# Complete Optimization Problem and Solution: match_season

## 1. Problem Context and Goals

### Context  
In the context of a sports league, the primary objective is to optimize the selection of players for a season to maximize the team's overall performance. The decision-making process involves selecting players, where each player is represented by a binary decision variable indicating whether they are chosen for the team. The operational parameters are designed to align with a linear objective, focusing on maximizing the total performance score. This score is calculated by summing the performance coefficients of the selected players. The business configuration includes several constraints: the maximum number of players allowed in a team, the maximum allowable sum of draft pick numbers, and the minimum number of players required in specific positions such as defenders, midfielders, and forwards. These constraints are essential to ensure a balanced and competitive team composition. The business logic is structured to maintain linear relationships, avoiding any nonlinear scenarios such as variable products or divisions. The configuration parameters are clearly defined to map to the expected coefficient sources, ensuring consistency and clarity in the optimization process.

### Goals  
The primary goal of this optimization problem is to maximize the total performance score of the team. This involves selecting players in such a way that the sum of their performance coefficients is maximized. The success of this optimization is measured by the alignment of the selected players' performance coefficients with the expected sources. The goal is articulated in natural language to emphasize the linear nature of the optimization objective, focusing on maximizing the total performance score through strategic player selection.

## 2. Constraints    

The optimization problem is subject to several constraints that are directly aligned with linear mathematical formulations. These constraints include:

- The total number of selected players must not exceed the maximum number allowed in a team, ensuring compliance with team size regulations.
- The sum of the draft pick numbers for the selected players must remain within the allowable limit, maintaining competitive balance.
- The team must include at least a minimum number of defenders, ensuring a solid defensive line.
- A minimum number of midfielders is required to provide sufficient midfield presence for control and transition.
- The team must also have a minimum number of forwards to maintain offensive pressure.

Each of these constraints is described in business terms that naturally lead to linear mathematical forms, avoiding any nonlinear relationships.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating new tables for performance coefficients and constraint bounds, modifying existing tables to include necessary columns, and updating business configuration logic for scalar parameters and formulas.

CREATE TABLE Player (
  player_id INTEGER,
  performance_coefficient FLOAT,
  selected BOOLEAN
);

CREATE TABLE PerformanceCoefficients (
  player_id INTEGER,
  coefficient FLOAT
);

CREATE TABLE ConstraintBounds (
  constraint_name STRING,
  value INTEGER
);
```

### Data Dictionary  
The data dictionary provides a comprehensive mapping of tables and columns to their business purposes and optimization roles:

- **Player Table**: This table stores information about each player, including their unique identifier, performance score, and selection status. The performance score is used in the objective function to maximize team performance, while the selection status serves as a decision variable in the optimization model.

- **PerformanceCoefficients Table**: This table contains the performance coefficients for each player, linking to the Player table through the player ID. These coefficients are crucial for the objective function, as they determine the contribution of each player to the total performance score.

- **ConstraintBounds Table**: This table holds the constraint bounds for team composition, including the maximum number of players, draft pick limits, and minimum position-specific requirements. These bounds are used in the constraint formulation to ensure the team meets all necessary criteria.

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical sports league configurations and realistic player performance metrics to ensure a balanced and competitive team composition.

-- Realistic data for Player
INSERT INTO Player (player_id, performance_coefficient, selected) VALUES (1, 0.88, False);
INSERT INTO Player (player_id, performance_coefficient, selected) VALUES (2, 0.92, True);
INSERT INTO Player (player_id, performance_coefficient, selected) VALUES (3, 0.85, False);

-- Realistic data for PerformanceCoefficients
INSERT INTO PerformanceCoefficients (player_id, coefficient) VALUES (1, 0.88);
INSERT INTO PerformanceCoefficients (player_id, coefficient) VALUES (2, 0.92);
INSERT INTO PerformanceCoefficients (player_id, coefficient) VALUES (3, 0.85);

-- Realistic data for ConstraintBounds
INSERT INTO ConstraintBounds (constraint_name, value) VALUES ('max_players_per_team', 25);
INSERT INTO ConstraintBounds (constraint_name, value) VALUES ('max_draft_pick_sum', 100);
INSERT INTO ConstraintBounds (constraint_name, value) VALUES ('min_defenders', 4);
INSERT INTO ConstraintBounds (constraint_name, value) VALUES ('min_midfielders', 3);
INSERT INTO ConstraintBounds (constraint_name, value) VALUES ('min_forwards', 3);
```

