## 4. Mathematical Optimization Formulation

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

#### Objective Function
Maximize the total satisfaction score:
\[
\text{Maximize } \sum_{(i,j) \in \text{SatisfactionScores}} \text{satisfaction\_score}_{i,j} \times x_{i,j}
\]
Where \(\text{satisfaction\_score}_{i,j}\) is the satisfaction score of participant \( i \) attending event \( j \).

#### Constraints
1. **Capacity Constraints**: Ensure that the number of participants assigned to each event does not exceed its capacity.
   \[
   \sum_{i} x_{i,j} \leq \text{capacity\_limit}_j \quad \forall j \in \text{EventCapacities}
   \]
   Where \(\text{capacity\_limit}_j\) is the maximum number of participants allowed for event \( j \).

2. **Binary Assignment Constraints**: Each participant can be assigned to an event or not, represented by binary variables.
   \[
   x_{i,j} \in \{0, 1\} \quad \forall (i,j) \in \text{SatisfactionScores}
   \]

Data Source Verification:
- The coefficients \(\text{satisfaction\_score}_{i,j}\) are sourced from the `SatisfactionScores` table, specifically from the `satisfaction_score` column.
- The capacity limits \(\text{capacity\_limit}_j\) are sourced from the `EventCapacities` table, specifically from the `capacity_limit` column.

## 5. Gurobipy Implementation

```python
# Complete GUROBIPY implementation

import gurobipy as gp
from gurobipy import GRB

def optimize_event_assignments():
    """Optimize participant assignments to events to maximize satisfaction scores."""
    
    # 1. MODEL & DATA SETUP
    model = gp.Model("local_govt_in_alabama")
    
    # Sample data based on the provided schema and data
    satisfaction_scores = {
        (1, 101): 0.7,
        (1, 102): 0.9,
        (2, 101): 0.6,
        (2, 103): 0.8,
        (3, 102): 0.5
    }
    
    event_capacities = {
        101: 50,
        102: 100,
        103: 150
    }
    
    participants = {1, 2, 3}
    events = {101, 102, 103}
    
    # CRITICAL: Validate array lengths before loops
    assert all(isinstance(satisfaction_scores[(i, j)], float) for i in participants for j in events if (i, j) in satisfaction_scores), "Satisfaction scores must be floats"
    assert all(isinstance(event_capacities[j], int) for j in events), "Event capacities must be integers"
    
    # 2. VARIABLES
    # Decision variables: x[i, j] = 1 if participant i is assigned to event j, 0 otherwise
    x = model.addVars(participants, events, vtype=GRB.BINARY, name="x")
    
    # 3. OBJECTIVE FUNCTION
    # Maximize total satisfaction score
    model.setObjective(gp.quicksum(satisfaction_scores[i, j] * x[i, j] for i, j in satisfaction_scores), GRB.MAXIMIZE)
    
    # 4. CONSTRAINTS - CORRECT SYNTAX PATTERNS
    
    # Capacity constraints: Ensure no event exceeds its capacity
    model.addConstrs((gp.quicksum(x[i, j] for i in participants) <= event_capacities[j] for j in events), name="capacity_limit")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in participants:
            for j in events:
                if x[i, j].x > 1e-6:
                    print(f"Participant {i} assigned to Event {j} with satisfaction score {satisfaction_scores[i, 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_event_assignments()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 3.5
**Execution Time**: 0.96 seconds
**Reliability**: high
**Analysis**: Gurobipy found an optimal solution quickly with the shortest execution time, indicating efficient performance and reliability.

## 6. DOCplex Implementation

```python
# Complete DOCPLEX implementation

from docplex.mp.model import Model

def optimize_event_participation():
    """Optimize participant assignments to events to maximize satisfaction scores."""
    
