Combined Compromise Solution :
CoCoSo Algorithm
The Combined Compromise Solution (CoCoSo) method integrates both the weighted sum and the weighted product models to determine a final ranking.
1. Normalization:
Benefit: \[ r_{ij} = \frac{x_{ij} – \min x_j}{\max x_j – \min x_j} \] Cost: \[ r_{ij} = \frac{\max x_j – x_{ij}}{\max x_j – \min x_j} \]
Benefit: \[ r_{ij} = \frac{x_{ij} – \min x_j}{\max x_j – \min x_j} \] Cost: \[ r_{ij} = \frac{\max x_j – x_{ij}}{\max x_j – \min x_j} \]
2. Weighted Sum (\(S_i\)) and Product (\(P_i\)):
\[ S_i = \sum_{j=1}^n (w_j r_{ij}), \quad P_i = \sum_{j=1}^n (r_{ij})^{w_j} \]
\[ S_i = \sum_{j=1}^n (w_j r_{ij}), \quad P_i = \sum_{j=1}^n (r_{ij})^{w_j} \]
3. Appraisal Scores:
\[ k_{ia} = \frac{P_i + S_i}{\sum (P_i + S_i)}, \quad k_{ib} = \frac{S_i}{\min S_i} + \frac{P_i}{\min P_i} \] \[ k_{ic} = \frac{\lambda S_i + (1-\lambda)P_i}{\lambda \max S_i + (1-\lambda) \max P_i} \]
\[ k_{ia} = \frac{P_i + S_i}{\sum (P_i + S_i)}, \quad k_{ib} = \frac{S_i}{\min S_i} + \frac{P_i}{\min P_i} \] \[ k_{ic} = \frac{\lambda S_i + (1-\lambda)P_i}{\lambda \max S_i + (1-\lambda) \max P_i} \]
4. Final Ranking (\(K_i\)):
\[ K_i = \sqrt[3]{k_{ia}k_{ib}k_{ic}} + \frac{1}{3}(k_{ia} + k_{ib} + k_{ic}) \]
\[ K_i = \sqrt[3]{k_{ia}k_{ib}k_{ic}} + \frac{1}{3}(k_{ia} + k_{ib} + k_{ic}) \]
Solved Example: Cloud Service Provider
Scenario: Selecting a provider based on: Speed (B), Reliability (B), Latency (C), and Price (C).
Weights: \(w = [0.3, 0.2, 0.3, 0.2]\). \(\lambda = 0.5\).
Step 1: Raw Matrix & Min/Max
| Alt | Speed | Rel. | Lat. | Price |
|---|---|---|---|---|
| P1 | 100 | 99 | 20 | 500 |
| P2 | 150 | 95 | 15 | 700 |
| P3 | 120 | 99.9 | 10 | 1000 |
| P4 | 140 | 98 | 25 | 400 |
| Max | 150 | 99.9 | 25 | 1000 |
| Min | 100 | 95 | 10 | 400 |
Step 2: Solved Normalization (P1)
Speed (B): \((100-100)/(150-100) = 0.00\)
Latency (C): \((25-20)/(25-10) = 0.33\)
Price (C): \((1000-500)/(1000-400) = 0.83\)
Speed (B): \((100-100)/(150-100) = 0.00\)
Latency (C): \((25-20)/(25-10) = 0.33\)
Price (C): \((1000-500)/(1000-400) = 0.83\)
Step 3: Calculating S and P (P1)
\(S_1 = (0.3 \times 0.0) + (0.2 \times 0.82) + (0.3 \times 0.33) + (0.2 \times 0.83) = 0.429\)
\(P_1 = (0.0^{0.3}) + (0.82^{0.2}) + (0.33^{0.3}) + (0.83^{0.2}) = 2.645\)
\(S_1 = (0.3 \times 0.0) + (0.2 \times 0.82) + (0.3 \times 0.33) + (0.2 \times 0.83) = 0.429\)
\(P_1 = (0.0^{0.3}) + (0.82^{0.2}) + (0.33^{0.3}) + (0.83^{0.2}) = 2.645\)
Step 4: Final Rankings
| Alt | S Score | P Score | K Score | Rank |
|---|---|---|---|---|
| P1 | 0.429 | 2.645 | 1.391 | 4 |
| P2 | 0.686 | 3.123 | 2.011 | 2 |
| P3 | 0.521 | 2.518 | 1.583 | 3 |
| P4 | 0.743 | 3.451 | 2.305 | 1 |
Interactive CoCoSo Calculator
| Weights | ||||
|---|---|---|---|---|
| Type | ||||
| A1 | ||||
| A2 | ||||
| A3 | ||||
| A4 |
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