## 4. Mathematical Optimization Formulation

#### Decision Variables
- \( x_{ijk} \): Binary variable that equals 1 if wedding \( k \) is held at church \( i \) with male participant \( j \) and female participant \( k \), and 0 otherwise.

#### Objective Function
Maximize the total number of weddings held:
\[ \text{Maximize} \quad \sum_{i,j,k} x_{ijk} \]

#### Constraints
1. **Church Capacity Constraints**: Ensure that the number of weddings at each church does not exceed its capacity.
   \[
   \sum_{j,k} x_{ijk} \leq \text{capacity}_i \quad \forall i
   \]
   - Data Source Verification: `ChurchCapacity.capacity`

2. **Unique Male Participation**: Each male participant can be involved in at most one wedding.
   \[
   \sum_{i,k} x_{ijk} \leq 1 \quad \forall j \text{ where } \text{ParticipantConstraints.constraint_type} = \text{'male'}
   \]
   - Data Source Verification: `ParticipantConstraints.participant_id`

3. **Unique Female Participation**: Each female participant can be involved in at most one wedding.
   \[
   \sum_{i,j} x_{ijk} \leq 1 \quad \forall k \text{ where } \text{ParticipantConstraints.constraint_type} = \text{'female'}
   \]
   - Data Source Verification: `ParticipantConstraints.participant_id`

#### Data Source Verification
- The coefficients for the church capacity constraints are derived from the `ChurchCapacity.capacity` column.
- The constraints for unique male and female participation are based on the `ParticipantConstraints.participant_id` and `ParticipantConstraints.constraint_type` columns.

This formulation provides a complete and immediately solvable linear mathematical model, ensuring that all constraints and the objective function are linear and adhere to the problem's requirements.

## 5. Gurobipy Implementation

```python
# Complete GUROBIPY implementation

import gurobipy as gp
from gurobipy import GRB

def optimize_weddings():
    """Optimize the allocation of weddings to churches."""
    
    # 1. MODEL & DATA SETUP
    model = gp.Model("wedding_optimization")
    
    # Data: Church capacities and participant constraints
    church_ids = [1, 2, 3]
    male_ids = [101, 102, 103]
    female_ids = [201, 202, 203]
    capacities = {1: 100, 2: 150, 3: 200}
    
    # Validate data lengths
    assert len(church_ids) == len(capacities), "Church data length mismatch"
    assert len(male_ids) == len(female_ids), "Participant data length mismatch"
    
    # 2. VARIABLES
    # Decision variables: x[i,j,k] = 1 if wedding with male j and female k is held at church i
    x = model.addVars(church_ids, male_ids, female_ids, vtype=GRB.BINARY, name="x")
    
    # 3. OBJECTIVE FUNCTION
    # Maximize the total number of weddings
    model.setObjective(gp.quicksum(x[i, j, k] for i in church_ids for j in male_ids for k in female_ids), GRB.MAXIMIZE)
    
    # 4. CONSTRAINTS
    
    # Church capacity constraints
    model.addConstrs((gp.quicksum(x[i, j, k] for j in male_ids for k in female_ids) <= capacities[i] for i in church_ids), name="church_capacity")
    
    # Unique male participation
    model.addConstrs((gp.quicksum(x[i, j, k] for i in church_ids for k in female_ids) <= 1 for j in male_ids), name="unique_male")
    
    # Unique female participation
    model.addConstrs((gp.quicksum(x[i, j, k] for i in church_ids for j in male_ids) <= 1 for k in female_ids), name="unique_female")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in church_ids:
            for j in male_ids:
                for k in female_ids:
                    if x[i, j, k].x > 1e-6:
                        print(f"Wedding at Church {i} with Male {j} and Female {k}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    
    return model

# Run the optimization
optimize_weddings()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 3.0
**Execution Time**: 1.07 seconds
**Reliability**: high
**Analysis**: Gurobipy found an optimal solution quickly with a consistent optimal value of 3.0. It is known for its efficiency and reliability in solving linear optimization problems.

## 6. DOCplex Implementation

```python
# Complete DOCPLEX implementation

from docplex.mp.model import Model

def optimize_weddings():
    """Optimize the allocation of weddings to churches."""
    
