## 4. Mathematical Optimization Formulation

#### Decision Variables
- Let \( x_{ij} \) be a binary decision variable where \( x_{ij} = 1 \) if artwork \( i \) is nominated at festival \( j \), and \( x_{ij} = 0 \) otherwise.

#### Objective Function
Maximize the total audience impact of successful nominations:
\[
\text{Maximize } \sum_{j} \text{Num\_of\_Audience}_j \times \sum_{i} x_{ij}
\]
where \(\text{Num\_of\_Audience}_j\) is the audience size for festival \( j \).

#### Constraints
1. **Maximum Nominations per Artwork:**
   \[
   \sum_{j} x_{ij} \leq \text{max\_nominations}_i \quad \forall i
   \]
   - This ensures that each artwork \( i \) is nominated at no more than its allowed maximum number of festivals.

2. **Maximum Capacity per Festival:**
   \[
   \sum_{i} x_{ij} \leq \text{max\_capacity}_j \quad \forall j
   \]
   - This ensures that each festival \( j \) does not exceed its nomination capacity.

3. **Binary Decision Variables:**
   \[
   x_{ij} \in \{0, 1\} \quad \forall i, j
   \]

Data Source Verification:
- \(\text{Num\_of\_Audience}_j\) comes from `festival_detail.Num_of_Audience`.
- \(\text{max\_nominations}_i\) comes from `artwork_constraints.max_nominations`.
- \(\text{max\_capacity}_j\) comes from `festival_constraints.max_capacity`.

This linear model is designed to maximize the visibility of artworks by strategically selecting nominations that maximize audience exposure, while adhering to the constraints on the number of nominations per artwork and the capacity of each festival.

## 5. Gurobipy Implementation

```python
# Complete GUROBIPY implementation

import gurobipy as gp
from gurobipy import GRB

def entertainment_awards_optimization():
    """Optimize artwork nominations for maximum audience impact."""
    
    # 1. MODEL & DATA SETUP
    model = gp.Model("entertainment_awards")
    
    # Data from the database
    num_of_audience = [1200, 4500, 9000]  # Audience sizes for each festival
    max_nominations = [3, 5, 7]  # Max nominations per artwork
    max_capacity = [50, 100, 150]  # Max capacity per festival
    
    num_artworks = len(max_nominations)
    num_festivals = len(num_of_audience)
    
    # CRITICAL: Validate array lengths before loops
    assert len(max_capacity) == num_festivals, "Array length mismatch for festivals"
    
    # 2. VARIABLES
    # Binary decision variables for nominations
    x = model.addVars(num_artworks, num_festivals, vtype=GRB.BINARY, name="x")
    
    # 3. OBJECTIVE FUNCTION
    # Maximize total audience impact
    model.setObjective(
        gp.quicksum(num_of_audience[j] * x[i, j] for i in range(num_artworks) for j in range(num_festivals)),
        GRB.MAXIMIZE
    )
    
    # 4. CONSTRAINTS - CORRECT SYNTAX PATTERNS
    
    # Maximum nominations per artwork
    model.addConstrs(
        (gp.quicksum(x[i, j] for j in range(num_festivals)) <= max_nominations[i] for i in range(num_artworks)),
        name="max_nominations_per_artwork"
    )
    
    # Maximum capacity per festival
    model.addConstrs(
        (gp.quicksum(x[i, j] for i in range(num_artworks)) <= max_capacity[j] for j in range(num_festivals)),
        name="max_capacity_per_festival"
    )
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in range(num_artworks):
            for j in range(num_festivals):
                if x[i, j].x > 1e-6:
                    print(f"x[{i},{j}] = {x[i, j].x:.3f}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    
    return model

# Run the optimization
entertainment_awards_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 44100.0
**Execution Time**: 1.77 seconds
**Reliability**: high
**Analysis**: Gurobipy found an optimal solution quickly with the shortest execution time, indicating efficient performance and reliability.

## 6. DOCplex Implementation

```python
# Complete DOCPLEX implementation

from docplex.mp.model import Model

def entertainment_awards_optimization():
    """Optimize artwork nominations for maximum audience impact."""
    
    # 1. MODEL & DATA SETUP
    mdl = Model(name="entertainment_awards")
    
    # Data from the database
    num_of_audience = [1200, 4500, 9000]  # Audience sizes for each festival
    max_nominations = [3, 5, 7]           # Max nominations per artwork
    max_capacity = [50, 100, 150]         # Max capacity per festival
    
    # Number of artworks and festivals
    num_artworks = len(max_nominations)
    num_festivals = len(num_of_audience)
    
    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(max_nominations) == num_artworks, "Array length mismatch for max_nominations"
    assert len(max_capacity) == num_festivals, "Array length mismatch for max_capacity"
    assert len(num_of_audience) == num_festivals, "Array length mismatch for num_of_audience"
    
    # Safe range for indexing
    safe_artworks = range(num_artworks)
    safe_festivals = range(num_festivals)
    
    # 2. VARIABLES
    # Binary decision variables for nominations
    x = {(i, j): mdl.binary_var(name=f"x_{i}_{j}") for i in safe_artworks for j in safe_festivals}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize the total audience impact of successful nominations
    objective = mdl.sum(num_of_audience[j] * x[i, j] for i in safe_artworks for j in safe_festivals)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS
    
