Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-25 22:48:34

Prompt:
You are a business analyst creating structured optimization problem documentation.

DATA SOURCES EXPLANATION:
- FINAL OR ANALYSIS: Final converged optimization problem from alternating process (iteration 1), contains business context and schema mapping evaluation
- DATABASE SCHEMA: Current database structure after iterative adjustments  
- DATA DICTIONARY: Business meanings and optimization roles of tables and columns
- CURRENT STORED VALUES: Realistic business data generated by triple expert (business + data + optimization)
- BUSINESS CONFIGURATION: Scalar parameters and business logic formulas separated from table data

CRITICAL REQUIREMENTS: 
- Ensure problem description naturally leads to LINEAR or MIXED-INTEGER optimization formulation
- Make business context consistent with the intended decision variables and objectives
- Align constraint descriptions with expected mathematical constraints
- Ensure data descriptions map clearly to expected coefficient sources
- Maintain business authenticity while fixing mathematical consistency issues
- Avoid business scenarios that would naturally require nonlinear relationships (variable products, divisions, etc.)

AUTO-EXTRACTED CONTEXT REQUIREMENTS:
- Business decisions match expected decision variables: x_ij: binary variable indicating if player i is assigned to position j in a team.
- Operational parameters align with expected linear objective: maximize ∑(w_ij * x_ij) where w_ij is the historical win rate of player i in position j and x_ij is a binary decision variable indicating if player i is assigned to position j in a team.
- Business configuration includes: Minimum number of players required for a team (used for Lower bound constraint for team size), Maximum number of players allowed for a team (used for Upper bound constraint for team size), Total number of players the league can accommodate (used for Upper bound constraint for total players)
- Business logic formulas to express in natural language: Historical win rate of player i in position j (calculation method for Coefficient in the objective function)
- Use natural language to precisely describe linear mathematical relationships
- NO mathematical formulas, equations, or symbolic notation
- Present data as current operational information
- Focus on precise operational decision-making that leads to linear formulations
- Resource limitations match expected linear constraints
- Avoid scenarios requiring variable products, divisions, or other nonlinear relationships
- Include specific operational parameters that map to expected coefficient sources
- Reference business configuration parameters where appropriate

FINAL OR ANALYSIS:
{
  "database_id": "match_season",
  "iteration": 1,
  "business_context": "A sports league aims to optimize the allocation of players to teams to maximize the overall performance of the league. The performance is measured by the total wins of all teams, considering the players' historical performance and their positions.",
  "optimization_problem_description": "Maximize the total wins across all teams by assigning players to teams based on their historical performance and positions. Constraints include ensuring each player is assigned to only one team, each team has a minimum and maximum number of players, and the total number of players assigned does not exceed the league's capacity.",
  "optimization_formulation": {
    "objective": "maximize \u2211(w_ij * x_ij) where w_ij is the historical win rate of player i in position j and x_ij is a binary decision variable indicating if player i is assigned to position j in a team.",
    "decision_variables": "x_ij: binary variable indicating if player i is assigned to position j in a team.",
    "constraints": "1. \u2211(x_ij) \u2264 1 for all i (each player is assigned to at most one position), 2. min_players_j \u2264 \u2211(x_ij) \u2264 max_players_j for all j (team size constraints), 3. \u2211(x_ij) \u2264 league_capacity (total players assigned does not exceed league capacity)."
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "w_ij[i,j]": {
        "currently_mapped_to": "player_position.historical_win_rate",
        "mapping_adequacy": "good",
        "description": "Historical win rate of player i in position j."
      }
    },
    "constraint_bounds": {
      "min_players_j[j]": {
        "currently_mapped_to": "team_constraints.min_players",
        "mapping_adequacy": "good",
        "description": "Minimum number of players required for team j."
      },
      "max_players_j[j]": {
        "currently_mapped_to": "team_constraints.max_players",
        "mapping_adequacy": "good",
        "description": "Maximum number of players allowed for team j."
      },
      "league_capacity": {
        "currently_mapped_to": "league_capacity.capacity",
        "mapping_adequacy": "good",
        "description": "Total number of players the league can accommodate."
      }
    },
    "decision_variables": {
      "x_ij[i,j]": {
        "currently_mapped_to": "player_position.player_assigned",
        "mapping_adequacy": "good",
        "description": "Binary decision variable indicating if player i is assigned to position j in a team.",
        "variable_type": "binary"
      }
    }
  },
  "missing_optimization_requirements": [],
  "iteration_status": {
    "complete": true,
    "confidence": "high",
    "next_focus": "Ready for convergence"
  }
}

FINAL 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
);


```

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);


