COPRAS (Complex Proportional Assessment)

COPRAS is a ranking method for when you already hold a decision table filled with numbers and want the alternatives arranged in a single order. Its output, for every alternative, is a benefit degree expressed as a percentage of the best alternative (the best alternative always scores 100) and the rank that degree produces. Unlike TOPSIS, it uses not distance but a proportional combination of benefit and cost totals. It produces no weights, taking them from outside. Zavadskas and Kaklauskas proposed it in 1996 for the problem of contractor selection in the construction sector; it has since been applied in many fields, such as supplier selection, public procurement and investment appraisal.

The idea behind COPRAS is this: find an alternative's value not through a single distance measure, but through two separate questions. The first is "how much benefit does it deliver," the second "how much cost does it incur to deliver that benefit." COPRAS finds these two answers separately, then combines them in a proportion. The weighted values on the benefit criteria (higher is better) are summed, and the weighted values on the cost criteria (lower is better) are summed separately. A correction term rewarding the alternative with the lower cost is then added to the benefit sum. This is the point that sets TOPSIS apart from COPRAS. TOPSIS measures geometric distance to two hypothetical extreme points (ideal/anti-ideal). COPRAS instead first sums benefit and cost separately, then strikes a proportional balance between them. This is a kind of multi-criteria benefit-cost logic.

This idea carries a philosophical consequence: COPRAS is also compensatory, but it presents its result on a directly interpretable scale. The output is expressed as a percentage of the best alternative; the best alternative is always 100. This makes a reading such as "the second alternative holds a relative superiority equal to 88 per cent of the best" possible. But this does not mean the alternative has an absolute "success percentage." This reading shows only a share relative to the best within this particular set.

The method proceeds through five steps.

First, linear sum normalisation. Every value in a column is divided by that column's total. Each cell thereby becomes its share of its criterion's overall magnitude, and the columns become comparable.

Second, weighting. Each column converted into shares is multiplied by its criterion's weight. Weights come from outside; they must sum to 1.

Third, benefit and cost totals. For every alternative, the weighted values on the benefit criteria (higher is better) are summed separately (the benefit total), and the weighted values on the cost criteria (lower is better) are summed separately (the cost total).

Fourth, relative significance. A correction term using the relationship among all the alternatives' cost totals is added to the benefit total. This term is constructed to reward the alternative whose cost total is low (that is, the relatively cheap one). This produces a single relative significance value for every alternative.

Fifth, the benefit degree. The relative significance values are divided by the highest one and converted into a percentage. The best alternative thereby becomes 100, and the others receive a percentage relative to it. Alternatives are ranked by this percentage 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 benefit degree tells you how far behind an alternative sits relative to the best alternative in this particular set; it says nothing more. A value of 88.46 does not mean "88.46 per cent good quality" or "88.46 per cent likely to be the best." It means that 88.46 per cent of the total benefit delivered by the best alternative has been reached. It cannot be compared with a COPRAS degree from a different analysis, because "best" is determined, in every analysis, from that analysis's own alternatives. Change the alternative set and the alternative scoring 100 per cent changes with it. A degree of 100 does not mean "perfect" but "the best in this set."

For that reason, instead of writing:

"COPRAS found this alternative to be 88 per cent successful"

it is correct to write:

"With these weights and this alternative set, this is the alternative with the highest benefit-cost balance; the second alternative reaches 88.46 per cent of the benefit this alternative delivers"

Classical COPRAS 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. COPRAS has fuzzy, grey, intuitionistic and other extensions. DecisionMind holds twenty COPRAS members alongside the base method.

You need the following: alternatives in rows, criteria in columns, one number per cell, and no empty cells. You also need direction information for every criterion ("higher is better" or "lower is better") and criterion weights summing to 1. COPRAS does not produce weights, it asks for them. Columns must contain no negative values, because linear sum normalisation rests on dividing by the column total. A minimum of two alternatives and two criteria is required; three to twelve criteria work comfortably. If the table contains no cost criterion at all (only benefit criteria), the relative significance calculation reduces directly to the benefit total. This is not an error but the method's natural boundary case.

COPRAS is a suitable choice if your criteria can be measured numerically, the table is completely filled, there are no negative values, and you want to report the result directly as a "percentage of the best." Its typical territory includes contractor and supplier selection, investment and project appraisal, and public procurement evaluation.

The case where it should not be used follows from its own philosophy. If you will not compromise on one criterion, COPRAS does not prevent this, because it is compensatory. It allows a low cost to mask a high benefit, or a high benefit to mask a high cost. If your data contains negative values (loss situations in a net profit/loss criterion, for instance), you must transform the data before feeding it into COPRAS. Where criteria are strongly linked to one another, that link needs handling first.

A numerical table, benefit/cost ratio logic, the result wanted as a percentage → COPRAS

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

Distance to the 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)

COPRAS's most important strength is that its output is directly interpretable. The best alternative always scores 100, and the others receive a percentage relative to it. This is easily conveyed to a decision-maker. Summing benefit and cost separately before combining them makes visible, at an intermediate step, whether an alternative gains from benefit or from low cost. Its computational burden is small, and it has a wide history of application in fields such as construction, procurement and investment appraisal (Zavadskas et al., 1994; Doğan and Yıldız, 2024).

Its limitations stem from the same structure. First, the assumption of full compensation: a low cost can more than mask a weak benefit. Second, like other ratio/distance-based ranking methods, COPRAS is exposed to rank reversal. An alternative added to or removed from the set changes the column totals, and hence every share (Aires and Ferreira, 2018). Third, its inability to work with negative values requires an additional transformation step for some financial indicators, such as net profit/loss. Fourth, the method treats criteria as independent of one another. Fifth, the quality of the weights lies outside the method. Even a flawless calculation built on poor weights gives a poor ranking.

The most common mistake is marking criterion direction wrongly. If a cost criterion is marked "higher is better," it is included in the benefit total, and the higher-cost alternative is rewarded. A second mistake is reading the benefit degree (88 per cent, say) as an absolute quality or success percentage. In fact, this is only a share relative to the best within the set. A third is feeding data into COPRAS without transformation despite negative values being present in the table. Dividing by the column total produces a meaningless result with negative values. A fourth is adding an alternative once the analysis is finished and being surprised the ranking shifts. A fifth is choosing a compensatory method for a situation where one criterion can never be traded away.

The governing principle is this:

A COPRAS result is a summary of the relative balance between the alternatives' benefit and cost totals. A percentage of the best is not an absolute measure of success, and the report must show this distinction.