## 4. Mathematical Optimization Formulation

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

#### Objective Function
- Minimize the maximum number of students in any classroom. Introduce an auxiliary variable \( M \) to represent this maximum number. The objective is to minimize \( M \).

\[
\text{Minimize } M
\]

#### Constraints
1. **Classroom Capacity Constraints**: Ensure that the number of students assigned to each classroom does not exceed its capacity.

   For each classroom \( j \):

   \[
   \sum_{i} x_{ij} \leq \text{max\_students}_j
   \]

   where \(\text{max\_students}_j\) is the maximum capacity of classroom \( j \) from the `ClassroomCapacity` table.

2. **Student Assignment Constraints**: Ensure each student is assigned to exactly one classroom.

   For each student \( i \):

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

3. **Maximum Students Constraint**: Ensure that the number of students in any classroom does not exceed \( M \).

   For each classroom \( j \):

   \[
   \sum_{i} x_{ij} \leq M
   \]

#### Data Source Verification:
- The coefficients for the classroom capacity constraints (\(\text{max\_students}_j\)) are sourced from the `ClassroomCapacity.max_students` column.
- The binary nature of the decision variables \( x_{ij} \) is derived from the `DecisionVariables.assignment` column, indicating whether a student is assigned to a classroom.

This formulation ensures that the optimization problem is linear and adheres to the constraints and objectives outlined in the business problem. The model can be solved using standard linear programming or mixed-integer programming solvers.

## 5. Gurobipy Implementation

```python
# Complete GUROBIPY implementation

import gurobipy as gp
from gurobipy import GRB

def optimize_classroom_assignment():
    """Optimize student assignments to classrooms to minimize the maximum number of students in any classroom."""
    
    # 1. MODEL & DATA SETUP
    model = gp.Model("classroom_assignment")
    
    # Example data setup
    students = [1, 2, 3, 4, 5]
    classrooms = [101, 102, 103]
    max_students = {101: 25, 102: 30, 103: 35}
    
    # CRITICAL: Validate array lengths before loops
    assert len(students) > 0, "No students available"
    assert len(classrooms) > 0, "No classrooms available"
    
    # 2. VARIABLES
    # Binary decision variables for student assignments
    x = model.addVars(students, classrooms, vtype=GRB.BINARY, name="x")
    
    # Auxiliary variable for the maximum number of students in any classroom
    M = model.addVar(vtype=GRB.CONTINUOUS, name="M", lb=0)
    
    # 3. OBJECTIVE FUNCTION
    # Minimize the maximum number of students in any classroom
    model.setObjective(M, GRB.MINIMIZE)
    
    # 4. CONSTRAINTS - CORRECT SYNTAX PATTERNS
    
    # Classroom capacity constraints
    model.addConstrs(
        (gp.quicksum(x[i, j] for i in students) <= max_students[j] for j in classrooms),
        name="classroom_capacity"
    )
    
    # Student assignment constraints
    model.addConstrs(
        (gp.quicksum(x[i, j] for j in classrooms) == 1 for i in students),
        name="student_assignment"
    )
    
    # Maximum students constraint
    model.addConstrs(
        (gp.quicksum(x[i, j] for i in students) <= M for j in classrooms),
        name="max_students"
    )
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in students:
            for j in classrooms:
                if x[i, j].x > 1e-6:
                    print(f"Student {i} assigned to classroom {j}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    
    return model

# Run the optimization
optimize_classroom_assignment()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 2.0
**Execution Time**: 0.16 seconds
**Reliability**: high
**Analysis**: Gurobipy found an optimal solution quickly with the lowest execution time, indicating efficient performance and reliability.

## 6. DOCplex Implementation

```python
# Complete DOCPLEX implementation

from docplex.mp.model import Model

def classroom_assignment_optimization():
    """Optimize student assignments to classrooms to minimize the maximum number of students in any classroom."""
    
    # 1. MODEL & DATA SETUP
    mdl = Model(name="classroom_assignment")
    
    # Data: Number of students and classrooms
    students = [1, 2, 3, 4, 5]
    classrooms = [101, 102, 103]
    
    # Classroom capacities
    classroom_capacities = {101: 25, 102: 30, 103: 35}
    
    # Validate data lengths
    assert len(classrooms) == len(classroom_capacities), "Mismatch in classroom data lengths"
    
    # Safe range for classrooms
    safe_classroom_range = range(len(classrooms))
    
    # 2. VARIABLES
    # Decision variables: x[i][j] = 1 if student i is assigned to classroom j, 0 otherwise
    x = {(i, j): mdl.binary_var(name=f"x_{i}_{j}") for i in students for j in classrooms}
    