    # 1. MODEL & DATA SETUP
    mdl = Model(name="wedding_optimization")
    
    # Data: Church capacities
    church_ids = [1, 2, 3]
    capacities = [100, 150, 200]
    
    # Data: Participants
    male_ids = [101, 102, 103]
    female_ids = [201, 202, 203]
    
    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(church_ids) == len(capacities), "Array length mismatch for church capacities"
    assert len(male_ids) == len(female_ids), "Array length mismatch for participants"
    
    # Safe indexing
    safe_church_range = range(len(church_ids))
    safe_participant_range = range(len(male_ids))
    
    # 2. VARIABLES
    # Decision variables: x[i][j][k] = 1 if wedding with male j and female k is held at church i
    x = {(i, j, k): mdl.binary_var(name=f"x_{i}_{j}_{k}") for i in safe_church_range for j in safe_participant_range for k in safe_participant_range}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize the total number of weddings
    objective = mdl.sum(x[i, j, k] for i in safe_church_range for j in safe_participant_range for k in safe_participant_range)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS
    
    # Church capacity constraints
    for i in safe_church_range:
        mdl.add_constraint(mdl.sum(x[i, j, k] for j in safe_participant_range for k in safe_participant_range) <= capacities[i], ctname=f"capacity_{i}")
    
    # Unique male participation
    for j in safe_participant_range:
        mdl.add_constraint(mdl.sum(x[i, j, k] for i in safe_church_range for k in safe_participant_range) <= 1, ctname=f"unique_male_{j}")
    
    # Unique female participation
    for k in safe_participant_range:
        mdl.add_constraint(mdl.sum(x[i, j, k] for i in safe_church_range for j in safe_participant_range) <= 1, ctname=f"unique_female_{k}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in safe_church_range:
            for j in safe_participant_range:
                for k in safe_participant_range:
                    value = solution.get_value(x[i, j, k])
                    if value > 1e-6:
                        print(f"x[{i}][{j}][{k}] = {value:.3f}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")
    
    return mdl

# Run the optimization
optimize_weddings()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 3.0
**Execution Time**: 6.71 seconds
**Reliability**: high
**Analysis**: DOCplex also found an optimal solution with the same optimal value of 3.0, albeit with a longer execution time. It is a robust solver for linear problems, especially within the IBM ecosystem.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation

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

def wedding_optimization():
    """Optimize the allocation of weddings to churches"""

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

    # 2. DATA SETUP
    # Example data based on the provided schema and realistic data
    church_ids = [1, 2, 3]
    male_ids = [101, 102, 103]
    female_ids = [201, 202, 203]
    capacities = {1: 100, 2: 150, 3: 200}

    # 3. SETS
    model.I = pyo.Set(initialize=church_ids)
    model.J = pyo.Set(initialize=male_ids)
    model.K = pyo.Set(initialize=female_ids)

    # 4. PARAMETERS
    model.capacity = pyo.Param(model.I, initialize=capacities)

    # 5. VARIABLES
    model.x = pyo.Var(model.I, model.J, model.K, within=pyo.Binary)

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

    # 7. CONSTRAINTS

    # Church Capacity Constraints
    def church_capacity_rule(model, i):
        return sum(model.x[i, j, k] for j in model.J for k in model.K) <= model.capacity[i]
    model.church_capacity_constraint = pyo.Constraint(model.I, rule=church_capacity_rule)

    # Unique Male Participation
    def unique_male_rule(model, j):
        return sum(model.x[i, j, k] for i in model.I for k in model.K) <= 1
    model.unique_male_constraint = pyo.Constraint(model.J, rule=unique_male_rule)

    # Unique Female Participation
    def unique_female_rule(model, k):
        return sum(model.x[i, j, k] for i in model.I for j in model.J) <= 1
    model.unique_female_constraint = pyo.Constraint(model.K, rule=unique_female_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
    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:
            for j in model.J:
                for k in model.K:
                    x_val = pyo.value(model.x[i, j, k])
                    if x_val > 1e-6:  # Only print non-zero values
                        print(f"x[{i},{j},{k}] = {int(x_val)}")

    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
wedding_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 3.0
**Execution Time**: 2.65 seconds
**Reliability**: high
**Analysis**: Pyomo achieved an optimal solution with the same optimal value of 3.0. It is a versatile tool that integrates well with various solvers, providing flexibility in modeling.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | OPTIMAL | 3.00 | 1.07s | N/A | N/A |
| Docplex | OPTIMAL | 3.00 | 6.71s | N/A | N/A |
| Pyomo | OPTIMAL | 3.00 | 2.65s | N/A | N/A |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Majority Vote Optimal Value**: 3.0

### Final Recommendation
**Recommended Optimal Value**: 3.0
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy is recommended due to its faster execution time and high reliability, making it suitable for operational environments where efficiency is crucial.

### Business Interpretation
**Overall Strategy**: The optimal solution indicates that a maximum of 3 weddings can be held given the constraints. This maximizes the use of available resources while adhering to participant and church capacity constraints.
**Objective Value Meaning**: The optimal objective value of 3.0 means that the maximum number of weddings that can be scheduled without violating any constraints is 3.
**Resource Allocation Summary**: Resources should be allocated to ensure that 3 weddings are scheduled, with each church and participant being utilized according to the constraints.
**Implementation Recommendations**: Ensure that the scheduling system is updated to reflect the optimal assignments, and communicate with all participants and churches to confirm the arrangements.