```

DATA DICTIONARY:
{
  "tables": {
    "player_position": {
      "business_purpose": "Mapping of players to positions and their historical win rates",
      "optimization_role": "decision_variables",
      "columns": {
        "player_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for a player",
          "optimization_purpose": "Identifier for player i",
          "sample_values": "1, 2, 3"
        },
        "position_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for a position",
          "optimization_purpose": "Identifier for position j",
          "sample_values": "1, 2, 3"
        },
        "historical_win_rate": {
          "data_type": "FLOAT",
          "business_meaning": "Historical win rate of player i in position j",
          "optimization_purpose": "Coefficient in the objective function",
          "sample_values": "0.75, 0.80, 0.85"
        },
        "player_assigned": {
          "data_type": "BOOLEAN",
          "business_meaning": "Indicates if player i is assigned to position j in a team",
          "optimization_purpose": "Binary decision variable x_ij",
          "sample_values": "true, false"
        }
      }
    },
    "team_constraints": {
      "business_purpose": "Minimum and maximum number of players allowed per team",
      "optimization_role": "constraint_bounds",
      "columns": {
        "team_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for a team",
          "optimization_purpose": "Identifier for team j",
          "sample_values": "1, 2, 3"
        },
        "min_players": {
          "data_type": "INTEGER",
          "business_meaning": "Minimum number of players required for team j",
          "optimization_purpose": "Lower bound constraint for team size",
          "sample_values": "5, 6, 7"
        },
        "max_players": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of players allowed for team j",
          "optimization_purpose": "Upper bound constraint for team size",
          "sample_values": "10, 11, 12"
        }
      }
    },
    "league_capacity": {
      "business_purpose": "Total number of players the league can accommodate",
      "optimization_role": "constraint_bounds",
      "columns": {
        "league_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for the league",
          "optimization_purpose": "Identifier for the league",
          "sample_values": "1"
        },
        "capacity": {
          "data_type": "INTEGER",
          "business_meaning": "Total number of players the league can accommodate",
          "optimization_purpose": "Upper bound constraint for total players",
          "sample_values": "100"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "min_players": {
    "data_type": "INTEGER",
    "business_meaning": "Minimum number of players required for a team",
    "optimization_role": "Lower bound constraint for team size",
    "configuration_type": "scalar_parameter",
    "value": 5,
    "business_justification": "A minimum of 5 players per team ensures that each team has enough members to compete effectively."
  },
  "max_players": {
    "data_type": "INTEGER",
    "business_meaning": "Maximum number of players allowed for a team",
    "optimization_role": "Upper bound constraint for team size",
    "configuration_type": "scalar_parameter",
    "value": 10,
    "business_justification": "A maximum of 10 players per team prevents overcrowding and maintains team balance."
  },
  "league_capacity": {
    "data_type": "INTEGER",
    "business_meaning": "Total number of players the league can accommodate",
    "optimization_role": "Upper bound constraint for total players",
    "configuration_type": "scalar_parameter",
    "value": 100,
    "business_justification": "A league capacity of 100 players allows for a reasonable number of teams and players, ensuring the league operates efficiently."
  },
  "historical_win_rate_formula": {
    "data_type": "STRING",
    "business_meaning": "Historical win rate of player i in position j",
    "optimization_role": "Coefficient in the objective function",
    "configuration_type": "business_logic_formula",
    "formula_expression": "w_ij = (wins_ij / total_games_ij)"
  }
}

Business Configuration Design: 
Our system separates business logic design from value determination:
- Configuration Logic (business_configuration_logic.json): Templates designed by data engineers with sample_value for scalars and actual formulas for business logic
- Configuration Values (business_configuration.json): Realistic values determined by domain experts for scalar parameters only
- Design Rationale: Ensures business logic consistency while allowing flexible parameter tuning


TASK: Create structured markdown documentation for SECTIONS 1-3 ONLY (Problem Description).

EXACT MARKDOWN STRUCTURE TO FOLLOW:

# Complete Optimization Problem and Solution: match_season

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: x_ij: binary variable indicating if player i is assigned to position j in a team.
- Operational parameters align with expected linear objective: maximize ∑(w_ij * x_ij) where w_ij is the historical win rate of player i in position j and x_ij is a binary decision variable indicating if player i is assigned to position j in a team.
- Business configuration includes: Minimum number of players required for a team (used for Lower bound constraint for team size), Maximum number of players allowed for a team (used for Upper bound constraint for team size), Total number of players the league can accommodate (used for Upper bound constraint for total players)
- Business logic formulas to express in natural language: Historical win rate of player i in position j (calculation method for Coefficient in the objective function)
- Use natural language to precisely describe linear mathematical relationships
- NO mathematical formulas, equations, or symbolic notation
- Present data as current operational information
- Focus on precise operational decision-making that leads to linear formulations
- Resource limitations match expected linear constraints
- Avoid scenarios requiring variable products, divisions, or other nonlinear relationships
- Include specific operational parameters that map to expected coefficient sources
- Reference business configuration parameters where appropriate
- CRITICAL: Include ALL business configuration information (scalar parameters AND business logic formulas) in natural business language

### Goals  
[Regenerate goals that clearly lead to LINEAR mathematical objective:]
- Optimization goal: maximize
- Metric to optimize: maximize ∑(w_ij * x_ij) where w_ij is the historical win rate of player i in position j and x_ij is a binary decision variable indicating if player i is assigned to position j in a team.
- Success measurement aligned with expected coefficient sources
- Use natural language to precisely describe linear optimization goal
- NO mathematical formulas, equations, or symbolic notation

## 2. Constraints    

[Regenerate constraints that directly match expected LINEAR mathematical constraints:]
- Expected constraint: 1. ∑(x_ij) ≤ 1 for all i (each player is assigned to at most one position), 2. min_players_j ≤ ∑(x_ij) ≤ max_players_j for all j (team size constraints), 3. ∑(x_ij) ≤ league_capacity (total players assigned does not exceed league capacity). (Form: Standard constraint form based on business requirements)

[Each constraint should be described in business terms that naturally lead to LINEAR mathematical forms (no variable products or divisions)]

## 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  
[Create comprehensive business-oriented data dictionary mapping tables and columns to their business purposes and optimization roles - NOT technical database terms. Base this on the data_dictionary provided but rewrite in business language that clearly connects to the expected linear mathematical formulation]

### 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);


```

STOP HERE - Do not include section 4 (Mathematical Optimization Formulation).

Response:
# 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);
```
