MARCOS Algorithm
The Measurement of Alternatives and Ranking according to Compromise Solution (MARCOS) method defines utility functions based on ideal (AI) and anti-ideal (AAI) values.
1. Define Ideal (\(AI\)) and Anti-Ideal (\(AAI\)) Solutions:
\[ AI = \max(x_{ij}) \text{ (Benefit)}, \min(x_{ij}) \text{ (Cost)} \] \[ AAI = \min(x_{ij}) \text{ (Benefit)}, \max(x_{ij}) \text{ (Cost)} \]
\[ AI = \max(x_{ij}) \text{ (Benefit)}, \min(x_{ij}) \text{ (Cost)} \] \[ AAI = \min(x_{ij}) \text{ (Benefit)}, \max(x_{ij}) \text{ (Cost)} \]
2. Normalize Decision Matrix:
Benefit: \( n_{ij} = \frac{x_{ij}}{AI_j} \quad \) Cost: \( n_{ij} = \frac{AI_j}{x_{ij}} \)
Benefit: \( n_{ij} = \frac{x_{ij}}{AI_j} \quad \) Cost: \( n_{ij} = \frac{AI_j}{x_{ij}} \)
3. Weighted Matrix and Sum (\(S_i\)):
\[ v_{ij} = n_{ij} \times w_j \quad \Rightarrow \quad S_i = \sum v_{ij} \]
\[ v_{ij} = n_{ij} \times w_j \quad \Rightarrow \quad S_i = \sum v_{ij} \]
4. Utility Degrees and Function:
\[ K_i^- = \frac{S_i}{S_{AAI}}, \quad K_i^+ = \frac{S_i}{S_{AI}} \] \[ f(K_i) = \frac{K_i^+ + K_i^-}{1 + \frac{1-f(K_i^+)}{f(K_i^+)} + \frac{1-f(K_i^-)}{f(K_i^-)}} \]
\[ K_i^- = \frac{S_i}{S_{AAI}}, \quad K_i^+ = \frac{S_i}{S_{AI}} \] \[ f(K_i) = \frac{K_i^+ + K_i^-}{1 + \frac{1-f(K_i^+)}{f(K_i^+)} + \frac{1-f(K_i^-)}{f(K_i^-)}} \]
Solved Example: Employee Hiring
Criteria: C1: Experience(B), C2: Skills(B), C3: Salary(C), C4: Interview(B). Weights: 0.25 each.
Step 1: Decision Matrix with AI and AAI
| Alt | C1(B) | C2(B) | C3(C) | C4(B) |
|---|---|---|---|---|
| AI | 10 | 90 | 2000 | 10 |
| Emp 1 | 8 | 80 | 2500 | 8 |
| Emp 2 | 10 | 70 | 2200 | 9 |
| Emp 3 | 7 | 90 | 3000 | 10 |
| Emp 4 | 9 | 85 | 2000 | 7 |
| AAI | 7 | 70 | 3000 | 7 |
Step 2: Normalization (Beneficial Logic)
Example Emp 1, C1: \( 8 / 10 = 0.80 \)
Example Emp 1, C3 (Cost): \( 2000 / 2500 = 0.80 \)
Step 3: Calculating Utility Degrees (K) for Emp 1
Sum of Weighted Normalized Values (\(S_i\)):
Emp 1: \(S_1 = 0.785\), AI: \(S_{AI} = 1.00\), AAI: \(S_{AAI} = 0.642\)
\( K_1^- = 0.785 / 0.642 = 1.22 \)
\( K_1^+ = 0.785 / 1.00 = 0.785 \)
Step 4: Final Utility Rank
After applying the utility function \(f(K_i)\), alternatives are ranked by highest value.
| Rank | Alt | f(Ki) Score |
|---|---|---|
| 1 | Emp 2 | 0.6842 |
| 2 | Emp 4 | 0.6511 |
| 3 | Emp 1 | 0.5920 |
Interactive MARCOS Calculator
| Weights | ||||
|---|---|---|---|---|
| Type | ||||
| A1 | ||||
| A2 | ||||
| A3 | ||||
| A4 |
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