## 4. Mathematical Optimization Formulation

#### Decision Variables
- Let \( x_{ij} \) be a binary decision variable where:
  - \( x_{ij} = 1 \) if singer \( i \) is assigned to concert in stadium \( j \),
  - \( x_{ij} = 0 \) otherwise.

#### Objective Function
Maximize the total audience capacity:
\[
\text{Maximize } Z = \sum_{i} \sum_{j} \text{stadium.Capacity}_j \times x_{ij}
\]
- **Coefficient Source**: `stadium.Capacity` from the `stadium` table.

#### Constraints
1. **Singer Assignment Limit**: Each singer \( i \) can participate in at most \( L \) concerts:
\[
\sum_{j} x_{ij} \leq L \quad \forall i
\]
- **Coefficient Source**: \( L \) is a scalar parameter from the business configuration logic.

2. **Stadium Capacity Limit**: Each concert in stadium \( j \) cannot exceed the stadium's capacity:
\[
\sum_{i} x_{ij} \leq \text{stadium.Capacity}_j \quad \forall j
\]
- **Coefficient Source**: `stadium.Capacity` from the `stadium` table.

#### Data Source Verification
- **stadium.Capacity**: From the `stadium` table.
- **L**: From the business configuration logic (scalar parameter for singer limits).

This formulation provides a complete, immediately solvable LINEAR mathematical model with all numerical coefficients derived from the provided data.

## 5. Gurobipy Implementation

```python
#!/usr/bin/env python3
"""
Gurobipy implementation for concert singer optimization problem
"""

import gurobipy as gp
from gurobipy import GRB

def optimize_concert_singer():
    # 1. MODEL & DATA SETUP
    model = gp.Model("concert_singer_optimization")
    
    # Data from the problem
    stadium_capacities = [10000, 20000, 30000]
    singer_assignments = [0, 1, 0, 1, 0]
    L = 2  # Maximum number of concerts a singer can participate in
    
    # Number of singers and stadiums
    n_singers = len(singer_assignments)
    n_stadiums = len(stadium_capacities)
    
    # CRITICAL: Validate array lengths before loops
    assert len(stadium_capacities) == n_stadiums, "Stadium capacities array length mismatch"
    assert len(singer_assignments) == n_singers, "Singer assignments array length mismatch"
    
    # 2. VARIABLES
    # Decision variables: x_ij (binary)
    x = model.addVars(n_singers, n_stadiums, vtype=GRB.BINARY, name="x")
    
    # 3. OBJECTIVE FUNCTION
    # Maximize total audience capacity
    model.setObjective(
        gp.quicksum(stadium_capacities[j] * x[i, j] for i in range(n_singers) for j in range(n_stadiums)),
        GRB.MAXIMIZE
    )
    
    # 4. CONSTRAINTS
    
    # Singer Assignment Limit: Each singer can participate in at most L concerts
    for i in range(n_singers):
        model.addConstr(
            gp.quicksum(x[i, j] for j in range(n_stadiums)) <= L,
            name=f"singer_limit_{i}"
        )
    
    # Stadium Capacity Limit: Each concert in stadium j cannot exceed the stadium's capacity
    for j in range(n_stadiums):
        model.addConstr(
            gp.quicksum(x[i, j] for i in range(n_singers)) <= stadium_capacities[j],
            name=f"stadium_capacity_{j}"
        )
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in range(n_singers):
            for j in range(n_stadiums):
                if x[i, j].x > 1e-6:
                    print(f"Singer {i} assigned to stadium {j}: {x[i, j].x:.0f}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    
    return model

# Execute the optimization
if __name__ == "__main__":
    optimize_concert_singer()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 250000.0
**Execution Time**: 0.19 seconds
**Reliability**: high
**Analysis**: Gurobipy found an optimal solution with the highest objective value and the fastest execution time, indicating high efficiency and reliability.

## 6. DOCplex Implementation

```python
#!/usr/bin/env python3
"""
DOCplex implementation for concert_singer optimization problem
"""

from docplex.mp.model import Model

def optimize_concert_singer():
    """Optimize singer assignments to concerts in stadiums to maximize audience capacity."""
    
    # 1. MODEL & DATA SETUP
    mdl = Model(name="concert_singer")
    
    # Data from the database
    stadium_capacities = [10000, 20000, 30000]
    singer_assignments = [0, 1, 0, 1, 0]
    
    # Business configuration parameter
    L = 2  # Maximum number of concerts a singer can participate in
    
    # Validate array lengths
    n_stadiums = len(stadium_capacities)
    n_assignments = len(singer_assignments)
    assert n_stadiums > 0 and n_assignments > 0, "Empty data arrays"
    
    # Safe range for indexing
    safe_range = range(min(n_stadiums, n_assignments))
    
    # 2. VARIABLES
    # Decision variables: x_ij = 1 if singer i is assigned to concert in stadium j, 0 otherwise
    x = {(i, j): mdl.binary_var(name=f"x_{i}_{j}") for i in safe_range for j in safe_range}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize total audience capacity
    objective = mdl.sum(stadium_capacities[j] * x[(i, j)] for i in safe_range for j in safe_range)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS
    
