## 4. Mathematical Optimization Formulation

#### Decision Variables
- Let \( x_i \) represent the number of races to be held at track \( i \).
- \( x_i \) is an integer variable for each track \( i \).

#### Objective Function
Maximize the total number of spectators across all tracks:
\[ \text{Maximize } Z = \sum_{i} (\text{Seating}_i \times x_i) \]

Data Source Verification:
- Coefficient \(\text{Seating}_i\) comes from the `track.Seating` column.

#### Constraints
1. Total number of races across all tracks must not exceed the total_races_limit:
   \[ \sum_{i} x_i \leq \text{total\_races\_limit} \]

2. For each track, the number of spectators must not exceed the maximum allowed:
   \[ \text{Seating}_i \times x_i \leq \text{max\_spectators}_i \quad \forall i \]

Data Source Verification:
- Coefficient \(\text{max\_spectators}_i\) comes from the `track.max_spectators` column.
- The parameter \(\text{total\_races\_limit}\) is a business configuration parameter.

#### Complete Model
Given the data:

- Track 1: Seating = 5000, max_spectators = 4500
- Track 2: Seating = 10000, max_spectators = 9500
- Track 3: Seating = 15000, max_spectators = 14500
- Assume \(\text{total\_races\_limit} = 10\) (for example)

The complete model is:

Maximize:
\[ Z = 5000x_1 + 10000x_2 + 15000x_3 \]

Subject to:
\[ x_1 + x_2 + x_3 \leq 10 \]
\[ 5000x_1 \leq 4500 \]
\[ 10000x_2 \leq 9500 \]
\[ 15000x_3 \leq 14500 \]

\[ x_1, x_2, x_3 \geq 0 \]
\[ x_1, x_2, x_3 \text{ are integers} \]

This model can be solved using a linear or mixed-integer programming solver to determine the optimal number of races to hold at each track to maximize spectator attendance.

## 5. Gurobipy Implementation

```python
# Complete GUROBIPY implementation

import gurobipy as gp
from gurobipy import GRB

def optimize_race_track():
    """Optimize the number of races to maximize spectators"""

    # 1. MODEL & DATA SETUP
    model = gp.Model("race_track")
    
    # Data from the problem statement
    seating = [5000, 10000, 15000]
    max_spectators = [4500, 9500, 14500]
    total_races_limit = 10
    n_tracks = len(seating)
    
    # CRITICAL: Validate array lengths before loops
    assert len(seating) == len(max_spectators) == n_tracks, "Array length mismatch"
    
    # 2. VARIABLES
    # Variable dictionaries for number of races at each track
    races = {i: model.addVar(vtype=GRB.INTEGER, name=f"races_{i}", lb=0) 
             for i in range(n_tracks)}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize the total number of spectators
    model.setObjective(gp.quicksum(seating[i] * races[i] for i in range(n_tracks)), GRB.MAXIMIZE)
    
    # 4. CONSTRAINTS - CORRECT SYNTAX PATTERNS
    
    # Total number of races constraint
    model.addConstr(gp.quicksum(races[i] for i in range(n_tracks)) <= total_races_limit, name="total_races_limit")
    
    # Maximum spectators constraint for each track
    for i in range(n_tracks):
        model.addConstr(seating[i] * races[i] <= max_spectators[i], name=f"max_spectators_{i}")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in range(n_tracks):
            if races[i].x > 1e-6:
                print(f"races[{i}] = {races[i].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
optimize_race_track()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: -0.0
**Execution Time**: 0.74 seconds
**Reliability**: high
**Analysis**: Gurobipy found an optimal solution with all decision variables set to zero, which is consistent with the constraints that prevent any races from being held due to the spectator limits.

## 6. DOCplex Implementation

```python
# Complete DOCPLEX implementation

from docplex.mp.model import Model

def optimize_race_track():
    """Optimize the number of races to maximize spectators"""
    
    # 1. MODEL & DATA SETUP
    mdl = Model(name="race_track_optimization")
    
    # Data from the problem statement
    seating = [5000, 10000, 15000]
    max_spectators = [4500, 9500, 14500]
    total_races_limit = 10
    
    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(seating) == len(max_spectators), "Array length mismatch"
    safe_range = range(min(len(seating), len(max_spectators)))  # Safe indexing
    
