## 4. Mathematical Optimization Formulation

#### Decision Variables
- Let \( x_{ij} \) be a binary decision variable where \( x_{ij} = 1 \) if actor \( i \) is assigned to musical \( j \), and \( x_{ij} = 0 \) otherwise.

#### Objective Function
Minimize the total age of the actors assigned to the musicals:

\[
\text{Minimize } \sum_{i} \sum_{j} \text{age}_i \times x_{ij}
\]

where \(\text{age}_i\) is the age of actor \( i \) from the `actor_details` table.

#### Constraints
1. **Musical Requirements**: Each musical must have exactly the number of actors it requires.

   For each musical \( j \):

   \[
   \sum_{i} x_{ij} = \text{required\_actors}_j
   \]

   where \(\text{required\_actors}_j\) is the number of actors required for musical \( j \) from the `musical_requirements` table.

2. **Actor Assignment**: Each actor can only be assigned to one musical.

   For each actor \( i \):

   \[
   \sum_{j} x_{ij} \leq 1
   \]

#### Data Source Verification:
- Coefficients \(\text{age}_i\) are sourced from `actor_details.age`.
- Coefficients \(\text{required\_actors}_j\) are sourced from `musical_requirements.required_actors`.

This linear mathematical model is immediately solvable using linear programming techniques, ensuring that the constraints and objective function are aligned with the business problem's requirements.

## 5. Gurobipy Implementation

```python
# Complete GUROBIPY implementation - Retry Attempt 4

import gurobipy as gp
from gurobipy import GRB

def optimize_casting():
    # 1. MODEL & DATA SETUP
    model = gp.Model("musical_casting")

    # Data from the database schema
    musical_requirements = {
        1: 3,
        2: 4,
        3: 5
    }
    
    actor_details = {
        101: 25,
        102: 30,
        103: 35,
        104: 28,
        105: 32
    }
    
    # Extracting lists of actors and musicals
    actors = list(actor_details.keys())
    musicals = list(musical_requirements.keys())
    
    # Validate lengths (not strictly necessary here, but good practice)
    assert len(actors) > 0, "No actors available"
    assert len(musicals) > 0, "No musicals available"
    
    # 2. VARIABLES
    # Binary decision variables for actor assignments
    x = model.addVars(actors, musicals, vtype=GRB.BINARY, name="x")
    
    # 3. OBJECTIVE FUNCTION
    # Minimize the total age of the actors assigned to the musicals
    model.setObjective(gp.quicksum(actor_details[i] * x[i, j] for i in actors for j in musicals), GRB.MINIMIZE)
    
    # 4. CONSTRAINTS
    
    # Each musical must have exactly the number of actors it requires
    for j in musicals:
        model.addConstr(gp.quicksum(x[i, j] for i in actors) == musical_requirements[j], name=f"musical_req_{j}")
    
    # Each actor can only be assigned to one musical
    for i in actors:
        model.addConstr(gp.quicksum(x[i, j] for j in musicals) <= 1, name=f"actor_assign_{i}")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in actors:
            for j in musicals:
                if x[i, j].x > 0.5:  # Binary variable, so check if it's 1
                    print(f"Actor {i} is assigned to musical {j}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    else:
        print("Optimization ended with status", model.status)

    return model

# Run the optimization
optimize_casting()
```

### Execution Results
**Status**: INFEASIBLE
**Error**: Problem is infeasible
**Analysis**: Gurobipy consistently reports the problem as infeasible, indicating a potential issue with the problem formulation or data.

## 6. DOCplex Implementation

```python
# Complete DOCPLEX implementation - Retry Attempt 4

from docplex.mp.model import Model

def optimize_casting():
    # 1. MODEL & DATA SETUP
    mdl = Model(name="musical_casting")

    # Data from the problem context
    actor_ages = {101: 25, 102: 30, 103: 35}
    musical_requirements = {1: 3, 2: 4, 3: 5}

    # Actors and musicals
    actors = list(actor_ages.keys())
    musicals = list(musical_requirements.keys())

    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(actors) > 0 and len(musicals) > 0, "Actors or musicals data is missing"

    # 2. VARIABLES
    # Binary decision variables for actor assignments
    x = {(i, j): mdl.binary_var(name=f"x_{i}_{j}") for i in actors for j in musicals}

    # 3. OBJECTIVE FUNCTION
    # Minimize the total age of the actors assigned to the musicals
    objective = mdl.sum(actor_ages[i] * x[i, j] for i in actors for j in musicals)
    mdl.minimize(objective)

    # 4. CONSTRAINTS
    # Musical requirements: Each musical must have exactly the number of actors it requires
    for j in musicals:
        mdl.add_constraint(mdl.sum(x[i, j] for i in actors) == musical_requirements[j], ctname=f"musical_req_{j}")

    # Actor assignment: Each actor can only be assigned to one musical
    for i in actors:
        mdl.add_constraint(mdl.sum(x[i, j] for j in musicals) <= 1, ctname=f"actor_assign_{i}")

    # 5. SOLVING & RESULTS
    solution = mdl.solve()

    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in actors:
            for j in musicals:
                if solution.get_value(x[i, j]) > 0.5:  # Binary variable threshold
                    print(f"Actor {i} is assigned to musical {j}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")

    return mdl

# Run the optimization
optimize_casting()
```

