Iteration final - PROBLEM_DESCRIPTION
Sequence: 7
Timestamp: 2025-07-28 00:28:53

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 2), 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: assignment[player_id, team_id] is a binary variable indicating if player_id is assigned to team_id
- Operational parameters align with expected linear objective: maximize total_performance_score = ∑(performance_coefficient[player_id] × assignment[player_id, team_id])
- Business configuration includes: Maximum number of players allowed in a team (used for Used as a constraint in the optimization model), Maximum total age allowed for players in a team (used for Used as a constraint in the optimization model)
- 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": "school_player",
  "iteration": 2,
  "business_context": "A school district is optimizing the allocation of players to different school teams to maximize overall team performance, considering constraints such as age limits, team size, and school enrollment capacity.",
  "optimization_problem_description": "The goal is to maximize the total performance score of all school teams by optimally assigning players to teams. The performance score is based on player attributes such as age and position. Constraints include maximum team size, age restrictions, and school enrollment limits.",
  "optimization_formulation": {
    "objective": "maximize total_performance_score = \u2211(performance_coefficient[player_id] \u00d7 assignment[player_id, team_id])",
    "decision_variables": "assignment[player_id, team_id] is a binary variable indicating if player_id is assigned to team_id",
    "constraints": [
      "\u2211(assignment[player_id, team_id]) \u2264 max_team_size for each team_id",
      "\u2211(age[player_id] \u00d7 assignment[player_id, team_id]) \u2264 max_age_limit for each team_id"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "performance_coefficient[player_id]": {
        "currently_mapped_to": "PlayerPerformance.performance_coefficient",
        "mapping_adequacy": "good",
        "description": "Performance contribution of the player"
      }
    },
    "constraint_bounds": {
      "max_team_size[team_id]": {
        "currently_mapped_to": "TeamConstraints.max_team_size",
        "mapping_adequacy": "good",
        "description": "Maximum number of players allowed in a team"
      },
      "max_age_limit[team_id]": {
        "currently_mapped_to": "TeamConstraints.max_age_limit",
        "mapping_adequacy": "good",
        "description": "Maximum total age allowed for players in a team"
      }
    },
    "decision_variables": {
      "assignment[player_id, team_id]": {
        "currently_mapped_to": "PlayerAssignment.assignment",
        "mapping_adequacy": "good",
        "description": "Indicates if a player is assigned to a team",
        "variable_type": "binary"
      }
    }
  },
  "missing_optimization_requirements": [],
  "iteration_status": {
    "complete": true,
    "confidence": "high",
    "next_focus": "Ready for convergence"
  }
}

FINAL DATABASE SCHEMA:
```sql
-- Iteration 2 Database Schema
-- Objective: Schema changes include creating a new table for decision variables, modifying existing tables to include missing data, and updating business configuration logic for scalar parameters and formulas.

CREATE TABLE PlayerPerformance (
  player_id INTEGER,
  performance_coefficient FLOAT,
  age INTEGER
);

CREATE TABLE TeamConstraints (
  team_id INTEGER,
  max_team_size INTEGER,
  max_age_limit INTEGER
);

CREATE TABLE PlayerAssignment (
  player_id INTEGER,
  team_id INTEGER,
  assignment BOOLEAN
);

CREATE TABLE PlayerAge (
  player_id INTEGER,
  age INTEGER
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical school team sizes and age distributions, ensuring that constraints are realistic and the optimization problem is solvable.

-- Realistic data for PlayerPerformance
INSERT INTO PlayerPerformance (player_id, performance_coefficient, age) VALUES (1, 1.8, 16);
INSERT INTO PlayerPerformance (player_id, performance_coefficient, age) VALUES (2, 2.3, 17);
INSERT INTO PlayerPerformance (player_id, performance_coefficient, age) VALUES (3, 2.7, 18);

-- Realistic data for TeamConstraints
INSERT INTO TeamConstraints (team_id, max_team_size, max_age_limit) VALUES (1, 15, 100);
INSERT INTO TeamConstraints (team_id, max_team_size, max_age_limit) VALUES (2, 18, 110);
INSERT INTO TeamConstraints (team_id, max_team_size, max_age_limit) VALUES (3, 20, 120);

-- Realistic data for PlayerAssignment
INSERT INTO PlayerAssignment (player_id, team_id, assignment) VALUES (1, 1, True);
INSERT INTO PlayerAssignment (player_id, team_id, assignment) VALUES (2, 2, True);
INSERT INTO PlayerAssignment (player_id, team_id, assignment) VALUES (3, 3, True);

-- Realistic data for PlayerAge
INSERT INTO PlayerAge (player_id, age) VALUES (1, 16);
INSERT INTO PlayerAge (player_id, age) VALUES (2, 17);
INSERT INTO PlayerAge (player_id, age) VALUES (3, 18);


