Iteration final - PROBLEM_DESCRIPTION
Sequence: 7
Timestamp: 2025-07-25 22:31:00

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: x_i ∈ {0,1} for each player i, indicating whether the player is selected
- Operational parameters align with expected linear objective: maximize ∑(Earnings_i × x_i) where x_i is a binary decision variable indicating whether player i is selected
- Business logic formulas to express in natural language: Ensures selected players represent at least three different nationalities (calculation method for Constraint to enforce diversity in nationalities)
- 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": "poker_player",
  "iteration": 2,
  "business_context": "A poker tournament organizer wants to maximize the total earnings of selected poker players while ensuring diversity in nationalities and limiting the number of players with low money ranks.",
  "optimization_problem_description": "Select a subset of poker players to maximize their total earnings, ensuring that the selected players represent at least three different nationalities and that no more than 20% of the selected players have a money rank below 50.",
  "optimization_formulation": {
    "objective": "maximize \u2211(Earnings_i \u00d7 x_i) where x_i is a binary decision variable indicating whether player i is selected",
    "decision_variables": "x_i \u2208 {0,1} for each player i, indicating whether the player is selected",
    "constraints": [
      "\u2211(x_i) \u2265 3 for each distinct nationality",
      "\u2211(x_i for Money_Rank_i < 50) \u2264 0.2 \u00d7 \u2211(x_i)"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "Earnings_i": {
        "currently_mapped_to": "poker_player.Earnings",
        "mapping_adequacy": "good",
        "description": "Earnings of each poker player used as coefficients in the objective function"
      }
    },
    "constraint_bounds": {
      "Nationality_Diversity": {
        "currently_mapped_to": "business_configuration_logic.nationality_diversity_formula",
        "mapping_adequacy": "good",
        "description": "Ensures selected players represent at least three different nationalities"
      },
      "Money_Rank_Limit": {
        "currently_mapped_to": "poker_player.Money_Rank",
        "mapping_adequacy": "good",
        "description": "Limits the number of selected players with a money rank below 50"
      }
    },
    "decision_variables": {
      "x_i": {
        "currently_mapped_to": "poker_player.Selected",
        "mapping_adequacy": "good",
        "description": "Binary decision variable indicating whether player i is selected",
        "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: Added Nationality column to poker_player table to enforce diversity constraint. Updated business configuration logic to include nationality diversity formula.

CREATE TABLE poker_player (
  Earnings FLOAT,
  Money_Rank INTEGER,
  Selected BOOLEAN,
  Nationality STRING
);

CREATE TABLE player_selection (
  Player_ID INTEGER,
  Selected BOOLEAN
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic poker tournament scenarios, ensuring diversity in nationalities and a mix of money ranks to make the optimization problem meaningful.

-- Realistic data for poker_player
INSERT INTO poker_player (Earnings, Money_Rank, Selected, Nationality) VALUES (1200.0, 45, False, 'USA');
INSERT INTO poker_player (Earnings, Money_Rank, Selected, Nationality) VALUES (1500.0, 55, False, 'Canada');
INSERT INTO poker_player (Earnings, Money_Rank, Selected, Nationality) VALUES (2000.0, 60, False, 'UK');
INSERT INTO poker_player (Earnings, Money_Rank, Selected, Nationality) VALUES (1000.0, 50, False, 'Australia');
INSERT INTO poker_player (Earnings, Money_Rank, Selected, Nationality) VALUES (1800.0, 58, False, 'Germany');

-- Realistic data for player_selection
INSERT INTO player_selection (Player_ID, Selected) VALUES (1, False);
INSERT INTO player_selection (Player_ID, Selected) VALUES (2, False);
INSERT INTO player_selection (Player_ID, Selected) VALUES (3, False);
INSERT INTO player_selection (Player_ID, Selected) VALUES (4, False);
INSERT INTO player_selection (Player_ID, Selected) VALUES (5, False);


```

DATA DICTIONARY:
{
  "tables": {
    "poker_player": {
      "business_purpose": "Stores information about poker players",
      "optimization_role": "objective_coefficients/constraint_bounds/business_data",
      "columns": {
        "Earnings": {
          "data_type": "FLOAT",
          "business_meaning": "Earnings of each poker player",
          "optimization_purpose": "Coefficient in the objective function",
          "sample_values": "1000.0, 1500.0, 2000.0"
        },
        "Money_Rank": {
          "data_type": "INTEGER",
          "business_meaning": "Money rank of each poker player",
          "optimization_purpose": "Used in constraint to limit players with money rank below 50",
          "sample_values": "45, 55, 60"
        },
        "Selected": {
          "data_type": "BOOLEAN",
          "business_meaning": "Binary decision variable indicating whether player is selected",
          "optimization_purpose": "Decision variable in the optimization model",
          "sample_values": "true, false"
        },
        "Nationality": {
          "data_type": "STRING",
          "business_meaning": "Nationality of each poker player",
          "optimization_purpose": "Used in constraint to enforce diversity in nationalities",
          "sample_values": "USA, