COPRAS - Complex Proportional Assessment

Overview

N_i ∈ (0,100]. The best alternative always receives N=100%. All others are expressed as percentages of the best. Q_i accounts for both benefit performance (S₊) and cost burden (S₋) proportionally. COPRAS is sensitive to the scale of cost criteria values - large absolute cost values amplify the S₋ denominator effect.

Output

utility percent, higher is better

Data

Crisp, complete numeric matrix

Weights

Needs a weight source

Size

2+ alternatives, 3-12 criteria works best

Used for

Alternative selection, Supplier evaluation

How it works

1. 1

   Linear sum normalisation r_ij = x_ij / Σ x_kj.

   $$r_{ij}=\frac{x_{ij}}{{\sum }_{k=1}^m{x}_{kj}}$$

   Zavadskas-Kaklauskas 1996, Eq.(2)

2. 2

   Weighted normalised matrix d_ij = w_j · r_ij.

   $$d_{ij}=w_j\cdot r_{ij}$$

   Zavadskas-Kaklauskas 1996, Eq.(3)

3. 3

   Sum benefit S+_i and cost S−_i.

   $$S_i^{+}=\sum _{j\in J^{+}}{d}_{ij},S_i^{-}=\sum _{j\in J^{-}}{d}_{ij}$$

   Zavadskas-Kaklauskas 1996, Eqs.(4)-(5)

4. 4

   Relative significance Q_i (Zavadskas-Kaklauskas compound).

   $$Q_i=S_i^{+}+\frac{{\min }_k{S}_k^{-}\cdot {\sum }_k{S}_k^{-}}{S_i^{-}\cdot {\sum }_k\frac{{\min }_k{S}_k^{-}}{S_k^{-}}}$$

   Zavadskas-Kaklauskas 1996, Eq.(6)

5. 5

   Utility degree N_i (percent of best) and descending ranking.

   $$N_i=\frac{Q_i}{{\max }_k{Q}_k}\times 100\%$$

   Zavadskas-Kaklauskas 1996, Eq.(7)

Look elsewhere when

Assumptions to verify

• Criteria preferences are independent (no synergistic interactions)
• Compensation is acceptable: high score on one criterion can offset low on another
• Decision matrix is complete (no missing values)

Edge cases and pitfalls

• S_minus_i = 0 for some alternative (all cost criteria are zero): division by zero in Q_i formula - handle degenerate case.

Works with

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