## 4. Mathematical Optimization Formulation

#### Decision Variables
- Let \( x_i \) be a binary decision variable for each player \( i \), where \( x_i = 1 \) if player \( i \) is selected for the team, and \( x_i = 0 \) otherwise.

#### Objective Function
Maximize the total performance score of the team:
\[ \text{Maximize } \sum_{i} \text{PerformanceCoefficients.coefficient}[i] \times x_i \]

Data Source Verification:
- Coefficients are sourced from the `PerformanceCoefficients.coefficient` column.

#### Constraints
1. **Maximum Number of Players:**
   \[ \sum_{i} x_i \leq \text{ConstraintBounds.value}[\text{'max_players_per_team'}] \]
   - Data Source Verification: `ConstraintBounds.value` where `constraint_name` = 'max_players_per_team'

2. **Maximum Draft Pick Sum:**
   \[ \sum_{i} \text{Player.draft_pick}[i] \times x_i \leq \text{ConstraintBounds.value}[\text{'max_draft_pick_sum'}] \]
   - Data Source Verification: `ConstraintBounds.value` where `constraint_name` = 'max_draft_pick_sum'

3. **Minimum Number of Defenders:**
   \[ \sum_{i \in \text{Defenders}} x_i \geq \text{ConstraintBounds.value}[\text{'min_defenders'}] \]
   - Data Source Verification: `ConstraintBounds.value` where `constraint_name` = 'min_defenders'

4. **Minimum Number of Midfielders:**
   \[ \sum_{i \in \text{Midfielders}} x_i \geq \text{ConstraintBounds.value}[\text{'min_midfielders'}] \]
   - Data Source Verification: `ConstraintBounds.value` where `constraint_name` = 'min_midfielders'

5. **Minimum Number of Forwards:**
   \[ \sum_{i \in \text{Forwards}} x_i \geq \text{ConstraintBounds.value}[\text{'min_forwards'}] \]
   - Data Source Verification: `ConstraintBounds.value` where `constraint_name` = 'min_forwards'

Note: The sets Defenders, Midfielders, and Forwards are assumed to be predefined based on player positions.

This linear mathematical model is immediately solvable using standard linear programming or mixed-integer programming solvers, ensuring all constraints and objectives are aligned with the provided data.

## 5. Gurobipy Implementation

```python
# Complete GUROBIPY implementation - Retry Attempt 4

import gurobipy as gp
from gurobipy import GRB

def optimize_team_selection():
    # 1. MODEL & DATA SETUP
    model = gp.Model("match_season")
    
    # Example data setup (replace with actual data retrieval logic)
    player_ids = [1, 2, 3]
    performance_coefficients = [0.88, 0.92, 0.85]
    draft_picks = [10, 20, 30]  # Example draft pick numbers
    positions = ['defender', 'midfielder', 'forward']  # Example positions
    
    # Constraint bounds (replace with actual data retrieval logic)
    max_players_per_team = 25
    max_draft_pick_sum = 100
    min_defenders = 4
    min_midfielders = 3
    min_forwards = 3
    
    # Validate array lengths
    assert len(player_ids) == len(performance_coefficients) == len(draft_picks) == len(positions), "Array length mismatch"
    
    # 2. VARIABLES
    x = {i: model.addVar(vtype=GRB.BINARY, name=f"x_{i}") for i in player_ids}
    
    # 3. OBJECTIVE FUNCTION
    model.setObjective(gp.quicksum(performance_coefficients[i] * x[player_ids[i]] for i in range(len(player_ids))), GRB.MAXIMIZE)
    
    # 4. CONSTRAINTS
    # Maximum number of players
    model.addConstr(gp.quicksum(x[player_ids[i]] for i in range(len(player_ids))) <= max_players_per_team, name="max_players")
    
    # Maximum draft pick sum
    model.addConstr(gp.quicksum(draft_picks[i] * x[player_ids[i]] for i in range(len(player_ids))) <= max_draft_pick_sum, name="max_draft_pick_sum")
    
    # Minimum number of defenders
    model.addConstr(gp.quicksum(x[player_ids[i]] for i in range(len(player_ids)) if positions[i] == 'defender') >= min_defenders, name="min_defenders")
    
    # Minimum number of midfielders
    model.addConstr(gp.quicksum(x[player_ids[i]] for i in range(len(player_ids)) if positions[i] == 'midfielder') >= min_midfielders, name="min_midfielders")
    
    # Minimum number of forwards
    model.addConstr(gp.quicksum(x[player_ids[i]] for i in range(len(player_ids)) if positions[i] == 'forward') >= min_forwards, name="min_forwards")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in player_ids:
            if x[i].x > 1e-6:
                print(f"Player {i} selected")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    
    return model

# Run the optimization
optimize_team_selection()
```