    # Singer Assignment Limit: Each singer can participate in at most L concerts
    for i in safe_range:
        mdl.add_constraint(mdl.sum(x[(i, j)] for j in safe_range) <= L, ctname=f"singer_limit_{i}")
    
    # Stadium Capacity Limit: Each concert in stadium j cannot exceed the stadium's capacity
    for j in safe_range:
        mdl.add_constraint(mdl.sum(x[(i, j)] for i in safe_range) <= stadium_capacities[j], ctname=f"stadium_capacity_{j}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in safe_range:
            for j in safe_range:
                value = solution.get_value(x[(i, j)])
                if value > 1e-6:
                    print(f"Singer {i} assigned to stadium {j}: {value:.0f}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")
    
    return mdl

# Execute the optimization
if __name__ == "__main__":
    optimize_concert_singer()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 150000.0
**Execution Time**: 1.41 seconds
**Reliability**: medium
**Analysis**: DOCplex found an optimal solution but with a significantly lower objective value compared to Gurobipy and Pyomo, suggesting potential issues with solver configuration or model interpretation.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation

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

def concert_singer_optimization():
    """Optimize singer assignments to concerts in stadiums to maximize audience capacity."""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Example data from the problem context
    stadium_capacities = [10000, 20000, 30000]  # Capacities of stadiums
    singer_assignments = [0, 1, 0, 1, 0]  # Initial assignments (binary)
    singer_limit = 2  # Maximum number of concerts a singer can participate in
    
    # CRITICAL: Validate array lengths before indexing
    n_stadiums = len(stadium_capacities)
    n_assignments = len(singer_assignments)
    assert n_stadiums > 0, "No stadiums provided"
    assert n_assignments > 0, "No singer assignments provided"
    
    # 3. SETS (Pyomo way to define indices)
    model.I = pyo.RangeSet(1, n_assignments)  # Singers (1-based indexing)
    model.J = pyo.RangeSet(1, n_stadiums)  # Stadiums (1-based indexing)
    
    # 4. PARAMETERS (data containers)
    model.stadium_capacity = pyo.Param(model.J, initialize={j: stadium_capacities[j-1] for j in model.J})
    model.singer_limit = pyo.Param(initialize=singer_limit)
    
    # 5. VARIABLES
    # Binary decision variable: x[i,j] = 1 if singer i is assigned to stadium j, 0 otherwise
    model.x = pyo.Var(model.I, model.J, within=pyo.Binary)
    
    # 6. OBJECTIVE FUNCTION
    # Maximize the total audience capacity
    def obj_rule(model):
        return sum(model.stadium_capacity[j] * 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
    
    # Constraint 1: Singer Assignment Limit
    def singer_limit_rule(model, i):
        return sum(model.x[i, j] for j in model.J) <= model.singer_limit
    model.singer_limit_constraint = pyo.Constraint(model.I, rule=singer_limit_rule)
    
    # Constraint 2: Stadium Capacity Limit
    def stadium_capacity_rule(model, j):
        return sum(model.x[i, j] for i in model.I) <= model.stadium_capacity[j]
    model.stadium_capacity_constraint = pyo.Constraint(model.J, rule=stadium_capacity_rule)
    
    # 8. SOLVING WITH GUROBI (your available solver)
    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:
                if pyo.value(model.x[i, j]) > 0.5:  # Only print assignments
                    print(f"Singer {i} assigned to Stadium {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
if __name__ == "__main__":
    concert_singer_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 250000.0
**Execution Time**: 0.99 seconds
**Reliability**: high
**Analysis**: Pyomo found an optimal solution with the same objective value as Gurobipy but with a longer execution time, indicating reliability but lower efficiency.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | OPTIMAL | 250000.00 | 0.19s | N/A | N/A |
| Docplex | OPTIMAL | 150000.00 | 1.41s | N/A | N/A |
| Pyomo | OPTIMAL | 250000.00 | 0.99s | N/A | N/A |

### Solver Consistency Analysis
**Result**: Solvers produced inconsistent results
**Consistent Solvers**: gurobipy, pyomo
**Inconsistent Solvers**: docplex
**Potential Issues**:
- Possible misconfiguration in DOCplex solver
- Different interpretations of the model constraints or objective function in DOCplex
**Majority Vote Optimal Value**: 250000.0

### Final Recommendation
**Recommended Optimal Value**: 250000.0
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy is recommended due to its high reliability, optimal solution, and fastest execution time. Pyomo also provides a consistent solution but is less efficient.

### Business Interpretation
**Overall Strategy**: The optimal solution maximizes the total audience capacity across all concerts, ensuring the best possible utilization of stadium resources.
**Objective Value Meaning**: The optimal objective value of 250,000 represents the maximum total audience capacity achievable under the given constraints.
**Resource Allocation Summary**: Resources should be allocated to ensure that each singer is assigned to concerts in a way that maximizes the total audience capacity without exceeding individual singer limits or stadium capacities.
**Implementation Recommendations**: Implement the optimal assignments of singers to concerts as determined by the solver. Ensure that the constraints on singer participation limits and stadium capacities are strictly adhered to during implementation.