## 4. Mathematical Optimization Formulation

#### Decision Variables
Let \( x_{i,j} \) be a binary decision variable where:
- \( x_{i,j} = 1 \) if member \( j \) participates in round \( i \)
- \( x_{i,j} = 0 \) otherwise

#### Objective Function
Maximize the total rank points earned by the college's members:
\[
\text{Maximize } Z = \sum_{i,j} \text{rank\_points}_{i,j} \times x_{i,j}
\]
where \(\text{rank\_points}_{i,j}\) is the rank points earned by member \( j \) in round \( i \).

Data Source Verification:
- \(\text{rank\_points}_{i,j}\) comes from `ObjectiveCoefficients.rank_points` for each combination of `round_id` and `member_id`.

#### Constraints
1. Each member can participate in a maximum of three rounds:
   \[
   \sum_{i} x_{i,j} \leq 3 \quad \forall j
   \]

2. Participation decision is binary:
   \[
   x_{i,j} \in \{0, 1\} \quad \forall i, j
   \]

Data Source Verification:
- The constraint on the number of rounds a member can participate in is derived from the business rule that each member can participate in up to three rounds.
- The binary nature of \( x_{i,j} \) is inherent in the problem description and aligns with `DecisionVariables.participation`.

This formulation provides a complete linear optimization model that can be solved using standard linear or mixed-integer programming solvers. The objective function and constraints are directly derived from the provided data and business rules, ensuring that the model is both accurate and actionable.

## 5. Gurobipy Implementation

```python
# Complete GUROBIPY implementation

import gurobipy as gp
from gurobipy import GRB

def decoration_competition_optimization():
    """Optimize member participation in decoration competition rounds to maximize rank points."""
    
    # 1. MODEL & DATA SETUP
    model = gp.Model("decoration_competition")
    
    # Sample data based on the provided schema and realistic data
    rounds = [1, 2, 3]
    members = [101, 102, 103]
    
    # Objective coefficients (rank points) from ObjectiveCoefficients table
    rank_points = {
        (1, 101): 25.0,
        (2, 102): 30.0,
        (3, 103): 0.0,
        (1, 102): 20.0,
        (2, 101): 0.0
    }
    
    # Validate data lengths
    assert all(isinstance(rank_points[key], float) for key in rank_points), "Rank points data type mismatch"
    
    # 2. VARIABLES
    # Binary decision variables for participation
    x = model.addVars(rounds, members, vtype=GRB.BINARY, name="participation")
    
    # 3. OBJECTIVE FUNCTION
    # Maximize total rank points
    model.setObjective(gp.quicksum(rank_points.get((i, j), 0) * x[i, j] for i in rounds for j in members), GRB.MAXIMIZE)
    
    # 4. CONSTRAINTS
    
    # Constraint: Each member can participate in a maximum of three rounds
    model.addConstrs((gp.quicksum(x[i, j] for i in rounds) <= 3 for j in members), name="max_rounds_per_member")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in rounds:
            for j in members:
                if x[i, j].x > 1e-6:
                    print(f"Member {j} participates in round {i}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    
    return model

# Run the optimization
decoration_competition_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 75.0
**Execution Time**: 0.53 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 decoration_competition_optimization():
    """Optimization for decoration competition to maximize rank points"""
    
    # 1. MODEL & DATA SETUP
    mdl = Model(name="decoration_competition")
    
    # Data from the problem context
    rounds = [1, 2, 3]
    members = [101, 102, 103]
    
    # Rank points data
    rank_points_data = {
        (1, 101): 25.0,
        (2, 102): 30.0,
        (3, 103): 0.0,
        (1, 102): 20.0,
        (2, 101): 0.0
    }
    
    # Validate data consistency
    assert all(isinstance(rank_points_data[key], float) for key in rank_points_data), "Rank points data must be float"
    
    # Safe range for indexing
    safe_rounds = range(len(rounds))
    safe_members = range(len(members))
    
    # 2. VARIABLES
    # Binary decision variables for participation
    x = {(i, j): mdl.binary_var(name=f"x_{i}_{j}") for i in rounds for j in members}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize total rank points
    objective = mdl.sum(rank_points_data.get((i, j), 0.0) * x[i, j] for i in rounds for j in members)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS
    # Each member can participate in a maximum of three rounds
    for j in members:
        mdl.add_constraint(mdl.sum(x[i, j] for i in rounds) <= 3, ctname=f"max_rounds_{j}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in rounds:
            for j in members:
                value = solution.get_value(x[i, j])
                if value > 1e-6:
                    print(f"Member {j} participates in round {i}: {value:.0f}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")

    return mdl

# Run the optimization
decoration_competition_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 75.0
**Execution Time**: 2.09 seconds
**Reliability**: high
**Analysis**: DOCplex also found an optimal solution, albeit with a longer execution time compared to Gurobipy, but still reliable.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation

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

def decoration_competition_optimization():
    """Optimization for decoration competition"""

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

    # 2. DATA SETUP
    # Example data based on the provided schema and realistic data
    rounds = [1, 2, 3]
    members = [101, 102, 103]
    
    # Rank points data from ObjectiveCoefficients table
    rank_points_data = {
        (1, 101): 25.0,
        (2, 102): 30.0,
        (3, 103): 0.0,
        (1, 102): 20.0,
        (2, 101): 0.0
    }
    
    # 3. SETS
    model.Rounds = pyo.Set(initialize=rounds)
    model.Members = pyo.Set(initialize=members)

    # 4. PARAMETERS
    model.rank_points = pyo.Param(model.Rounds, model.Members, initialize=rank_points_data, default=0.0)

    # 5. VARIABLES
    model.x = pyo.Var(model.Rounds, model.Members, within=pyo.Binary)

    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.rank_points[i, j] * model.x[i, j] for i in model.Rounds for j in model.Members)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)

    # 7. CONSTRAINTS
    # Constraint: Each member can participate in a maximum of three rounds
    def participation_limit_rule(model, j):
        return sum(model.x[i, j] for i in model.Rounds) <= 3
    model.participation_limit = pyo.Constraint(model.Members, rule=participation_limit_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("\nParticipation decisions:")
        for i in model.Rounds:
            for j in model.Members:
                x_val = pyo.value(model.x[i, j])
                if x_val > 0.5:  # Only print participation decisions
                    print(f"Member {j} participates in Round {i}")
        
    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
decoration_competition_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 75.0
**Execution Time**: 1.29 seconds
**Reliability**: high
**Analysis**: Pyomo achieved the optimal solution with a moderate execution time, demonstrating reliability and consistency with other solvers.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | OPTIMAL | 75.00 | 0.53s | N/A | N/A |
| Docplex | OPTIMAL | 75.00 | 2.09s | N/A | N/A |
| Pyomo | OPTIMAL | 75.00 | 1.29s | N/A | N/A |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Majority Vote Optimal Value**: 75.0

### Final Recommendation
**Recommended Optimal Value**: 75.0
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy is preferred due to its fastest execution time and high reliability, making it suitable for scenarios requiring quick decision-making.

### Business Interpretation
**Overall Strategy**: The optimal allocation of members to rounds maximizes the total rank points, ensuring the best performance for the college.
**Objective Value Meaning**: The optimal objective value of 75.0 indicates the maximum rank points achievable under the given constraints.
**Resource Allocation Summary**: Members should be strategically allocated to rounds to maximize rank points, adhering to the constraint of maximum three participations per member.
**Implementation Recommendations**: Implement the solution by scheduling members according to the optimal decision variables, ensuring compliance with participation constraints.