    # Auxiliary variable for the maximum number of students in any classroom
    M = mdl.continuous_var(name="M", lb=0)
    
    # 3. OBJECTIVE FUNCTION
    # Minimize the maximum number of students in any classroom
    mdl.minimize(M)
    
    # 4. CONSTRAINTS
    
    # Classroom capacity constraints
    for j in classrooms:
        mdl.add_constraint(mdl.sum(x[i, j] for i in students) <= classroom_capacities[j], ctname=f"capacity_{j}")
    
    # Student assignment constraints
    for i in students:
        mdl.add_constraint(mdl.sum(x[i, j] for j in classrooms) == 1, ctname=f"assignment_{i}")
    
    # Maximum students constraint
    for j in classrooms:
        mdl.add_constraint(mdl.sum(x[i, j] for i in students) <= M, ctname=f"max_students_{j}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in students:
            for j in classrooms:
                if solution.get_value(x[i, j]) > 0.5:  # Check if student i is assigned to classroom j
                    print(f"Student {i} is assigned to classroom {j}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")
    
    return mdl

# Run the optimization
classroom_assignment_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 2.0
**Execution Time**: 1.70 seconds
**Reliability**: high
**Analysis**: DOCplex also found an optimal solution, though it took longer than Gurobipy. The results are consistent with other solvers, indicating reliability.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation

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

def classroom_assignment_optimization():
    """Optimize student assignments to classrooms to minimize the maximum number of students in any classroom."""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Example data based on the provided database schema and realistic data
    students = [1, 2, 3, 4, 5]
    classrooms = [101, 102, 103]
    max_students = {101: 25, 102: 30, 103: 35}
    
    # CRITICAL: Validate data lengths before indexing
    assert len(students) > 0, "No students available"
    assert len(classrooms) > 0, "No classrooms available"
    
    # 3. SETS
    model.I = pyo.Set(initialize=students)
    model.J = pyo.Set(initialize=classrooms)
    
    # 4. PARAMETERS
    model.max_students = pyo.Param(model.J, initialize=max_students)
    
    # 5. VARIABLES
    model.x = pyo.Var(model.I, model.J, within=pyo.Binary)
    model.M = pyo.Var(within=pyo.NonNegativeIntegers)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return model.M
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.minimize)
    
    # 7. CONSTRAINTS
    
    # Classroom Capacity Constraints
    def capacity_rule(model, j):
        return sum(model.x[i, j] for i in model.I) <= model.max_students[j]
    model.capacity_constraint = pyo.Constraint(model.J, rule=capacity_rule)
    
    # Student Assignment Constraints
    def assignment_rule(model, i):
        return sum(model.x[i, j] for j in model.J) == 1
    model.assignment_constraint = pyo.Constraint(model.I, rule=assignment_rule)
    
    # Maximum Students Constraint
    def max_students_rule(model, j):
        return sum(model.x[i, j] for i in model.I) <= model.M
    model.max_students_constraint = pyo.Constraint(model.J, rule=max_students_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
    # 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("\nAssignment of students to classrooms:")
        for i in model.I:
            for j in model.J:
                if pyo.value(model.x[i, j]) > 0.5:  # Binary variable check
                    print(f"Student {i} assigned to Classroom {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
classroom_assignment_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 2.0
**Execution Time**: 1.98 seconds
**Reliability**: high
**Analysis**: Pyomo achieved an optimal solution with the longest execution time among the solvers, but the results are consistent, indicating reliability.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | OPTIMAL | 2.00 | 0.16s | N/A | N/A |
| Docplex | OPTIMAL | 2.00 | 1.70s | N/A | N/A |
| Pyomo | OPTIMAL | 2.00 | 1.98s | N/A | N/A |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Majority Vote Optimal Value**: 2.0

### Final Recommendation
**Recommended Optimal Value**: 2.0
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy is preferred due to its fastest execution time and consistent results, making it efficient for large-scale problems.

### Business Interpretation
**Overall Strategy**: The optimal value of 2.0 indicates that the maximum number of students in any classroom is minimized to 2, ensuring balanced distribution across classrooms.
**Objective Value Meaning**: The optimal objective value of 2.0 means that no classroom will have more than 2 students, optimizing space utilization and maintaining manageable class sizes.
**Resource Allocation Summary**: Students are evenly distributed across available classrooms, ensuring no classroom exceeds its capacity.
**Implementation Recommendations**: Implement the solution by assigning students to classrooms as per the solver's output, ensuring compliance with capacity constraints.