# Complete Optimization Problem and Solution: poker_player

## 1. Problem Context and Goals

### Context  
In the context of a poker tournament, the organizer aims to distribute prize money among players in a way that maximizes overall player satisfaction. The decision revolves around how much prize money to allocate to each player, represented as a continuous variable for each player. The satisfaction of each player is influenced by their performance metrics, which are quantified through satisfaction coefficients. The objective is to maximize the total satisfaction of all players by allocating prize money in proportion to these coefficients.

The operational parameters include a total budget for prize money allocation, which serves as a constraint to ensure that the total distributed prize money does not exceed this budget. Additionally, there is a maximum limit on the prize money that can be allocated to any single player, ensuring fairness and preventing any player from receiving an excessively large portion of the budget. These parameters are critical in shaping the linear optimization model, guiding the allocation process within defined limits.

### Goals  
The primary goal of this optimization problem is to maximize the total satisfaction of players participating in the tournament. This is achieved by strategically allocating prize money based on each player's satisfaction coefficient, which reflects their performance. The success of this optimization is measured by the total satisfaction score, calculated as the sum of the product of each player's satisfaction coefficient and their allocated prize money. The aim is to achieve the highest possible total satisfaction while adhering to the budgetary and individual prize constraints.

## 2. Constraints    

The constraints in this optimization problem are designed to ensure that the prize money allocation remains within practical and fair limits. The first constraint ensures that the total prize money allocated to all players does not exceed the available budget. This is a critical resource limitation that aligns with the tournament's financial capabilities. The second constraint sets a maximum limit on the prize money that can be allocated to any single player, promoting fairness and preventing disproportionate allocations. These constraints are expressed in linear terms, directly guiding the allocation process within the defined boundaries.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating new tables for decision variables and constraint bounds, and moving scalar parameters to configuration logic. This addresses the OR expert's mapping gaps and missing requirements.

CREATE TABLE PlayerSatisfactionCoefficients (
  player_id INTEGER,
  coefficient FLOAT
);

CREATE TABLE PrizeMoneyAllocation (
  player_id INTEGER,
  prize_money FLOAT
);
```

### Data Dictionary  
The data dictionary provides a business-oriented overview of the tables and columns used in the optimization model:

- **PlayerSatisfactionCoefficients Table**: This table stores the satisfaction coefficients for each player, which are used to determine the impact of prize money on player satisfaction. Each player is uniquely identified by a player ID, and their satisfaction coefficient reflects their performance metrics.

  - **player_id**: A unique identifier for each player, linking satisfaction coefficients to individual players.
  - **coefficient**: Represents the satisfaction coefficient for each player, used in calculating the total satisfaction score.

- **PrizeMoneyAllocation Table**: This table records the prize money allocated to each player, which is the decision variable in the optimization model. The allocation is determined based on the satisfaction coefficients and the constraints of the model.

  - **player_id**: A unique identifier for each player, linking prize money allocations to individual players.
  - **prize_money**: The amount of prize money allocated to each player, serving as the decision variable in the optimization process.

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical poker tournament prize distributions and player performance metrics, ensuring a balance between competitive payouts and budget constraints.

-- Realistic data for PlayerSatisfactionCoefficients
INSERT INTO PlayerSatisfactionCoefficients (player_id, coefficient) VALUES (1, 1.1);
INSERT INTO PlayerSatisfactionCoefficients (player_id, coefficient) VALUES (2, 0.9);
INSERT INTO PlayerSatisfactionCoefficients (player_id, coefficient) VALUES (3, 1.3);

-- Realistic data for PrizeMoneyAllocation
INSERT INTO PrizeMoneyAllocation (player_id, prize_money) VALUES (1, 8000);
INSERT INTO PrizeMoneyAllocation (player_id, prize_money) VALUES (2, 6000);
INSERT INTO PrizeMoneyAllocation (player_id, prize_money) VALUES (3, 10000);
```

## 4. Mathematical Optimization Formulation

#### Decision Variables
Let \( x_i \) be the prize money allocated to player \( i \). These are continuous decision variables representing the amount of prize money each player receives.

#### Objective Function
Maximize the total satisfaction of all players, which is calculated as the sum of the product of each player's satisfaction coefficient and their allocated prize money:

\[
\text{Maximize } \sum_{i} c_i \times x_i
\]

where:
- \( c_i \) is the satisfaction coefficient for player \( i \).

