## 4. Mathematical Optimization Formulation

#### Decision Variables
Let \( x_{s,c} \) be a binary decision variable where:
- \( x_{s,c} = 1 \) if student \( s \) is assigned to club \( c \),
- \( x_{s,c} = 0 \) otherwise.

#### Objective Function
Maximize the total engagement score:
\[
\text{Maximize} \quad \sum_{s} \sum_{c} \text{Engagement\_Scores.score}_{s,c} \times x_{s,c}
\]

#### Constraints
1. **Club Capacity Constraint**: The number of students assigned to each club \( c \) must not exceed its capacity:
\[
\sum_{s} x_{s,c} \leq \text{Club\_Capacities.capacity}_{c} \quad \forall c
\]

2. **Student Assignment Constraint**: Each student \( s \) can be assigned to at most one club:
\[
\sum_{c} x_{s,c} \leq 1 \quad \forall s
\]

3. **Binary Constraint**: The decision variables must be binary:
\[
x_{s,c} \in \{0, 1\} \quad \forall s, c
\]

#### Data Source Verification
- **Engagement\_Scores.score**: The coefficient for the objective function comes from the `Engagement_Scores.score` column.
- **Club\_Capacities.capacity**: The right-hand side of the club capacity constraint comes from the `Club_Capacities.capacity` column.

### Complete Numerical Model
Using the provided data, the numerical model is as follows:

#### Decision Variables
\( x_{1,101}, x_{2,102}, x_{3,103} \)

#### Objective Function
\[
\text{Maximize} \quad 0.6 \times x_{1,101} + 0.8 \times x_{2,102} + 0.4 \times x_{3,103}
\]

#### Constraints
1. **Club Capacity Constraints**:
\[
x_{1,101} \leq 20
\]
\[
x_{2,102} \leq 25
\]
\[
x_{3,103} \leq 15
\]

2. **Student Assignment Constraints**:
\[
x_{1,101} \leq 1
\]
\[
x_{2,102} \leq 1
\]
\[
x_{3,103} \leq 1
\]

3. **Binary Constraints**:
\[
x_{1,101}, x_{2,102}, x_{3,103} \in \{0, 1\}
\]

This is a complete, immediately solvable Mixed-Integer Linear Programming (MIP) model.

## 5. Gurobipy Implementation

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

import gurobipy as gp
from gurobipy import GRB

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

    # Data from the problem
    students = [1, 2, 3]
    clubs = [101, 102, 103]
    
    engagement_scores = {
        (1, 101): 0.6,
        (2, 102): 0.8,
        (3, 103): 0.4
    }
    
    club_capacities = {
        101: 20,
        102: 25,
        103: 15
    }

    # CRITICAL: Validate array lengths before loops
    assert len(engagement_scores) == len(students) * len(clubs), "Engagement scores length mismatch"
    assert len(club_capacities) == len(clubs), "Club capacities length mismatch"

    # 2. VARIABLES
    x = model.addVars(students, clubs, vtype=GRB.BINARY, name="x")

    # 3. OBJECTIVE FUNCTION
    model.setObjective(
        gp.quicksum(engagement_scores[s, c] * x[s, c] for s in students for c in clubs),
        GRB.MAXIMIZE
    )

    # 4. CONSTRAINTS

    # Club Capacity Constraint
    for c in clubs:
        model.addConstr(
            gp.quicksum(x[s, c] for s in students) <= club_capacities[c],
            name=f"club_capacity_{c}"
        )

    # Student Assignment Constraint
    for s in students:
        model.addConstr(
            gp.quicksum(x[s, c] for c in clubs) <= 1,
            name=f"student_assignment_{s}"
        )

    # 5. SOLVING & RESULTS
    model.optimize()

    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for s in students:
            for c in clubs:
                if x[s, c].x > 0.5:
                    print(f"Student {s} is assigned to Club {c}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")

    return model

# Execute the optimization
optimize_club_assignment()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmpfr5_v0yn.py", line 72, in <module>
    optimize_club_assignment()
  File "/tmp/tmpfr5_v0yn.py", line 27, in optimize_club_assignment
    assert len(engagement_scores) == len(students) * len(clubs), "Engagement scores length mismatch"
AssertionError: Engagement scores length mismatch

**Analysis**: Gurobipy encountered an error due to a mismatch in the length of engagement scores. This suggests a data preparation issue rather than a solver limitation.