    # Maximum nominations per artwork
    for i in safe_artworks:
        mdl.add_constraint(mdl.sum(x[i, j] for j in safe_festivals) <= max_nominations[i], ctname=f"max_nominations_artwork_{i}")
    
    # Maximum capacity per festival
    for j in safe_festivals:
        mdl.add_constraint(mdl.sum(x[i, j] for i in safe_artworks) <= max_capacity[j], ctname=f"max_capacity_festival_{j}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in safe_artworks:
            for j in safe_festivals:
                value = solution.get_value(x[i, j])
                if value > 1e-6:
                    print(f"x[{i},{j}] = {value:.3f}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")
    
    return mdl

# Run the optimization
entertainment_awards_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 44100.0
**Execution Time**: 11.44 seconds
**Reliability**: high
**Analysis**: DOCplex also found an optimal solution, but with a longer execution time compared to Gurobipy, suggesting it is reliable but less efficient.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation

import pyomo.environ as pyo
from pyomo.opt import SolverFactory

def entertainment_awards_optimization():
    """Optimize artwork nominations for maximum audience impact."""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Example data based on the provided problem context
    num_artworks = 3
    num_festivals = 3
    audience_sizes = [1200, 4500, 9000]
    max_nominations_per_artwork = [3, 5, 7]
    max_capacity_per_festival = [50, 100, 150]
    
    # CRITICAL: Validate array lengths before indexing
    assert len(audience_sizes) == num_festivals, "Audience sizes length mismatch"
    assert len(max_nominations_per_artwork) == num_artworks, "Max nominations length mismatch"
    assert len(max_capacity_per_festival) == num_festivals, "Max capacity length mismatch"
    
    # 3. SETS
    model.I = pyo.RangeSet(1, num_artworks)  # Artworks
    model.J = pyo.RangeSet(1, num_festivals)  # Festivals
    
    # 4. PARAMETERS
    model.audience_size = pyo.Param(model.J, initialize={j+1: audience_sizes[j] for j in range(num_festivals)})
    model.max_nominations = pyo.Param(model.I, initialize={i+1: max_nominations_per_artwork[i] for i in range(num_artworks)})
    model.max_capacity = pyo.Param(model.J, initialize={j+1: max_capacity_per_festival[j] for j in range(num_festivals)})
    
    # 5. VARIABLES
    model.x = pyo.Var(model.I, model.J, within=pyo.Binary)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.audience_size[j] * sum(model.x[i, j] for i in model.I) for j in model.J)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)
    
    # 7. CONSTRAINTS
    
    # Maximum nominations per artwork
    def max_nominations_rule(model, i):
        return sum(model.x[i, j] for j in model.J) <= model.max_nominations[i]
    model.max_nominations_constraint = pyo.Constraint(model.I, rule=max_nominations_rule)
    
    # Maximum capacity per festival
    def max_capacity_rule(model, j):
        return sum(model.x[i, j] for i in model.I) <= model.max_capacity[j]
    model.max_capacity_constraint = pyo.Constraint(model.J, rule=max_capacity_rule)
    
    # 8. SOLVING WITH GUROBI
    solver = SolverFactory('gurobi')
    
    # Optional: Set solver options
    solver.options['TimeLimit'] = 300  # 5 minutes
    solver.options['MIPGap'] = 0.01    # 1% gap
    
    # Solve the model
    results = solver.solve(model, tee=True)  # tee=True shows solver output
    
    # 9. RESULT PROCESSING
    # Check solver status
    if results.solver.termination_condition == pyo.TerminationCondition.optimal:
        print("Optimal solution found!")
        print(f"Optimal value: {pyo.value(model.objective)}")
        
        # Extract variable values
        print("\nVariable values:")
        for i in model.I:
            for j in model.J:
                x_val = pyo.value(model.x[i, j])
                if x_val > 1e-6:  # Only print non-zero values
                    print(f"x[{i},{j}] = {int(x_val)}")
        
    elif results.solver.termination_condition == pyo.TerminationCondition.infeasible:
        print("Problem is infeasible")
    elif results.solver.termination_condition == pyo.TerminationCondition.unbounded:
        print("Problem is unbounded")
    else:
        print(f"Solver terminated with condition: {results.solver.termination_condition}")
    
    return model

# Run the optimization
entertainment_awards_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 44100.0
**Execution Time**: 3.18 seconds
**Reliability**: high
**Analysis**: Pyomo achieved an optimal solution with a moderate execution time, demonstrating reliability and reasonable efficiency.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | OPTIMAL | 44100.00 | 1.77s | N/A | N/A |
| Docplex | OPTIMAL | 44100.00 | 11.44s | N/A | N/A |
| Pyomo | OPTIMAL | 44100.00 | 3.18s | N/A | N/A |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Majority Vote Optimal Value**: 44100.0

### Final Recommendation
**Recommended Optimal Value**: 44100.0
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy is preferred due to its high reliability and the shortest execution time, making it the most efficient choice for this problem.

### Business Interpretation
**Overall Strategy**: The optimal solution maximizes audience exposure for the nominated artworks, ensuring maximum visibility and impact.
**Objective Value Meaning**: The optimal objective value of 44100.0 indicates the maximum potential audience impact achievable under the given constraints.
**Resource Allocation Summary**: Resources should be allocated to ensure that artworks are nominated at festivals with the largest audiences, within the constraints of maximum nominations and festival capacities.
**Implementation Recommendations**: Implement the solution by coordinating with festival organizers to secure nominations for the selected artworks, ensuring compliance with the constraints.