CODAS Algorithm
The Combinative Distance-based ASsessment (CODAS) method determines the desirability of alternatives by calculating their Euclidean and Taxicab distances from the negative-ideal solution.
1. Normalization:
For Benefit: \( r_{ij} = \frac{x_{ij}}{\max_i x_{ij}} \) | For Cost: \( r_{ij} = \frac{\min_i x_{ij}}{x_{ij}} \)
For Benefit: \( r_{ij} = \frac{x_{ij}}{\max_i x_{ij}} \) | For Cost: \( r_{ij} = \frac{\min_i x_{ij}}{x_{ij}} \)
2. Weighted Matrix:
\[ v_{ij} = w_j \times r_{ij} \]
\[ v_{ij} = w_j \times r_{ij} \]
3. Negative-Ideal Solution (\(ns\)):
\[ ns_j = \min_i v_{ij} \]
\[ ns_j = \min_i v_{ij} \]
4. Distance Measures:
Euclidean: \( E_i = \sqrt{\sum (v_{ij} – ns_j)^2} \)
Taxicab: \( T_i = \sum |v_{ij} – ns_j| \)
Euclidean: \( E_i = \sqrt{\sum (v_{ij} – ns_j)^2} \)
Taxicab: \( T_i = \sum |v_{ij} – ns_j| \)
5. Assessment Score (\(AS_i\)):
\[ AS_i = \sum_{k=1}^m (E_i – E_k) + \psi(E_i – E_k) \times (T_i – T_k) \] Where \(\psi\) is a threshold function (usually \(\tau = 0.02\)).
\[ AS_i = \sum_{k=1}^m (E_i – E_k) + \psi(E_i – E_k) \times (T_i – T_k) \] Where \(\psi\) is a threshold function (usually \(\tau = 0.02\)).
Solved Example: Industrial Robot Selection
Criteria: C1 Speed (B), C2 Load (B), C3 Price (C), C4 Reliability (B).
Weights: 0.3, 0.2, 0.3, 0.2.
Step 1: Decision Matrix
| Alt | C1 (B) | C2 (B) | C3 (C) | C4 (B) |
|---|---|---|---|---|
| R1 | 10 | 50 | 8000 | 0.95 |
| R2 | 12 | 40 | 9000 | 0.98 |
| R3 | 9 | 60 | 7500 | 0.90 |
| R4 | 11 | 45 | 8500 | 0.92 |
Step 2: Solved Normalization & Weighting
Example R1-C1: \(10 / 12 = 0.833\) | Weighted: \(0.833 \times 0.3 = 0.250\)
Example R1-C3 (Cost): \(7500 / 8000 = 0.937\) | Weighted: \(0.937 \times 0.3 = 0.281\)
Step 3: Distances from Negative-Ideal (ns)
After finding \(ns = [0.225, 0.133, 0.250, 0.183]\):
| Alt | Euclidean (E) | Taxicab (T) |
|---|---|---|
| R1 | 0.046 | 0.081 |
| R2 | 0.082 | 0.141 |
| R3 | 0.053 | 0.066 |
| R4 | 0.054 | 0.091 |
Step 4: Final Assessment Score (AS)
R2 has the highest Euclidean distance from the “worst” case and highest \(AS\).
Rank 1: Robot 2
Interactive CODAS Calculator
| Weights | ||||
|---|---|---|---|---|
| Type | ||||
| A1 | ||||
| A2 | ||||
| A3 | ||||
| A4 |
Threshold (\(\tau\)):
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