### Execution Results
**Status**: INFEASIBLE
**Error**: Problem is infeasible
**Analysis**: DOCplex also reports infeasibility, suggesting that the constraints cannot be satisfied with the given data.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation - Retry Attempt 4

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

def optimize_casting():
    """Optimize the casting process for musicals to minimize total age of the cast."""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Example data based on the problem description
    actor_ids = [101, 102, 103, 104, 105]
    musical_ids = [1, 2, 3]
    ages = {101: 25, 102: 30, 103: 35, 104: 28, 105: 32}
    required_actors = {1: 3, 2: 4, 3: 5}
    
    # CRITICAL: Validate data consistency
    assert all(actor_id in ages for actor_id in actor_ids), "Missing age data for some actors"
    assert all(musical_id in required_actors for musical_id in musical_ids), "Missing requirement data for some musicals"
    
    # 3. SETS
    model.Actors = pyo.Set(initialize=actor_ids)
    model.Musicals = pyo.Set(initialize=musical_ids)
    
    # 4. PARAMETERS
    model.age = pyo.Param(model.Actors, initialize=ages)
    model.required_actors = pyo.Param(model.Musicals, initialize=required_actors)
    
    # 5. VARIABLES
    model.x = pyo.Var(model.Actors, model.Musicals, within=pyo.Binary)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.age[i] * model.x[i, j] for i in model.Actors for j in model.Musicals)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.minimize)
    
    # 7. CONSTRAINTS
    # Musical Requirements Constraint
    def musical_requirement_rule(model, j):
        return sum(model.x[i, j] for i in model.Actors) == model.required_actors[j]
    model.musical_requirement = pyo.Constraint(model.Musicals, rule=musical_requirement_rule)
    
    # Actor Assignment Constraint
    def actor_assignment_rule(model, i):
        return sum(model.x[i, j] for j in model.Musicals) <= 1
    model.actor_assignment = pyo.Constraint(model.Actors, rule=actor_assignment_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)
    
    # 9. RESULT PROCESSING
    if results.solver.termination_condition == pyo.TerminationCondition.optimal:
        print("Optimal solution found!")
        print(f"Optimal value: {pyo.value(model.objective)}")
        
        # Extract variable values
        print("\nAssignment of actors to musicals:")
        for i in model.Actors:
            for j in model.Musicals:
                if pyo.value(model.x[i, j]) > 0.5:  # Binary variable, check if assigned
                    print(f"Actor {i} assigned to Musical {j}")
        
    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
optimize_casting()
```

### Execution Results
**Status**: INFEASIBLE
**Error**: Problem is infeasible
**Analysis**: Pyomo confirms the infeasibility, reinforcing the need to review the problem setup and data.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | INFEASIBLE | N/A | 0.64s | N/A | 4 |
| Docplex | INFEASIBLE | N/A | 8.01s | N/A | 4 |
| Pyomo | INFEASIBLE | N/A | 3.92s | N/A | 4 |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Solver Retry Summary**: gurobipy: 4 attempts, docplex: 4 attempts, pyomo: 4 attempts

### Final Recommendation
**Confidence Level**: LOW
**Preferred Solver(s)**: multiple
**Reasoning**: All solvers consistently report infeasibility, suggesting a fundamental issue with the problem setup rather than solver-specific errors.

### Business Interpretation
**Overall Strategy**: The current problem setup cannot be solved as it stands, indicating a need for data or model adjustments.
**Objective Value Meaning**: The objective value represents the total age of actors assigned to musicals, which is currently not achievable due to infeasibility.
**Resource Allocation Summary**: Actors cannot be allocated to musicals as per the current requirements and constraints.
**Implementation Recommendations**: Review and verify the data in `actor_details` and `musical_requirements`. Ensure that the total number of actors matches the total required actors across all musicals. Re-evaluate the constraints to ensure they accurately reflect the business requirements.