## 4. Mathematical Optimization Formulation

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

#### Objective Function
Maximize the total compatibility score from player-coach assignments:

\[
\text{Maximize } \sum_{(i,j) \in \text{compatibility\_scores}} \text{compatibility\_score}_{ij} \times x_{ij}
\]

Data Source Verification:
- Coefficients \(\text{compatibility\_score}_{ij}\) are from the `compatibility_scores` table.

#### Constraints

1. **Coach Capacity Constraints:**
   Each coach can manage a limited number of players:

   \[
   \sum_{i} x_{ij} \leq \text{max\_players}_j \quad \forall j
   \]

   Data Source Verification:
   - \(\text{max\_players}_j\) are from the `coach_constraints` table.

2. **Player Assignment Constraints:**
   Each player can be assigned to at most one coach:

   \[
   \sum_{j} x_{ij} \leq 1 \quad \forall i
   \]

3. **Compatibility Threshold Constraints:**
   A player-coach assignment is only valid if the compatibility score meets or exceeds a minimum threshold (assume threshold is \( \text{min\_compatibility\_score} \)):

   \[
   x_{ij} = 0 \quad \text{if } \text{compatibility\_score}_{ij} < \text{min\_compatibility\_score}
   \]

   Data Source Verification:
   - \(\text{compatibility\_score}_{ij}\) are from the `compatibility_scores` table.
   - \(\text{min\_compatibility\_score}\) is a business configuration parameter.

#### Additional Notes
- The binary nature of \( x_{ij} \) ensures that the assignments are either made or not, aligning with the business requirement for binary decisions.
- The constraints ensure that no coach is overburdened and that each player is assigned to at most one coach, maintaining operational feasibility and quality of training.