### Execution Results
**Status**: INFEASIBLE
**Error**: Problem is infeasible
**Analysis**: Gurobipy consistently reported the problem as infeasible across multiple retries, indicating a likely issue with the constraints or data.

## 6. DOCplex Implementation

```python
# Complete DOCPLEX implementation - Retry Attempt 4

from docplex.mp.model import Model

def optimize_team_selection():
    # 1. MODEL & DATA SETUP
    mdl = Model(name="match_season")
    
    # Example data setup (replace with actual data retrieval logic)
    player_ids = [1, 2, 3]
    performance_coefficients = [0.88, 0.92, 0.85]
    draft_picks = [10, 20, 30]  # Example draft pick numbers
    positions = ['defender', 'midfielder', 'forward']  # Example positions
    
    # Constraint bounds (replace with actual data retrieval logic)
    max_players_per_team = 25
    max_draft_pick_sum = 100
    min_defenders = 4
    min_midfielders = 3
    min_forwards = 3
    
    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(player_ids) == len(performance_coefficients) == len(draft_picks) == len(positions), "Array length mismatch"
    safe_range = range(len(player_ids))  # Safe indexing
    
    # 2. VARIABLES
    x = {i: mdl.binary_var(name=f"x_{player_ids[i]}") for i in safe_range}
    
    # 3. OBJECTIVE FUNCTION
    objective = mdl.sum(performance_coefficients[i] * x[i] for i in safe_range)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS - CORRECT SYNTAX PATTERNS
    
    # Maximum number of players
    mdl.add_constraint(mdl.sum(x[i] for i in safe_range) <= max_players_per_team, ctname="max_players")
    
    # Maximum draft pick sum
    mdl.add_constraint(mdl.sum(draft_picks[i] * x[i] for i in safe_range) <= max_draft_pick_sum, ctname="max_draft_pick_sum")
    
    # Minimum number of defenders
    mdl.add_constraint(mdl.sum(x[i] for i in safe_range if positions[i] == 'defender') >= min_defenders, ctname="min_defenders")
    
    # Minimum number of midfielders
    mdl.add_constraint(mdl.sum(x[i] for i in safe_range if positions[i] == 'midfielder') >= min_midfielders, ctname="min_midfielders")
    
    # Minimum number of forwards
    mdl.add_constraint(mdl.sum(x[i] for i in safe_range if positions[i] == 'forward') >= min_forwards, ctname="min_forwards")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in safe_range:
            value = solution.get_value(x[i])
            if value > 1e-6:
                print(f"Player {player_ids[i]} selected with performance coefficient {performance_coefficients[i]}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")

    return mdl

# Execute the optimization
optimize_team_selection()
```

### Execution Results
**Status**: INFEASIBLE
**Error**: Problem is infeasible
**Analysis**: DOCplex also reported infeasibility, suggesting that the problem formulation or data constraints are not satisfiable.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation - Retry Attempt 4