```

DATA DICTIONARY:
{
  "tables": {
    "PlayerPerformance": {
      "business_purpose": "Stores performance coefficients and age for each player",
      "optimization_role": "objective_coefficients",
      "columns": {
        "player_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each player",
          "optimization_purpose": "Links player to their performance coefficient",
          "sample_values": "1, 2, 3"
        },
        "performance_coefficient": {
          "data_type": "FLOAT",
          "business_meaning": "Performance contribution of the player",
          "optimization_purpose": "Used in the objective function",
          "sample_values": "1.5, 2.0, 2.5"
        },
        "age": {
          "data_type": "INTEGER",
          "business_meaning": "Age of the player",
          "optimization_purpose": "Used in age-based constraints",
          "sample_values": "16, 17, 18"
        }
      }
    },
    "TeamConstraints": {
      "business_purpose": "Stores maximum team size and age limits for each team",
      "optimization_role": "constraint_bounds",
      "columns": {
        "team_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each team",
          "optimization_purpose": "Links constraints to specific teams",
          "sample_values": "1, 2, 3"
        },
        "max_team_size": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of players allowed in a team",
          "optimization_purpose": "Constraint in optimization",
          "sample_values": "15, 20, 25"
        },
        "max_age_limit": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum total age allowed for players in a team",
          "optimization_purpose": "Constraint in optimization",
          "sample_values": "100, 120, 140"
        }
      }
    },
    "PlayerAssignment": {
      "business_purpose": "Stores binary decision variables for player assignments",
      "optimization_role": "decision_variables",
      "columns": {
        "player_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each player",
          "optimization_purpose": "Links player to their team assignment",
          "sample_values": "1, 2, 3"
        },
        "team_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each team",
          "optimization_purpose": "Links team to player assignments",
          "sample_values": "1, 2, 3"
        },
        "assignment": {
          "data_type": "BOOLEAN",
          "business_meaning": "Indicates if a player is assigned to a team",
          "optimization_purpose": "Binary decision variable in optimization",
          "sample_values": "true, false"
        }
      }
    },
    "PlayerAge": {
      "business_purpose": "Stores age information for each player",
      "optimization_role": "business_data",
      "columns": {
        "player_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each player",
          "optimization_purpose": "Links player to their age",
          "sample_values": "1, 2, 3"
        },
        "age": {
          "data_type": "INTEGER",
          "business_meaning": "Age of the player",
          "optimization_purpose": "Used in age-based constraints",
          "sample_values": "16, 17, 18"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "max_team_size": {
    "data_type": "INTEGER",
    "business_meaning": "Maximum number of players allowed in a team",
    "optimization_role": "Used as a constraint in the optimization model",
    "configuration_type": "scalar_parameter",
    "value": 18,
    "business_justification": "A team size of 18 allows for flexibility in player assignments while maintaining a manageable team size."
  },
  "max_age_limit": {
    "data_type": "INTEGER",
    "business_meaning": "Maximum total age allowed for players in a team",
    "optimization_role": "Used as a constraint in the optimization model",
    "configuration_type": "scalar_parameter",
    "value": 110,
    "business_justification": "An age limit of 110 ensures that teams can include a mix of younger and older players, balancing experience and potential."
  }
}

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: school_player

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: assignment[player_id, team_id] is a binary variable indicating if player_id is assigned to team_id
- Operational parameters align with expected linear objective: maximize total_performance_score = ∑(performance_coefficient[player_id] × assignment[player_id, team_id])
- Business configuration includes: Maximum number of players allowed in a team (used for Used as a constraint in the optimization model), Maximum total age allowed for players in a team (used for Used as a constraint in the optimization model)
- 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 total_performance_score = ∑(performance_coefficient[player_id] × assignment[player_id, team_id])
- 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: ['∑(assignment[player_id, team_id]) ≤ max_team_size for each team_id', '∑(age[player_id] × assignment[player_id, team_id]) ≤ max_age_limit for each team_id'] (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 2 Database Schema
-- Objective: Schema changes include creating a new table for decision variables, modifying existing tables to include missing data, and updating business configuration logic for scalar parameters and formulas.

CREATE TABLE PlayerPerformance (
  player_id INTEGER,
  performance_coefficient FLOAT,
  age INTEGER
);

CREATE TABLE TeamConstraints (
  team_id INTEGER,
  max_team_size INTEGER,
  max_age_limit INTEGER
);

CREATE TABLE PlayerAssignment (
  player_id INTEGER,
  team_id INTEGER,
  assignment BOOLEAN
);

CREATE TABLE PlayerAge (
  player_id INTEGER,
  age 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 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical school team sizes and age distributions, ensuring that constraints are realistic and the optimization problem is solvable.

