-relative, balanced); this information is lost if the strategies disagree. The fourth mistake is reading the magnitude of the combined score as a percentage or a probability. Treating a score above 1, such as 2.39, as a percentage is wrong; the score is meant only for ranking within this alternative set. The fifth mistake is assigning equal weights without justification and failing to present this as a choice.

The governing principle is this:

A CoCoSo result is a weighted combination of three different compromise viewpoints; if these three viewpoints do not give the same order, the result is fragile, and the report must show this.