    # 1. MODEL & DATA SETUP
    mdl = Model(name="local_govt_in_alabama")
    
    # Data from the database (mocked for this example)
    satisfaction_scores = [
        (1, 101, 0.7),
        (1, 102, 0.9),
        (2, 101, 0.6),
        (2, 103, 0.8),
        (3, 102, 0.5)
    ]
    
    event_capacities = {
        101: 50,
        102: 100,
        103: 150
    }
    
    # Extract unique participants and events
    participants = set(participant_id for participant_id, _, _ in satisfaction_scores)
    events = set(event_id for _, event_id, _ in satisfaction_scores)
    
    # CRITICAL: Validate data consistency
    assert all(event_id in event_capacities for _, event_id, _ in satisfaction_scores), "Event ID mismatch"
    
    # 2. VARIABLES
    # Binary decision variables for participant-event assignments
    x = {(i, j): mdl.binary_var(name=f"x_{i}_{j}") for i, j, _ in satisfaction_scores}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize total satisfaction score
    objective = mdl.sum(score * x[i, j] for i, j, score in satisfaction_scores)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS
    
    # Capacity constraints for each event
    for j in events:
        mdl.add_constraint(mdl.sum(x[i, j] for i, _, _ in satisfaction_scores if j == _) <= event_capacities[j], ctname=f"capacity_{j}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i, j, _ in satisfaction_scores:
            if solution.get_value(x[i, j]) > 0.5:  # Binary variable threshold
                print(f"Participant {i} assigned to Event {j}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")

    return mdl

# Run the optimization
optimize_event_participation()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 3.5
**Execution Time**: 8.65 seconds
**Reliability**: high
**Analysis**: DOCplex also found an optimal solution, but with a longer execution time compared to Gurobipy, suggesting it is reliable but less efficient.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation

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

def local_govt_in_alabama_optimization():
    """Optimize participant assignments to maximize satisfaction scores."""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Sample data based on the provided database schema and current stored values
    participants = [1, 2, 3]
    events = [101, 102, 103]
    
    satisfaction_scores = {
        (1, 101): 0.7,
        (1, 102): 0.9,
        (2, 101): 0.6,
        (2, 103): 0.8,
        (3, 102): 0.5
    }
    
    capacity_limits = {
        101: 50,
        102: 100,
        103: 150
    }
    
    # 3. SETS
    model.Participants = pyo.Set(initialize=participants)
    model.Events = pyo.Set(initialize=events)
    
    # 4. PARAMETERS
    model.satisfaction_score = pyo.Param(model.Participants, model.Events, initialize=satisfaction_scores, default=0)
    model.capacity_limit = pyo.Param(model.Events, initialize=capacity_limits)
    
    # 5. VARIABLES
    model.x = pyo.Var(model.Participants, model.Events, within=pyo.Binary)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.satisfaction_score[i, j] * model.x[i, j] for i in model.Participants for j in model.Events)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)
    
    # 7. CONSTRAINTS
    
    # Capacity constraints
    def capacity_rule(model, j):
        return sum(model.x[i, j] for i in model.Participants) <= model.capacity_limit[j]
    model.capacity_constraint = pyo.Constraint(model.Events, rule=capacity_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("\nAssignment of participants to events:")
        for i in model.Participants:
            for j in model.Events:
                x_val = pyo.value(model.x[i, j])
                if x_val > 0.5:  # Only print assignments
                    print(f"Participant {i} assigned to Event {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
local_govt_in_alabama_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 3.5
**Execution Time**: 4.91 seconds
**Reliability**: high
**Analysis**: Pyomo achieved an optimal solution with a moderate execution time, indicating a balance between reliability and efficiency.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | OPTIMAL | 3.50 | 0.96s | N/A | N/A |
| Docplex | OPTIMAL | 3.50 | 8.65s | N/A | N/A |
| Pyomo | OPTIMAL | 3.50 | 4.91s | N/A | N/A |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Majority Vote Optimal Value**: 3.5

### Final Recommendation
**Recommended Optimal Value**: 3.5
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy is recommended due to its high reliability and the shortest execution time, making it the most efficient choice for this problem.

### Business Interpretation
**Overall Strategy**: The optimal satisfaction score of 3.5 indicates the best possible allocation of participants to events, maximizing overall satisfaction.
**Objective Value Meaning**: The optimal objective value of 3.5 represents the maximum achievable satisfaction score by optimally assigning participants to events.
**Resource Allocation Summary**: Participants should be allocated to events in a manner that maximizes the total satisfaction score while respecting event capacity constraints.
**Implementation Recommendations**: Implement the solution by assigning participants to events as per the optimal decision variables, ensuring adherence to capacity constraints.