-- Realistic data for PlayerPerformance
INSERT INTO PlayerPerformance (player_id, performance_coefficient, age) VALUES (1, 1.8, 16);
INSERT INTO PlayerPerformance (player_id, performance_coefficient, age) VALUES (2, 2.3, 17);
INSERT INTO PlayerPerformance (player_id, performance_coefficient, age) VALUES (3, 2.7, 18);

-- Realistic data for TeamConstraints
INSERT INTO TeamConstraints (team_id, max_team_size, max_age_limit) VALUES (1, 15, 100);
INSERT INTO TeamConstraints (team_id, max_team_size, max_age_limit) VALUES (2, 18, 110);
INSERT INTO TeamConstraints (team_id, max_team_size, max_age_limit) VALUES (3, 20, 120);

-- Realistic data for PlayerAssignment
INSERT INTO PlayerAssignment (player_id, team_id, assignment) VALUES (1, 1, True);
INSERT INTO PlayerAssignment (player_id, team_id, assignment) VALUES (2, 2, True);
INSERT INTO PlayerAssignment (player_id, team_id, assignment) VALUES (3, 3, True);

-- Realistic data for PlayerAge
INSERT INTO PlayerAge (player_id, age) VALUES (1, 16);
INSERT INTO PlayerAge (player_id, age) VALUES (2, 17);
INSERT INTO PlayerAge (player_id, age) VALUES (3, 18);


```

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

Response:
# Complete Optimization Problem and Solution: school_player

## 1. Problem Context and Goals

### Context  
The school district is focused on optimizing the allocation of players to various school teams to enhance overall team performance. The decision-making process involves assigning players to teams, where each assignment is represented by a binary decision variable indicating whether a player is assigned to a specific team. The primary objective is to maximize the total performance score of all teams, which is calculated based on the performance contributions of individual players. 

Operational parameters include constraints such as the maximum number of players allowed in each team and the total age limit for players within a team. These constraints ensure that teams are balanced and adhere to school policies. The maximum team size is set to allow flexibility in player assignments while maintaining a manageable team size, and the age limit ensures a mix of younger and older players, balancing experience and potential.

### Goals  
The optimization goal is to maximize the total performance score of the school teams. This score is determined by summing the performance contributions of players assigned to teams. The success of this optimization is measured by the total performance score, which reflects the effectiveness of player assignments in enhancing team performance. The goal is articulated in natural language to emphasize the linear nature of the optimization objective, focusing on maximizing the sum of performance contributions from assigned players.

## 2. Constraints    

The optimization problem is subject to several constraints that ensure feasible and balanced team compositions:

- Each team has a maximum number of players that can be assigned, ensuring that team sizes remain manageable and within school policy limits.
- The total age of players assigned to each team must not exceed a specified limit, promoting a balanced mix of player ages within teams.

These constraints are described in business terms that naturally align with linear mathematical formulations, ensuring that the optimization problem remains linear and solvable.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 2 Database Schema
-- Objective: Schema changes include creating a new table for decision variables, modifying existing tables to include missing data, and updating business configuration logic for scalar parameters and formulas.

CREATE TABLE PlayerPerformance (
  player_id INTEGER,
  performance_coefficient FLOAT,
  age INTEGER
);

CREATE TABLE TeamConstraints (
  team_id INTEGER,
  max_team_size INTEGER,
  max_age_limit INTEGER
);

CREATE TABLE PlayerAssignment (
  player_id INTEGER,
  team_id INTEGER,
  assignment BOOLEAN
);

CREATE TABLE PlayerAge (
  player_id INTEGER,
  age INTEGER
);
```

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

- **PlayerPerformance**: This table captures the performance contributions and ages of players. The performance coefficient represents each player's contribution to the team's performance, which is used in calculating the total performance score.
  
- **TeamConstraints**: This table defines the constraints for each team, including the maximum number of players allowed and the total age limit. These constraints ensure that teams are balanced and adhere to school policies.

- **PlayerAssignment**: This table records the binary decision variables indicating player assignments to teams. Each entry specifies whether a player is assigned to a particular team, forming the basis for the optimization decision-making process.

- **PlayerAge**: This table stores the age information for each player, which is used in age-based constraints to ensure a balanced mix of player ages within teams.

### Current Stored Values  
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical school team sizes and age distributions, ensuring that constraints are realistic and the optimization problem is solvable.

-- Realistic data for PlayerPerformance
INSERT INTO PlayerPerformance (player_id, performance_coefficient, age) VALUES (1, 1.8, 16);
INSERT INTO PlayerPerformance (player_id, performance_coefficient, age) VALUES (2, 2.3, 17);
INSERT INTO PlayerPerformance (player_id, performance_coefficient, age) VALUES (3, 2.7, 18);

-- Realistic data for TeamConstraints
INSERT INTO TeamConstraints (team_id, max_team_size, max_age_limit) VALUES (1, 15, 100);
INSERT INTO TeamConstraints (team_id, max_team_size, max_age_limit) VALUES (2, 18, 110);
INSERT INTO TeamConstraints (team_id, max_team_size, max_age_limit) VALUES (3, 20, 120);

-- Realistic data for PlayerAssignment
INSERT INTO PlayerAssignment (player_id, team_id, assignment) VALUES (1, 1, True);
INSERT INTO PlayerAssignment (player_id, team_id, assignment) VALUES (2, 2, True);
INSERT INTO PlayerAssignment (player_id, team_id, assignment) VALUES (3, 3, True);

-- Realistic data for PlayerAge
INSERT INTO PlayerAge (player_id, age) VALUES (1, 16);
INSERT INTO PlayerAge (player_id, age) VALUES (2, 17);
INSERT INTO PlayerAge (player_id, age) VALUES (3, 18);
```
