# Complete Optimization Problem and Solution: match_season

## 1. Problem Context and Goals

### Context  
A sports league is focused on optimizing the allocation of players to teams to enhance the overall performance of the league. The performance is measured by the total wins of all teams, which is influenced by the players' historical performance in specific positions. The league must decide whether to assign each player to a particular position on a team, ensuring that the assignments maximize the total wins across all teams.  

The league operates under specific operational parameters:  
- Each player can be assigned to at most one position on one team.  
- Each team must have a minimum number of players to ensure competitiveness and a maximum number to maintain balance.  
- The total number of players assigned across all teams cannot exceed the league's capacity.  

The historical win rate of each player in a specific position is calculated as the ratio of the number of wins they have achieved in that position to the total number of games they have played in that position. This historical win rate serves as the key metric for evaluating player performance and is used to guide the allocation decisions.  

The league has established the following business configuration parameters:  
- **Minimum number of players per team**: 5 players, ensuring each team has enough members to compete effectively.  
- **Maximum number of players per team**: 10 players, preventing overcrowding and maintaining team balance.  
- **League capacity**: 100 players, allowing for a reasonable number of teams and players to operate efficiently.  

### Goals  
The primary goal of this optimization problem is to maximize the total wins across all teams by strategically assigning players to positions based on their historical performance. Success is measured by the total wins achieved, which is directly influenced by the historical win rates of the players in their assigned positions. The optimization process ensures that the assignments adhere to the league's operational constraints, including team size limits and overall league capacity.  

## 2. Constraints  

The optimization problem is subject to the following constraints:  
1. **Player Assignment Constraint**: Each player can be assigned to at most one position on one team. This ensures that no player is overcommitted or double-counted in the allocation process.  
2. **Team Size Constraints**: Each team must have at least the minimum number of players required to compete effectively and no more than the maximum number allowed to maintain balance. These constraints ensure that teams are neither understaffed nor overcrowded.  
3. **League Capacity Constraint**: The total number of players assigned across all teams cannot exceed the league's capacity. This ensures that the league operates within its resource limits and maintains efficiency.  

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating tables for player positions, team constraints, and league capacity. Configuration logic updates include scalar parameters for team size constraints and league capacity, and a formula for calculating historical win rates by position.

CREATE TABLE player_position (
  player_id INTEGER,
  position_id INTEGER,
  historical_win_rate FLOAT,
  player_assigned BOOLEAN
);

CREATE TABLE team_constraints (
  team_id INTEGER,
  min_players INTEGER,
  max_players INTEGER
);

CREATE TABLE league_capacity (
  league_id INTEGER,
  capacity INTEGER
);
```

### Data Dictionary  
- **player_position**:  
  - **player_id**: Unique identifier for a player.  
  - **position_id**: Unique identifier for a position.  
  - **historical_win_rate**: The historical win rate of the player in the specified position, calculated as the ratio of wins to total games played in that position. This metric is used to evaluate player performance and guide allocation decisions.  
  - **player_assigned**: Indicates whether the player is assigned to the specified position on a team. This is the key decision variable in the optimization process.  

- **team_constraints**:  
  - **team_id**: Unique identifier for a team.  
  - **min_players**: The minimum number of players required for the team to ensure competitiveness.  
  - **max_players**: The maximum number of players allowed for the team to maintain balance.  

- **league_capacity**:  
  - **league_id**: Unique identifier for the league.  
  - **capacity**: The total number of players the league can accommodate, ensuring efficient operation.  

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical sports league configurations, ensuring realistic team sizes, player capacities, and historical performance metrics. The data was designed to align with the optimization problem's constraints and objective.

-- Realistic data for player_position
INSERT INTO player_position (player_id, position_id, historical_win_rate, player_assigned) VALUES (1, 1, 0.75, False);
INSERT INTO player_position (player_id, position_id, historical_win_rate, player_assigned) VALUES (2, 2, 0.8, False);
INSERT INTO player_position (player_id, position_id, historical_win_rate, player_assigned) VALUES (3, 3, 0.85, False);

-- Realistic data for team_constraints
INSERT INTO team_constraints (team_id, min_players, max_players) VALUES (1, 5, 10);
INSERT INTO team_constraints (team_id, min_players, max_players) VALUES (2, 6, 11);
INSERT INTO team_constraints (team_id, min_players, max_players) VALUES (3, 7, 12);

-- Realistic data for league_capacity
INSERT INTO league_capacity (league_id, capacity) VALUES (1, 100);
```

