COPRAS: Complex Proportional Assessment

• COPRAS is a multi-criteria decision-making method that explicitly separates benefit and cost criteria to rank alternatives.
• It employs sum-based normalization, weighted aggregation, and a distinctive ratio adjustment step to compute performance scores.
• The method is computationally moderate and offers clear benefit-cost insights, though it may exhibit rank reversal and sensitivity to extreme values.

Searching arXiv for the specified chapter and COPRAS to ground the article in the provided source. Complex Proportional Assessment (COPRAS) is an aggregation-type multi-criteria decision-making (MCDM) method that converts an alternatives-criteria matrix (ACM) into a single performance score per alternative for ranking. In the chapter "Chapter 8 Multi-Criteria Decision-Making: Aggregation-Type Methods," COPRAS is presented alongside Simple Additive Weighting (SAW), Multiplicative Exponent Weighting (MEW), Analytic Hierarchy Process (AHP), Analytic Network Process (ANP), Multi-Objective Optimization on the basis of Ratio Analysis (MOORA), Faire Un Choix Adequat (FUCA) and Weighted Aggregated Sum Product Assessment (WASPAS). Within that setting, COPRAS is characterized by additive aggregation with an explicit separation of benefit and cost criteria and a ratio adjustment in the final score computation (Wang et al., 8 Sep 2025).

1. Position within aggregation-type MCDM

The chapter describes selected aggregation-type MCDM methods that convert an alternatives-criteria matrix into a single performance score per alternative through additive, multiplicative or hybrid manipulations, for ranking the alternatives. COPRAS belongs to this class and is one of the eight methods detailed step-by-step with a common ACM example and full numerical calculations (Wang et al., 8 Sep 2025).

The practical role of COPRAS is framed through three properties stated in the chapter. First, it uses an explicit separation of benefit and cost criteria. Second, it employs relatively simple computations: normalization, weighted sums, and ratio adjustment. Third, it is presented as a method that has been widely applied in engineering, management and supply-chain problems. This combination places COPRAS between plain weighted-sum procedures and more structurally elaborate approaches such as AHP or ANP.

The chapter’s comparative summary also situates COPRAS relative to other aggregation methods. COPRAS is described as more sensitive than FUCA and MOORA to outliers, but often less extreme than the pure multiplicative MEW. Its computational effort is described as moderate: more work than SAW or FUCA but far less than AHP or ANP. Its interpretability is described as high for those who like explicit benefit/cost splits, while being less transparent than SAW and easier to explain than eigenvector-based AHP.

2. Formal problem setup and notation

The COPRAS formulation in the chapter uses the following problem setup (Wang et al., 8 Sep 2025):

Symbol	Meaning
$m$	number of alternatives, indexed by $i = 1,\ldots,m$
$n$	number of criteria, indexed by $j = 1,\ldots,n$
$X = [x_{ij}]$	alternatives-criteria matrix, where $x_{ij}$ is the performance of alternative $i$ on criterion $j$
$w_j \ge 0$	normalized weight of criterion $j$, with $i = 1,\ldots,m$0
$i = 1,\ldots,m$1	set of benefit (to-be-maximized) criteria
$i = 1,\ldots,m$2	set of cost (to-be-minimized) criteria

The stated goal is to compute a single score $i = 1,\ldots,m$3 for each alternative $i = 1,\ldots,m$4 and then rank the alternatives in descending order of $i = 1,\ldots,m$5.

The notation used in the algorithm is also fixed. The normalized matrix is denoted $i = 1,\ldots,m$6, the weighted normalized matrix is denoted $i = 1,\ldots,m$7, the sum of weighted values for benefit criteria is denoted $i = 1,\ldots,m$8, the sum of weighted values for cost criteria is denoted $i = 1,\ldots,m$9, and the final COPRAS score is denoted $n$0. This notation makes the benefit/cost decomposition explicit at every stage of the procedure.

3. Algorithmic procedure

The algorithm is given as a five-step procedure (Wang et al., 8 Sep 2025).

Step 1. Sum-based normalization. For every criterion $n$1 and alternative $n$2,

$n$3

so that $n$4 for each $n$5.

Step 2. Weighted normalization. Each normalized value is multiplied by the corresponding criterion weight:

$n$6

Step 3. Compute partial sums. Benefit and cost criteria are separated. For each alternative $n$7,

$n$8

Step 4. Compute relative significance $n$9. If both benefit and cost criteria exist, COPRAS proposes

$j = 1,\ldots,n$0

This step is the distinctive ratio-adjustment stage. The score combines a direct contribution from benefit criteria through $j = 1,\ldots,n$1 and an inverse dependence on the cost aggregate through the factor involving $j = 1,\ldots,n$2. A plausible implication is that alternatives with smaller weighted cost sums receive a larger second term, other things equal.

Step 5. Ranking. Alternatives are ranked by descending $j = 1,\ldots,n$3. The highest $j = 1,\ldots,n$4 indicates the most preferred alternative.

The chapter also states two special cases. If all criteria are benefits, so that $j = 1,\ldots,n$5, then $j = 1,\ldots,n$6. If all criteria are costs, so that $j = 1,\ldots,n$7, sometimes one takes $j = 1,\ldots,n$8 or normalizes $j = 1,\ldots,n$9 so that larger is better.