#### Constraints
1. **Total Budget Constraint**: The total prize money allocated to all players must not exceed the available budget.

   \[
   \sum_{i} x_i \leq \text{Total Budget}
   \]

2. **Individual Prize Limit Constraint**: The prize money allocated to any single player must not exceed a specified maximum limit.

   \[
   x_i \leq \text{Max Prize per Player} \quad \forall i
   \]

3. **Non-negativity Constraint**: The prize money allocated to each player must be non-negative.

   \[
   x_i \geq 0 \quad \forall i
   \]

Data Source Verification:
- \( c_i \) values are sourced from the `PlayerSatisfactionCoefficients.coefficient` column.
- The total budget and maximum prize per player are assumed to be provided as business configuration parameters, not explicitly listed in the database schema.

Given the provided data, the specific formulation with numerical coefficients is:

#### Numerical Example
Using the provided data:

- Player 1: \( c_1 = 1.1 \)
- Player 2: \( c_2 = 0.9 \)
- Player 3: \( c_3 = 1.3 \)

Assuming:
- Total Budget = 24000
- Max Prize per Player = 10000

The linear optimization model becomes:

\[
\text{Maximize } 1.1x_1 + 0.9x_2 + 1.3x_3
\]

Subject to:

\[
x_1 + x_2 + x_3 \leq 24000
\]

\[
x_1 \leq 10000
\]

\[
x_2 \leq 10000
\]

\[
x_3 \leq 10000
\]

\[
x_1, x_2, x_3 \geq 0
\]

This model can be solved using linear programming techniques to determine the optimal allocation of prize money that maximizes player satisfaction while adhering to the constraints.

## 5. Gurobipy Implementation

```python
# Complete GUROBIPY implementation

import gurobipy as gp
from gurobipy import GRB

def poker_player_optimization():
    """Optimize prize money allocation for poker players to maximize satisfaction."""
    
    # 1. MODEL & DATA SETUP
    model = gp.Model("poker_player")
    
    # Data: Satisfaction coefficients and constraints
    player_ids = [1, 2, 3]
    coefficients = [1.1, 0.9, 1.3]
    total_budget = 24000
    max_prize_per_player = 10000
    
    # CRITICAL: Validate array lengths before loops
    assert len(player_ids) == len(coefficients), "Array length mismatch"
    
    # 2. VARIABLES
    # Prize money allocation variables for each player
    prize_money = {i: model.addVar(vtype=GRB.CONTINUOUS, name=f"x_{i}", lb=0) 
                   for i in player_ids}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize total satisfaction
    model.setObjective(gp.quicksum(coefficients[i-1] * prize_money[i] for i in player_ids), GRB.MAXIMIZE)
    
    # 4. CONSTRAINTS - CORRECT SYNTAX PATTERNS
    
    # Total budget constraint
    model.addConstr(gp.quicksum(prize_money[i] for i in player_ids) <= total_budget, name="total_budget")
    
    # Individual prize limit constraints
    for i in player_ids:
        model.addConstr(prize_money[i] <= max_prize_per_player, name=f"max_prize_{i}")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in player_ids:
            if prize_money[i].x > 1e-6:
                print(f"Prize money for player {i}: {prize_money[i].x:.2f}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    
    return model

# Run the optimization
poker_player_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 27600.0
**Execution Time**: 0.40 seconds
**Reliability**: high
**Analysis**: Gurobipy found an optimal solution quickly with the highest reliability due to its robust performance and industry reputation.

## 6. DOCplex Implementation

```python
# Complete DOCPLEX implementation

from docplex.mp.model import Model

def poker_player_optimization():
    """Optimize prize money allocation to maximize player satisfaction."""
    