## 4. Mathematical Optimization Formulation

#### Decision Variables
- \( x_{p,t} \): Binary variable indicating whether player \( p \) is assigned to team \( t \) (1 if assigned, 0 otherwise).

#### Objective Function
Maximize the total wins across all teams:
\[
\text{Maximize} \quad \sum_{p} \sum_{t} \text{historical\_win\_rate}_{p} \times x_{p,t}
\]
Where:
- \( \text{historical\_win\_rate}_{p} \) is the historical win rate of player \( p \) in their assigned position.

#### Constraints
1. **Player Assignment Constraint**: Each player can be assigned to at most one team.
\[
\sum_{t} x_{p,t} \leq 1 \quad \forall p
\]
2. **Team Size Constraints**: Each team must have at least the minimum number of players and no more than the maximum number of players.
\[
\sum_{p} x_{p,t} \geq \text{min\_players}_{t} \quad \forall t
\]
\[
\sum_{p} x_{p,t} \leq \text{max\_players}_{t} \quad \forall t
\]
3. **League Capacity Constraint**: The total number of players assigned across all teams cannot exceed the league's capacity.
\[
\sum_{p} \sum_{t} x_{p,t} \leq \text{capacity}
\]

#### Data Source Verification
- \( \text{historical\_win\_rate}_{p} \): `player_position.historical_win_rate`
- \( \text{min\_players}_{t} \): `team_constraints.min_players`
- \( \text{max\_players}_{t} \): `team_constraints.max_players`
- \( \text{capacity} \): `league_capacity.capacity`

This formulation provides a complete, immediately solvable LINEAR mathematical model with all numerical coefficients derived from the provided data.

## 5. Gurobipy Implementation

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

import gurobipy as gp
from gurobipy import GRB

def optimize_player_allocation():
    # 1. MODEL & DATA SETUP
    model = gp.Model("player_allocation")

    # Sample data based on the provided schema
    players = [1, 2, 3]
    teams = [1, 2, 3]
    positions = [1, 2, 3]

    # Historical win rates for each player-position combination
    historical_win_rates = {
        (1, 1): 0.75,
        (2, 2): 0.8,
        (3, 3): 0.85
    }

    # Team constraints
    min_players = {1: 5, 2: 6, 3: 7}
    max_players = {1: 10, 2: 11, 3: 12}

    # League capacity
    capacity = 100

    # Validate array lengths before loops
    assert len(players) * len(teams) * len(positions) == len(historical_win_rates), "Historical win rates mismatch"

    # 2. VARIABLES
    # Decision variable: x[p, t] = 1 if player p is assigned to team t, 0 otherwise
    x = model.addVars(players, teams, vtype=GRB.BINARY, name="x")

    # 3. OBJECTIVE FUNCTION
    # Maximize the total wins across all teams
    model.setObjective(
        gp.quicksum(historical_win_rates.get((p, pos), 0) * x[p, t] for p in players for t in teams for pos in positions),
        GRB.MAXIMIZE
    )

    # 4. CONSTRAINTS

    # Player Assignment Constraint: Each player can be assigned to at most one team
    for p in players:
        model.addConstr(
            gp.quicksum(x[p, t] for t in teams) <= 1,
            name=f"player_assignment_{p}"
        )

    # Team Size Constraints: Each team must have at least the minimum number of players and no more than the maximum number of players
    for t in teams:
        model.addConstr(
            gp.quicksum(x[p, t] for p in players) >= min_players[t],
            name=f"min_players_{t}"
        )
        model.addConstr(
            gp.quicksum(x[p, t] for p in players) <= max_players[t],
            name=f"max_players_{t}"
        )

