## 4. Mathematical Optimization Formulation

#### Decision Variables
- \( x_{ij} \): Binary decision variable indicating whether member \( i \) is assigned to event \( j \).
  - \( x_{ij} = 1 \) if member \( i \) is assigned to event \( j \), 0 otherwise.

#### Objective Function
Maximize the total effectiveness of the party events:
\[ \text{Maximize} \quad \sum_{i,j} \text{efficiency\_score}_{ij} \times x_{ij} \]

Data Source Verification:
- Coefficients \(\text{efficiency\_score}_{ij}\) are sourced from `effectiveness_scores.efficiency_score`.

#### Constraints
1. **Event Capacity Constraints**: Ensure that the number of members assigned to each event does not exceed the event's capacity.
   \[
   \sum_{i} x_{ij} \leq \text{max\_members\_per\_event}_j \quad \forall j
   \]

   Data Source Verification:
   - \(\text{max\_members\_per\_event}_j\) is sourced from `party_events.max_members_per_event`.

2. **Member Availability Constraints**: Members can only be assigned to events if they are available.
   \[
   x_{ij} \leq \text{is\_available}_{ij} \quad \forall i, j
   \]

   Data Source Verification:
   - \(\text{is\_available}_{ij}\) is sourced from `member_availability.is_available`.

3. **Binary Constraints**: Ensure that decision variables are binary.
   \[
   x_{ij} \in \{0, 1\} \quad \forall i, j
   \]

This formulation provides a complete and immediately solvable linear mathematical model, ensuring that all coefficients and constraints are derived from the provided data.

## 5. Gurobipy Implementation

```python
# Complete GUROBIPY implementation - Retry Attempt 2

import gurobipy as gp
from gurobipy import GRB

def optimize_party_people():
    # 1. MODEL & DATA SETUP
    model = gp.Model("party_people")
    
    # Sample data based on the provided schema and realistic data
    member_ids = [1, 2, 3]
    event_ids = [101, 102, 103]
    
    # Effectiveness scores
    effectiveness_scores = {
        (1, 101): 0.85,
        (2, 102): 0.95,
        (3, 103): 0.75
    }
    
    # Member availability
    member_availability = {
        (1, 101): True,
        (2, 102): False,
        (3, 103): True
    }
    
    # Event capacities
    max_members_per_event = {
        101: 50,
        102: 100,
        103: 150
    }
    
    # CRITICAL: Validate array lengths before loops
    assert len(effectiveness_scores) == len(member_availability), "Array length mismatch"
    
    # 2. VARIABLES
    # Decision variables: x[i, j] = 1 if member i is assigned to event j, 0 otherwise
    x = model.addVars(member_ids, event_ids, vtype=GRB.BINARY, name="x")
    
    # 3. OBJECTIVE FUNCTION
    # Maximize total effectiveness
    model.setObjective(
        gp.quicksum(effectiveness_scores[i, j] * x[i, j] for i in member_ids for j in event_ids if (i, j) in effectiveness_scores),
        GRB.MAXIMIZE
    )
    
    # 4. CONSTRAINTS
    # Event capacity constraints
    model.addConstrs(
        (gp.quicksum(x[i, j] for i in member_ids) <= max_members_per_event[j] for j in event_ids),
        name="event_capacity"
    )
    
    # Member availability constraints
    model.addConstrs(
        (x[i, j] <= member_availability[i, j] for i in member_ids for j in event_ids if (i, j) in member_availability),
        name="member_availability"
    )
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in member_ids:
            for j in event_ids:
                if x[i, j].x > 0.5:  # Binary variable, so check if it's 1
                    print(f"Member {i} assigned to Event {j}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")

# Run the optimization
optimize_party_people()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 1.6
**Execution Time**: 0.36 seconds
**Reliability**: high
**Retry Attempt**: 2
**Analysis**: Gurobipy successfully found an optimal solution with a value of 1.6 in a short execution time. The solver's reliability is high due to its consistent performance and optimal status.

## 6. DOCplex Implementation

```python
# Complete DOCPLEX implementation - Retry Attempt 2

from docplex.mp.model import Model

def optimize_party_people():
    # 1. MODEL & DATA SETUP
    mdl = Model(name="party_people")
    
    # Data from the problem statement
    member_ids = [1, 2, 3]
    event_ids = [101, 102, 103]
    
    # Effectiveness scores
    efficiency_scores = {
        (1, 101): 0.85,
        (2, 102): 0.95,
        (3, 103): 0.75
    }
    
    # Member availability
    is_available = {
        (1, 101): True,
        (2, 102): False,
        (3, 103): True
    }
    
    # Event capacities
    max_members_per_event = {
        101: 50,
        102: 100,
        103: 150
    }
    
    # 2. VARIABLES
    x = {(i, j): mdl.binary_var(name=f"x_{i}_{j}") for i in member_ids for j in event_ids}
    
    # 3. OBJECTIVE FUNCTION
    objective = mdl.sum(efficiency_scores.get((i, j), 0) * x[(i, j)] for i in member_ids for j in event_ids)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS
    # Event Capacity Constraints
    for j in event_ids:
        mdl.add_constraint(mdl.sum(x[(i, j)] for i in member_ids) <= max_members_per_event[j], ctname=f"capacity_{j}")
    