## 5. Gurobipy Implementation

```python
# Complete GUROBIPY implementation

import gurobipy as gp
from gurobipy import GRB

def optimize_riding_club():
    """Optimize player-coach assignments for the riding club"""

    # 1. MODEL & DATA SETUP
    model = gp.Model("riding_club")

    # Data from the database
    compatibility_scores = [
        (1, 1, 0.85),
        (1, 2, 0.65),
        (2, 1, 0.75),
        (2, 3, 0.6),
        (3, 2, 0.9),
        (3, 3, 0.7)
    ]
    
    coach_constraints = {
        1: 5,
        2: 4,
        3: 6
    }
    
    min_compatibility_score = 0.7

    # Extract unique players and coaches
    players = set(player for player, _, _ in compatibility_scores)
    coaches = set(coach for _, coach, _ in compatibility_scores)

    # CRITICAL: Validate data consistency
    assert all(isinstance(score, float) for _, _, score in compatibility_scores), "Invalid score type"

    # 2. VARIABLES
    # Decision variables: x[i, j] = 1 if player i is assigned to coach j
    x = model.addVars(players, coaches, vtype=GRB.BINARY, name="x")

    # 3. OBJECTIVE FUNCTION
    # Maximize total compatibility score
    model.setObjective(
        gp.quicksum(score * x[i, j] for i, j, score in compatibility_scores),
        GRB.MAXIMIZE
    )

    # 4. CONSTRAINTS

    # Coach capacity constraints
    model.addConstrs(
        (gp.quicksum(x[i, j] for i in players) <= coach_constraints[j] for j in coaches),
        name="coach_capacity"
    )

    # Player assignment constraints
    model.addConstrs(
        (gp.quicksum(x[i, j] for j in coaches) <= 1 for i in players),
        name="player_assignment"
    )

    # Compatibility threshold constraints
    model.addConstrs(
        (x[i, j] == 0 for i, j, score in compatibility_scores if score < min_compatibility_score),
        name="compatibility_threshold"
    )

    # 5. SOLVING & RESULTS
    model.optimize()

    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i, j in x.keys():
            if x[i, j].x > 1e-6:
                print(f"Player {i} assigned to Coach {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_riding_club()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 2.5
**Execution Time**: 2.44 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 riding_club_optimization():
    """Riding Club Optimization using DOCPLEX"""
    
    # 1. MODEL & DATA SETUP
    mdl = Model(name="riding_club")
    
    # Data from the problem statement
    compatibility_scores = [
        (1, 1, 0.85),
        (1, 2, 0.65),
        (2, 1, 0.75),
        (2, 3, 0.6),
        (3, 2, 0.9),
        (3, 3, 0.7)
    ]
    
    coach_constraints = {
        1: 5,
        2: 4,
        3: 6
    }
    
    min_compatibility_score = 0.7
    
    # Extract unique player and coach IDs
    player_ids = set(cs[0] for cs in compatibility_scores)
    coach_ids = set(cs[1] for cs in compatibility_scores)
    
    # 2. VARIABLES
    # Binary decision variables for player-coach assignments
    x = {(i, j): mdl.binary_var(name=f"x_{i}_{j}") for i in player_ids for j in coach_ids}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize the total compatibility score from player-coach assignments
    objective = mdl.sum(cs[2] * x[cs[0], cs[1]] for cs in compatibility_scores)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS
    
    # Coach Capacity Constraints
    for j in coach_ids:
        mdl.add_constraint(mdl.sum(x[i, j] for i in player_ids if (i, j) in x) <= coach_constraints[j], ctname=f"coach_capacity_{j}")
    
    # Player Assignment Constraints
    for i in player_ids:
        mdl.add_constraint(mdl.sum(x[i, j] for j in coach_ids if (i, j) in x) <= 1, ctname=f"player_assignment_{i}")
    
    # Compatibility Threshold Constraints
    for cs in compatibility_scores:
        i, j, score = cs
        if score < min_compatibility_score:
            mdl.add_constraint(x[i, j] == 0, ctname=f"compatibility_threshold_{i}_{j}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for (i, j) in x:
            value = solution.get_value(x[i, j])
            if value > 1e-6:
                print(f"x[{i},{j}] = {value:.3f}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")
    
    return mdl

# Run the optimization
riding_club_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 2.5
**Execution Time**: 10.12 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 riding_club_optimization():
    """Optimization model for the riding club player-coach assignment problem."""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Compatibility scores and coach constraints data
    compatibility_scores = {
        (1, 1): 0.85,
        (1, 2): 0.65,
        (2, 1): 0.75,
        (2, 3): 0.6,
        (3, 2): 0.9,
        (3, 3): 0.7
    }
    coach_constraints = {
        1: 5,
        2: 4,
        3: 6
    }
    min_compatibility_score = 0.6  # Example threshold
    
    # 3. SETS
    model.Players = pyo.Set(initialize=set(i for i, j in compatibility_scores))
    model.Coaches = pyo.Set(initialize=set(j for i, j in compatibility_scores))
    
    # 4. PARAMETERS
    model.compatibility_score = pyo.Param(model.Players, model.Coaches, initialize=compatibility_scores, default=0)
    model.max_players = pyo.Param(model.Coaches, initialize=coach_constraints)
    
    # 5. VARIABLES
    model.x = pyo.Var(model.Players, model.Coaches, within=pyo.Binary)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.compatibility_score[i, j] * model.x[i, j] for i in model.Players for j in model.Coaches)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)
    
    # 7. CONSTRAINTS
    
    # Coach capacity constraints
    def coach_capacity_rule(model, j):
        return sum(model.x[i, j] for i in model.Players) <= model.max_players[j]
    model.coach_capacity_constraint = pyo.Constraint(model.Coaches, rule=coach_capacity_rule)
    
    # Player assignment constraints
    def player_assignment_rule(model, i):
        return sum(model.x[i, j] for j in model.Coaches) <= 1
    model.player_assignment_constraint = pyo.Constraint(model.Players, rule=player_assignment_rule)
    
    # Compatibility threshold constraints
    def compatibility_threshold_rule(model, i, j):
        if model.compatibility_score[i, j] < min_compatibility_score:
            return model.x[i, j] == 0
        return pyo.Constraint.Skip
    model.compatibility_threshold_constraint = pyo.Constraint(model.Players, model.Coaches, rule=compatibility_threshold_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 decisions:")
        for i in model.Players:
            for j in model.Coaches:
                x_val = pyo.value(model.x[i, j])
                if x_val > 0.5:  # Only print assignments that are made
                    print(f"Player {i} assigned to Coach {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
riding_club_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 2.5
**Execution Time**: 4.56 seconds
**Reliability**: high
**Analysis**: Pyomo achieved an optimal solution with a moderate execution time, demonstrating reliability and reasonable efficiency.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | OPTIMAL | 2.50 | 2.44s | N/A | N/A |
| Docplex | OPTIMAL | 2.50 | 10.12s | N/A | N/A |
| Pyomo | OPTIMAL | 2.50 | 4.56s | N/A | N/A |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Majority Vote Optimal Value**: 2.5

### Final Recommendation
**Recommended Optimal Value**: 2.5
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy is preferred due to its high reliability and the shortest execution time, making it the most efficient choice.

### Business Interpretation
**Overall Strategy**: The optimal assignment maximizes compatibility scores, ensuring the best player-coach pairings.
**Objective Value Meaning**: The optimal objective value of 2.5 represents the maximum achievable compatibility score for the player-coach assignments.
**Resource Allocation Summary**: Players should be assigned to coaches based on the optimal solution to maximize compatibility and training effectiveness.
**Implementation Recommendations**: Implement the solution using Gurobipy for efficient execution, and ensure all assignments meet the compatibility threshold.