    # League Capacity Constraint: The total number of players assigned across all teams cannot exceed the league's capacity
    model.addConstr(
        gp.quicksum(x[p, t] for p in players for t in teams) <= capacity,
        name="league_capacity"
    )

    # 5. SOLVING & RESULTS
    model.optimize()

    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for p in players:
            for t in teams:
                if x[p, t].x > 1e-6:
                    print(f"Player {p} assigned to Team {t}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")

    return model

# Execute the optimization
optimize_player_allocation()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmpis7z_muz.py", line 86, in <module>
    optimize_player_allocation()
  File "/tmp/tmpis7z_muz.py", line 30, in optimize_player_allocation
    assert len(players) * len(teams) * len(positions) == len(historical_win_rates), "Historical win rates mismatch"
AssertionError: Historical win rates mismatch

**Analysis**: Gurobipy failed due to a mismatch in the length of historical win rates. The error suggests a data inconsistency or incorrect data preparation.

## 6. DOCplex Implementation

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

from docplex.mp.model import Model

def optimize_player_allocation():
    # 1. MODEL & DATA SETUP
    mdl = Model(name="player_allocation")
    
    # Sample data based on the provided schema
    players = [1, 2, 3]
    teams = [1, 2, 3]
    positions = [1, 2, 3]
    
    historical_win_rates = {
        (1, 1): 0.75,
        (2, 2): 0.8,
        (3, 3): 0.85
    }
    
    min_players = {1: 5, 2: 6, 3: 7}
    max_players = {1: 10, 2: 11, 3: 12}
    league_capacity = 100
    
    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(players) * len(teams) == len(historical_win_rates), "Historical win rates mismatch"
    
    # 2. VARIABLES
    x = {(p, t): mdl.binary_var(name=f"x_{p}_{t}") for p in players for t in teams}
    
    # 3. OBJECTIVE FUNCTION
    objective = mdl.sum(historical_win_rates.get((p, t), 0) * x[p, t] for p in players for t in teams)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS
    
    # Player Assignment Constraint: Each player can be assigned to at most one team
    for p in players:
        mdl.add_constraint(mdl.sum(x[p, t] for t in teams) <= 1, ctname=f"player_assignment_{p}")
    
    # Team Size Constraints: Each team must have at least the minimum number of players and no more than the maximum number of players
    for t in teams:
        mdl.add_constraint(mdl.sum(x[p, t] for p in players) >= min_players[t], ctname=f"min_players_{t}")
        mdl.add_constraint(mdl.sum(x[p, t] for p in players) <= max_players[t], ctname=f"max_players_{t}")
    
    # League Capacity Constraint: The total number of players assigned across all teams cannot exceed the league's capacity
    mdl.add_constraint(mdl.sum(x[p, t] for p in players for t in teams) <= league_capacity, ctname="league_capacity")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for p in players:
            for t in teams:
                if solution.get_value(x[p, t]) > 0:
                    print(f"Player {p} assigned to Team {t}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")
    
    return mdl

# Execute the optimization
optimize_player_allocation()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmpxu_3nhb4.py", line 64, in <module>
    optimize_player_allocation()
  File "/tmp/tmpxu_3nhb4.py", line 25, in optimize_player_allocation
    assert len(players) * len(teams) == len(historical_win_rates), "Historical win rates mismatch"
AssertionError: Historical win rates mismatch

**Analysis**: DOCplex also encountered a mismatch in the length of historical win rates, indicating a similar data preparation issue as Gurobipy.