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

def match_season_optimization():
    """Optimize player selection for maximum performance score"""

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

    # 2. DATA SETUP
    # Example data based on the problem context
    player_ids = [1, 2, 3]
    performance_coefficients = {1: 0.88, 2: 0.92, 3: 0.85}
    draft_picks = {1: 10, 2: 20, 3: 30}  # Example draft pick numbers
    max_players_per_team = 25
    max_draft_pick_sum = 100
    min_defenders = 4
    min_midfielders = 3
    min_forwards = 3

    # Validate data lengths
    assert len(player_ids) == len(performance_coefficients) == len(draft_picks), "Data length mismatch"

    # 3. SETS
    model.Players = pyo.Set(initialize=player_ids)

    # 4. PARAMETERS
    model.performance_coefficient = pyo.Param(model.Players, initialize=performance_coefficients)
    model.draft_pick = pyo.Param(model.Players, initialize=draft_picks)

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

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

    # 7. CONSTRAINTS
    # Maximum number of players
    def max_players_rule(model):
        return sum(model.x[i] for i in model.Players) <= max_players_per_team
    model.max_players_constraint = pyo.Constraint(rule=max_players_rule)

    # Maximum draft pick sum
    def max_draft_pick_rule(model):
        return sum(model.draft_pick[i] * model.x[i] for i in model.Players) <= max_draft_pick_sum
    model.max_draft_pick_constraint = pyo.Constraint(rule=max_draft_pick_rule)

    # Minimum number of defenders
    # Assuming player positions are predefined
    defenders = [1]  # Example defender IDs
    def min_defenders_rule(model):
        return sum(model.x[i] for i in defenders) >= min_defenders
    model.min_defenders_constraint = pyo.Constraint(rule=min_defenders_rule)

    # Minimum number of midfielders
    midfielders = [2]  # Example midfielder IDs
    def min_midfielders_rule(model):
        return sum(model.x[i] for i in midfielders) >= min_midfielders
    model.min_midfielders_constraint = pyo.Constraint(rule=min_midfielders_rule)

    # Minimum number of forwards
    forwards = [3]  # Example forward IDs
    def min_forwards_rule(model):
        return sum(model.x[i] for i in forwards) >= min_forwards
    model.min_forwards_constraint = pyo.Constraint(rule=min_forwards_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)}")

        # Extract variable values
        print("\nSelected players:")
        for i in model.Players:
            if pyo.value(model.x[i]) > 0.5:  # Binary variable, check if selected
                print(f"Player {i} selected")
    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

# Execute the optimization
match_season_optimization()
```

### Execution Results
**Status**: INFEASIBLE
**Error**: Problem is infeasible
**Analysis**: Pyomo's results align with the other solvers, indicating a consistent infeasibility across different solver technologies.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | INFEASIBLE | N/A | 0.17s | N/A | 4 |
| Docplex | INFEASIBLE | N/A | 1.51s | N/A | 4 |
| Pyomo | INFEASIBLE | N/A | 0.92s | N/A | 4 |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Solver Retry Summary**: gurobipy: 4 attempts, docplex: 4 attempts, pyomo: 4 attempts

### Final Recommendation
**Confidence Level**: LOW
**Preferred Solver(s)**: multiple
**Reasoning**: All solvers consistently report infeasibility, indicating a problem with the model or data rather than the solvers themselves.

### Business Interpretation
**Overall Strategy**: The current model cannot produce a feasible team selection under the given constraints. This suggests a need to revisit the constraints or data inputs.
**Objective Value Meaning**: The optimal objective value would represent the maximum achievable performance score for the team, but it is currently unattainable due to infeasibility.
**Resource Allocation Summary**: Resources cannot be allocated effectively until the constraints are adjusted to allow for a feasible solution.
**Implementation Recommendations**: Review and potentially relax constraints, verify data accuracy, and ensure player classifications are correct. Consider consulting with domain experts to validate the model assumptions.