## 6. DOCplex Implementation

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

from docplex.mp.model import Model

def optimize_club_assignment():
    # 1. MODEL & DATA SETUP
    mdl = Model(name="club_assignment")
    
    # Data from the database
    engagement_scores = [
        (1, 101, 0.6),
        (2, 102, 0.8),
        (3, 103, 0.4)
    ]
    
    club_capacities = [
        (101, 20),
        (102, 25),
        (103, 15)
    ]
    
    # Extract unique students and clubs
    students = list(set(score[0] for score in engagement_scores))
    clubs = list(set(capacity[0] for capacity in club_capacities))
    
    # Create dictionaries for engagement scores and capacities
    engagement_dict = {(s, c): score for s, c, score in engagement_scores}
    capacity_dict = {c: cap for c, cap in club_capacities}
    
    # CRITICAL: Validate array lengths
    assert len(engagement_scores) == len(students) * len(clubs), "Engagement scores length mismatch"
    
    # 2. VARIABLES
    x = mdl.binary_var_dict(keys=[(s, c) for s in students for c in clubs], name="x")
    
    # 3. OBJECTIVE FUNCTION
    objective = mdl.sum(engagement_dict[(s, c)] * x[(s, c)] for s in students for c in clubs)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS
    
    # Club Capacity Constraint
    for c in clubs:
        mdl.add_constraint(mdl.sum(x[(s, c)] for s in students) <= capacity_dict[c], ctname=f"club_capacity_{c}")
    
    # Student Assignment Constraint
    for s in students:
        mdl.add_constraint(mdl.sum(x[(s, c)] for c in clubs) <= 1, ctname=f"student_assignment_{s}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for s in students:
            for c in clubs:
                if solution.get_value(x[(s, c)]) > 0.5:
                    print(f"Student {s} assigned to Club {c}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")

# Execute the optimization
optimize_club_assignment()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmpv33fx7al.py", line 64, in <module>
    optimize_club_assignment()
  File "/tmp/tmpv33fx7al.py", line 31, in optimize_club_assignment
    assert len(engagement_scores) == len(students) * len(clubs), "Engagement scores length mismatch"
AssertionError: Engagement scores length mismatch

**Analysis**: DOCplex also failed with the same error, indicating a systemic issue with the input data structure.

## 7. Pyomo Implementation

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

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

def optimize_club_assignment():
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Example data from the problem statement
    students = [1, 2, 3]
    clubs = [101, 102, 103]
    
    engagement_scores = {
        (1, 101): 0.6,
        (2, 102): 0.8,
        (3, 103): 0.4
    }
    
    club_capacities = {
        101: 20,
        102: 25,
        103: 15
    }
    
    # CRITICAL: Validate array lengths before indexing
    assert len(engagement_scores) == len(students) * len(clubs), "Engagement scores length mismatch"
    
    # 3. SETS
    model.students = pyo.Set(initialize=students)
    model.clubs = pyo.Set(initialize=clubs)
    
    # 4. PARAMETERS
    model.engagement_scores = pyo.Param(model.students, model.clubs, initialize=engagement_scores)
    model.club_capacities = pyo.Param(model.clubs, initialize=club_capacities)
    
    # 5. VARIABLES
    model.x = pyo.Var(model.students, model.clubs, within=pyo.Binary)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.engagement_scores[s, c] * model.x[s, c] for s in model.students for c in model.clubs)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)
    
    # 7. CONSTRAINTS
    # Club Capacity Constraint
    def club_capacity_rule(model, c):
        return sum(model.x[s, c] for s in model.students) <= model.club_capacities[c]
    model.club_capacity_constraint = pyo.Constraint(model.clubs, rule=club_capacity_rule)
    
    # Student Assignment Constraint
    def student_assignment_rule(model, s):
        return sum(model.x[s, c] for c in model.clubs) <= 1
    model.student_assignment_constraint = pyo.Constraint(model.students, rule=student_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)  # 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 s in model.students:
            for c in model.clubs:
                if pyo.value(model.x[s, c]) > 0:
                    print(f"Student {s} is assigned to Club {c}")
        
    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

# Execute the optimization
optimize_club_assignment()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmppkgkwvbk.py", line 90, in <module>
    optimize_club_assignment()
  File "/tmp/tmppkgkwvbk.py", line 28, in optimize_club_assignment
    assert len(engagement_scores) == len(students) * len(clubs), "Engagement scores length mismatch"
AssertionError: Engagement scores length mismatch

**Analysis**: Pyomo's failure mirrors the other solvers, reinforcing the conclusion that the problem lies in the data preparation phase.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | ERROR | N/A | 0.23s | N/A | 4 |
| Docplex | ERROR | N/A | 1.02s | N/A | 4 |
| Pyomo | ERROR | N/A | 1.01s | N/A | 4 |

### Solver Consistency Analysis
**Result**: Solvers produced inconsistent results
**Inconsistent Solvers**: gurobipy, docplex, pyomo
**Potential Issues**:
- Data preparation error: Engagement scores length mismatch
**Solver Retry Summary**: gurobipy: 4 attempts, docplex: 4 attempts, pyomo: 4 attempts

### Final Recommendation
**Confidence Level**: LOW
**Preferred Solver(s)**: none
**Reasoning**: All solvers failed due to the same data preparation issue, indicating that the problem must be addressed before any solver can be effectively used.

### Business Interpretation
**Overall Strategy**: The optimization process cannot proceed due to a critical data preparation error, preventing any meaningful business decision.
**Objective Value Meaning**: No objective value could be determined due to the error in data preparation.
**Resource Allocation Summary**: Resource allocation cannot be determined until the data preparation issue is resolved.
**Implementation Recommendations**: Review and correct the data preparation process, ensuring that the engagement scores are correctly formatted and match the expected dimensions before reattempting the optimization.