    # 2. VARIABLES
    # Integer variables for the number of races at each track
    races = {i: mdl.integer_var(name=f"races_{i}", lb=0) for i in safe_range}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize the total number of spectators
    objective = mdl.sum(seating[i] * races[i] for i in safe_range)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS - CORRECT SYNTAX PATTERNS
    
    # Total number of races constraint
    total_races = mdl.sum(races[i] for i in safe_range)
    mdl.add_constraint(total_races <= total_races_limit, ctname="total_races_limit")
    
    # Spectator capacity constraints for each track
    for i in safe_range:
        mdl.add_constraint(seating[i] * races[i] <= max_spectators[i], ctname=f"capacity_{i}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in safe_range:
            value = solution.get_value(races[i])
            if value > 1e-6:
                print(f"races[{i}] = {value:.0f}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")
    
    return mdl

# Run the optimization
optimize_race_track()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 0.0
**Execution Time**: 3.76 seconds
**Reliability**: high
**Analysis**: DOCplex also found an optimal solution with all decision variables set to zero, aligning with the constraints.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation

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

def race_track_optimization():
    """Optimize the number of races to maximize spectators"""

    # 1. MODEL CREATION
    model = pyo.ConcreteModel()

    # 2. DATA SETUP
    seating = [5000, 10000, 15000]
    max_spectators = [4500, 9500, 14500]
    total_races_limit = 10

    # CRITICAL: Validate array lengths before indexing
    assert len(seating) == len(max_spectators), "Array length mismatch"

    # 3. SETS (Pyomo way to define indices)
    model.I = pyo.RangeSet(1, len(seating))  # 1-based indexing

    # 4. PARAMETERS (data containers)
    model.seating = pyo.Param(model.I, initialize={i+1: seating[i] for i in range(len(seating))})
    model.max_spectators = pyo.Param(model.I, initialize={i+1: max_spectators[i] for i in range(len(max_spectators))})

    # 5. VARIABLES
    model.x = pyo.Var(model.I, within=pyo.NonNegativeIntegers)

    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.seating[i] * model.x[i] for i in model.I)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)

    # 7. CONSTRAINTS

    # Total races constraint
    def total_races_rule(model):
        return sum(model.x[i] for i in model.I) <= total_races_limit
    model.total_races_constraint = pyo.Constraint(rule=total_races_rule)

    # Spectator capacity constraints
    def spectator_capacity_rule(model, i):
        return model.seating[i] * model.x[i] <= model.max_spectators[i]
    model.spectator_capacity_constraint = pyo.Constraint(model.I, rule=spectator_capacity_rule)

    # 8. SOLVING WITH GUROBI
    solver = SolverFactory('gurobi')

    # 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("\nVariable values:")
        for i in model.I:
            x_val = pyo.value(model.x[i])
            if x_val > 1e-6:  # Only print non-zero values
                print(f"x[{i}] = {x_val:.3f}")

    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
race_track_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 0.0
**Execution Time**: 3.83 seconds
**Reliability**: high
**Analysis**: Pyomo's results are consistent with the other solvers, indicating no races can be held due to the constraints.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | OPTIMAL | -0.00 | 0.74s | N/A | N/A |
| Docplex | OPTIMAL | 0.00 | 3.76s | N/A | N/A |
| Pyomo | OPTIMAL | 0.00 | 3.83s | N/A | N/A |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Majority Vote Optimal Value**: 0.0

### Final Recommendation
**Recommended Optimal Value**: 0.0
**Confidence Level**: HIGH
**Preferred Solver(s)**: multiple
**Reasoning**: All solvers provided consistent results, confirming the constraints prevent any races from being held.

### Optimal Decision Variables
- **x_1** = 0.000
  - *Business Meaning*: Number of races at Track 2, which is zero due to spectator constraints.
- **x_2** = 0.000
  - *Business Meaning*: Number of races at Track 3, which is zero due to spectator constraints.
- **x_3** = 0.000
  - *Business Meaning*: Resource allocation for x_3

### Business Interpretation
**Overall Strategy**: No races can be held at any track due to the spectator constraints, resulting in zero total spectators.
**Objective Value Meaning**: The optimal objective value of 0 indicates no spectators can be accommodated given the constraints.
**Resource Allocation Summary**: No resources should be allocated to holding races as the constraints prevent any races from occurring.
**Implementation Recommendations**: Review and potentially adjust the spectator constraints or total race limits to allow for feasible race scheduling.