4. Worked numerical example

The chapter provides a worked example with $X = [x_{ij}]$0 alternatives and $X = [x_{ij}]$1 criteria (Wang et al., 8 Sep 2025). Criteria $X = [x_{ij}]$2 and $X = [x_{ij}]$3 are benefits, and $X = [x_{ij}]$4 is a cost. The criterion weights are $X = [x_{ij}]$5, $X = [x_{ij}]$6, and $X = [x_{ij}]$7. The original ACM is:

Alternative	$X = [x_{ij}]$8	$X = [x_{ij}]$9	$x_{ij}$0
$x_{ij}$1	0.93	600	15
$x_{ij}$2	0.51	700	20
$x_{ij}$3	0.77	500	10
$x_{ij}$4	0.82	400	9

Under sum-normalization, the criterion sums are $x_{ij}$5 for $x_{ij}$6, $x_{ij}$7 for $x_{ij}$8, and $x_{ij}$9 for $i$0. The resulting normalized values are:

• $i$1: $i$2
• $i$3: $i$4
• $i$5: $i$6
• $i$7: $i$8

After weighted normalization, the values become:

• $i$9: $j$0
• $j$1: $j$2
• $j$3: $j$4
• $j$5: $j$6

The benefit and cost sums are then:

• $j$7: $j$8, $j$9
• $w_j \ge 0$0: $w_j \ge 0$1, $w_j \ge 0$2
• $w_j \ge 0$3: $w_j \ge 0$4, $w_j \ge 0$5
• $w_j \ge 0$6: $w_j \ge 0$7, $w_j \ge 0$8

For the ratio-adjustment term, the minimum weighted cost is

$w_j \ge 0$9

and the sum of weighted costs is

$j$0

Thus,

$j$1

The resulting COPRAS scores are:

• $j$2
• $j$3
• $j$4
• $j$5

The ranking by descending $j$6 is:

$j$7

The chapter notes that these numbers illustrate the procedure and that small rounding differences may occur. The example is also significant because it shows a ranking in which the alternative with the smallest cost aggregate, $j$8, attains the highest overall score despite not having the largest benefit sum. This suggests how strongly the ratio-adjustment term can influence the final ordering when cost differences are material.

5. Strengths, weaknesses, and comparative properties

The chapter outlines three practical strengths of COPRAS (Wang et al., 8 Sep 2025). It provides explicit separation of benefit and cost criteria, allowing decision-makers to see exactly how each contributes. It uses relatively simple computations: normalization, weighted sums, and ratio adjustment. It has been widely applied in engineering, management and supply-chain problems.

The weaknesses are also stated directly. Extremely high or low weighted values can dominate the sum, potentially masking balanced performance. Like most MCDM methods, COPRAS can exhibit rank reversal when alternatives are added or removed. The final $j$9 depends on $i = 1,\ldots,m$00 and $i = 1,\ldots,m$01, which can be non-intuitive to stakeholders unfamiliar with the ratio adjustment step.

The comparative discussion positions COPRAS between several neighboring methods. Relative to FUCA, which is rank-based, and MOORA, which is difference-based, COPRAS is described as more sensitive to outliers. Relative to MEW, it is often less extreme than the pure multiplicative formulation. Relative to SAW or FUCA, its computational effort is higher, but it remains far less demanding than AHP or ANP. Relative to SAW, COPRAS is less transparent than a plain weighted sum; relative to AHP, it is easier to explain than an eigenvector-based procedure. In the chapter’s own summary, COPRAS offers a middle ground between purely additive and purely multiplicative aggregation, with an intuitive separation of beneficial and adverse effects, moderate computational cost, and clear ranking outcomes.

6. Interpretation, edge cases, and methodological cautions

The structure of COPRAS rests on the decomposition of performance into beneficial and adverse effects. The additive term $i = 1,\ldots,m$02 represents the weighted contribution of benefit criteria, while the second term modifies the score according to the relative magnitude of the weighted cost sum. Because this second term depends on both $i = 1,\ldots,m$03 and $i = 1,\ldots,m$04, the method is not only alternative-specific but also set-dependent. This suggests why the chapter identifies rank reversal as a possible phenomenon when alternatives are added or removed.

The two special cases clarify the method’s boundaries. When all criteria are benefits, the method collapses to

$i = 1,\ldots,m$05

When all criteria are costs, the chapter states that sometimes one takes $i = 1,\ldots,m$06 or normalizes $i = 1,\ldots,m$07 so that larger is better. These cases indicate that the usual COPRAS formula is intended for mixed benefit-cost settings and that the cost-only case may require a convention rather than a single universally fixed expression.

A common misunderstanding is to treat COPRAS as a plain weighted-sum method. The chapter’s formulation does not support that interpretation. The weighted sums $i = 1,\ldots,m$08 and $i = 1,\ldots,m$09 are intermediate quantities, but the final ranking in the mixed-criteria case is determined by

$i = 1,\ldots,m$10

not by a single direct sum over all criteria. Another possible misunderstanding is to assume that explicit benefit/cost separation automatically guarantees transparency for all audiences. The chapter instead states that interpretability is high for those who like explicit benefit/cost splits, while the dependence of $i = 1,\ldots,m$11 on $i = 1,\ldots,m$12 and $i = 1,\ldots,m$13 can still be non-intuitive.

Within aggregation-type MCDM, COPRAS is therefore best understood as a method that combines sum-based normalization, weighted aggregation, and a cost-sensitive ratio adjustment. Its procedural simplicity coexists with sensitivity to outliers, possible rank reversal, and a final score construction that is straightforward computationally but not always immediately intuitive (Wang et al., 8 Sep 2025).