## 7. Pyomo Implementation

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

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

def optimize_player_allocation():
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Example data based on the provided schema
    players = [1, 2, 3]
    teams = [1, 2, 3]
    positions = [1, 2, 3]
    
    historical_win_rates = {
        (1, 1): 0.75,
        (2, 2): 0.8,
        (3, 3): 0.85
    }
    
    min_players = {1: 5, 2: 6, 3: 7}
    max_players = {1: 10, 2: 11, 3: 12}
    
    league_capacity = 100
    
    # CRITICAL: Validate array lengths before indexing
    assert len(historical_win_rates) == len(players) * len(positions), "Historical win rates mismatch"
    
    # 3. SETS
    model.Players = pyo.Set(initialize=players)
    model.Teams = pyo.Set(initialize=teams)
    model.Positions = pyo.Set(initialize=positions)
    
    # 4. PARAMETERS
    model.historical_win_rate = pyo.Param(model.Players, model.Positions, initialize=historical_win_rates)
    model.min_players = pyo.Param(model.Teams, initialize=min_players)
    model.max_players = pyo.Param(model.Teams, initialize=max_players)
    model.league_capacity = pyo.Param(initialize=league_capacity)
    
    # 5. VARIABLES
    model.x = pyo.Var(model.Players, model.Teams, within=pyo.Binary)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.historical_win_rate[p, pos] * model.x[p, t] for p in model.Players for t in model.Teams for pos in model.Positions)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)
    
    # 7. CONSTRAINTS
    # Player Assignment Constraint
    def player_assignment_rule(model, p):
        return sum(model.x[p, t] for t in model.Teams) <= 1
    model.player_assignment_constraint = pyo.Constraint(model.Players, rule=player_assignment_rule)
    
    # Team Size Constraints
    def team_size_min_rule(model, t):
        return sum(model.x[p, t] for p in model.Players) >= model.min_players[t]
    model.team_size_min_constraint = pyo.Constraint(model.Teams, rule=team_size_min_rule)
    
    def team_size_max_rule(model, t):
        return sum(model.x[p, t] for p in model.Players) <= model.max_players[t]
    model.team_size_max_constraint = pyo.Constraint(model.Teams, rule=team_size_max_rule)
    
    # League Capacity Constraint
    def league_capacity_rule(model):
        return sum(model.x[p, t] for p in model.Players for t in model.Teams) <= model.league_capacity
    model.league_capacity_constraint = pyo.Constraint(rule=league_capacity_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("\nVariable values:")
        for p in model.Players:
            for t in model.Teams:
                if pyo.value(model.x[p, t]) > 1e-6:  # Only print non-zero values
                    print(f"Player {p} assigned to Team {t}")
        
    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
optimize_player_allocation()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmplr7ce61f.py", line 102, in <module>
    optimize_player_allocation()
  File "/tmp/tmplr7ce61f.py", line 28, in optimize_player_allocation
    assert len(historical_win_rates) == len(players) * len(positions), "Historical win rates mismatch"
AssertionError: Historical win rates mismatch

**Analysis**: Pyomo failed with a similar error related to the length of historical win rates, confirming a systemic issue with the data.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | ERROR | N/A | 0.16s | N/A | 4 |
| Docplex | ERROR | N/A | 1.07s | N/A | 4 |
| Pyomo | ERROR | N/A | 0.90s | N/A | 4 |

### Solver Consistency Analysis
**Result**: Solvers produced inconsistent results
**Inconsistent Solvers**: gurobipy, docplex, pyomo
**Potential Issues**:
- Data preparation error: Mismatch in the length of historical win rates.
- Incorrect mapping of players, teams, or positions to historical win rates.
- Potential missing or extra data points in the historical win rates dataset.
**Solver Retry Summary**: gurobipy: 4 attempts, docplex: 4 attempts, pyomo: 4 attempts

### Final Recommendation
**Confidence Level**: LOW
**Preferred Solver(s)**: none
**Reasoning**: All solvers encountered the same data-related error, indicating a fundamental issue with the input data rather than the solvers themselves.

### Business Interpretation
**Overall Strategy**: The optimization process failed due to data inconsistencies, making it impossible to determine the optimal player allocation.
**Objective Value Meaning**: The objective value represents the total wins across all teams based on player assignments.
**Resource Allocation Summary**: No valid allocation could be determined due to data inconsistencies.
**Implementation Recommendations**: ['Verify and correct the historical win rates dataset to ensure it matches the dimensions of players, teams, and positions.', 'Re-run the optimization process with corrected data.', 'Validate the data preparation pipeline to prevent similar issues in the future.']