    # 1. MODEL & DATA SETUP
    mdl = Model(name="poker_player")
    
    # Data: Satisfaction coefficients and constraints
    player_ids = [1, 2, 3]
    satisfaction_coefficients = [1.1, 0.9, 1.3]
    total_budget = 24000
    max_prize_per_player = 10000
    
    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(player_ids) == len(satisfaction_coefficients), "Array length mismatch"
    safe_range = range(len(player_ids))  # Safe indexing
    
    # 2. VARIABLES
    # Prize money allocation for each player
    prize_money = {i: mdl.continuous_var(name=f"x_{player_ids[i]}", lb=0) for i in safe_range}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize total satisfaction
    objective = mdl.sum(satisfaction_coefficients[i] * prize_money[i] for i in safe_range)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS - CORRECT SYNTAX PATTERNS
    
    # Total budget constraint
    total_allocation = mdl.sum(prize_money[i] for i in safe_range)
    mdl.add_constraint(total_allocation <= total_budget, ctname="total_budget")
    
    # Individual prize limit constraints
    for i in safe_range:
        mdl.add_constraint(prize_money[i] <= max_prize_per_player, ctname=f"max_prize_{player_ids[i]}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in safe_range:
            value = solution.get_value(prize_money[i])
            if value > 1e-6:
                print(f"Prize money for player {player_ids[i]}: {value:.2f}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")
    
    return mdl

# Run the optimization
poker_player_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 27600.0
**Execution Time**: 1.63 seconds
**Reliability**: high
**Analysis**: DOCplex also found the 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 poker_player_optimization():
    """Optimize prize money allocation for poker players to maximize satisfaction."""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Player satisfaction coefficients
    satisfaction_coefficients = [1.1, 0.9, 1.3]
    n_players = len(satisfaction_coefficients)
    
    # Budget and maximum prize constraints
    total_budget = 24000
    max_prize_per_player = 10000
    
    # CRITICAL: Validate array lengths before indexing
    assert len(satisfaction_coefficients) == n_players, "Array length mismatch"
    
    # 3. SETS (Pyomo way to define indices)
    model.I = pyo.RangeSet(1, n_players)  # 1-based indexing
    
    # 4. PARAMETERS (data containers)
    model.coefficient = pyo.Param(model.I, initialize={i+1: satisfaction_coefficients[i] for i in range(n_players)})
    
    # 5. VARIABLES
    model.x = pyo.Var(model.I, within=pyo.NonNegativeReals, bounds=(0, max_prize_per_player))
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.coefficient[i] * model.x[i] for i in model.I)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)
    
    # 7. CONSTRAINTS - CORRECT SYNTAX PATTERNS
    
    # Total budget constraint
    def budget_rule(model):
        return sum(model.x[i] for i in model.I) <= total_budget
    model.budget_constraint = pyo.Constraint(rule=budget_rule)
    
    # 8. SOLVING WITH GUROBI (your available solver)
    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):.3f}")
        
        # Extract variable values
        print("\nPrize Money Allocation:")
        for i in model.I:
            x_val = pyo.value(model.x[i])
            if x_val > 1e-6:  # Only print non-zero values
                print(f"Player {i}: Prize Money = {x_val:.2f}")
        
    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
poker_player_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 27600.0
**Execution Time**: 1.39 seconds
**Reliability**: high
**Analysis**: Pyomo achieved the optimal solution with a slightly faster execution time than DOCplex, indicating good performance.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | OPTIMAL | 27600.00 | 0.40s | N/A | N/A |
| Docplex | OPTIMAL | 27600.00 | 1.63s | N/A | N/A |
| Pyomo | OPTIMAL | 27600.00 | 1.39s | N/A | N/A |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Majority Vote Optimal Value**: 27600.0

### Final Recommendation
**Recommended Optimal Value**: 27600.0
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy is preferred due to its faster execution time and high reliability, making it suitable for real-time decision-making.

### Optimal Decision Variables
- **x_1** = 10000.000
  - *Business Meaning*: Prize money allocated to Player 2, maximizing their satisfaction coefficient.
- **x_2** = 10000.000
  - *Business Meaning*: Prize money allocated to Player 3, maximizing their satisfaction coefficient.
- **x_3** = 4000.000
  - *Business Meaning*: Resource allocation for x_3

### Business Interpretation
**Overall Strategy**: The optimal allocation maximizes player satisfaction within the budget constraints, ensuring the best use of resources.
**Objective Value Meaning**: The optimal objective value of 27600 represents the maximum total satisfaction achievable given the constraints.
**Resource Allocation Summary**: Allocate the maximum allowable prize to Players 1 and 2, and the remaining budget to Player 3.
**Implementation Recommendations**: Implement the prize distribution as calculated, ensuring compliance with budgetary and individual limits.