    # Member Availability Constraints
    for i in member_ids:
        for j in event_ids:
            if not is_available.get((i, j), False):
                mdl.add_constraint(x[(i, j)] == 0, ctname=f"availability_{i}_{j}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in member_ids:
            for j in event_ids:
                if solution.get_value(x[(i, j)]) > 0.5:  # Check if assigned
                    print(f"Member {i} assigned to Event {j}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")

# Run the optimization
optimize_party_people()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 1.6
**Execution Time**: 8.21 seconds
**Reliability**: high
**Retry Attempt**: 2
**Analysis**: DOCplex also found an optimal solution with the same value of 1.6, albeit with a longer execution time compared to Gurobipy. The reliability is high as it reached the same optimal value.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation - Retry Attempt 2

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

def party_people_optimization():
    """Optimize member assignments to maximize event effectiveness."""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Example data based on the provided schema and realistic data
    members = [1, 2, 3]
    events = [101, 102, 103]
    effectiveness_scores = {
        (1, 101): 0.85,
        (2, 102): 0.95,
        (3, 103): 0.75
    }
    member_availability = {
        (1, 101): True,
        (2, 102): False,
        (3, 103): True
    }
    max_members_per_event = {
        101: 50,
        102: 100,
        103: 150
    }
    
    # 3. SETS
    model.M = pyo.Set(initialize=members)
    model.E = pyo.Set(initialize=events)
    
    # 4. PARAMETERS
    model.efficiency_score = pyo.Param(model.M, model.E, initialize=effectiveness_scores, default=0)
    model.is_available = pyo.Param(model.M, model.E, initialize=member_availability, within=pyo.Boolean)
    model.max_members_per_event = pyo.Param(model.E, initialize=max_members_per_event)
    
    # 5. VARIABLES
    model.x = pyo.Var(model.M, model.E, within=pyo.Binary)
    
    # 6. OBJECTIVE FUNCTION
    def objective_rule(model):
        return sum(model.efficiency_score[m, e] * model.x[m, e] for m in model.M for e in model.E)
    model.objective = pyo.Objective(rule=objective_rule, sense=pyo.maximize)
    
    # 7. CONSTRAINTS
    # Event Capacity Constraints
    def event_capacity_rule(model, e):
        return sum(model.x[m, e] for m in model.M) <= model.max_members_per_event[e]
    model.event_capacity_constraint = pyo.Constraint(model.E, rule=event_capacity_rule)
    
    # Member Availability Constraints
    def member_availability_rule(model, m, e):
        return model.x[m, e] <= model.is_available[m, e]
    model.member_availability_constraint = pyo.Constraint(model.M, model.E, rule=member_availability_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):.2f}")
        
        # Extract variable values
        print("\nAssignment decisions:")
        for m in model.M:
            for e in model.E:
                if pyo.value(model.x[m, e]) > 0.5:  # Only print assigned members
                    print(f"Member {m} assigned to Event {e}")
        
    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}")

# Execute the optimization
party_people_optimization()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmpvdmz0x23.py", line 86, in <module>
    party_people_optimization()
  File "/tmp/tmpvdmz0x23.py", line 58, in party_people_optimization
    model.member_availability_constraint = pyo.Constraint(model.M, model.E, rule=member_availability_rule)
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/block.py", line 571, in __setattr__
    self.add_component(name, val)
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/block.py", line 1101, in add_component
    val.construct(data)
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/constraint.py", line 722, in construct
    self._setitem_when_not_present(index, rule(block, index))
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/initializer.py", line 349, in __call__
    return self._fcn(parent, *idx)
  File "/tmp/tmpvdmz0x23.py", line 57, in member_availability_rule
    return model.x[m, e] <= model.is_available[m, e]
TypeError: '<=' not supported between instances of 'VarData' and 'bool'

**Analysis**: Pyomo encountered an error related to constraint definition, specifically a TypeError in handling binary constraints. This indicates a potential issue in the model setup or data handling, leading to low reliability.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | OPTIMAL | 1.60 | 0.36s | N/A | 2 |
| Docplex | OPTIMAL | 1.60 | 8.21s | N/A | 2 |
| Pyomo | ERROR | N/A | 5.79s | N/A | 2 |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex
**Majority Vote Optimal Value**: 1.6
**Solver Retry Summary**: gurobipy: 2 attempts, docplex: 2 attempts, pyomo: 2 attempts

### Final Recommendation
**Recommended Optimal Value**: 1.6
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy is preferred due to its faster execution time and high reliability in achieving the optimal solution.

### Business Interpretation
**Overall Strategy**: The optimal value of 1.6 indicates the maximum achievable effectiveness score for the event assignments given the constraints.
**Objective Value Meaning**: The optimal objective value of 1.6 represents the highest total effectiveness score achievable by optimally assigning members to events within the given constraints.
**Resource Allocation Summary**: Members should be allocated to events where their presence maximizes the effectiveness score, respecting capacity and availability constraints.
**Implementation Recommendations**: Ensure data integrity for availability and capacity constraints, and use Gurobipy for solving similar optimization problems due to its efficiency and reliability.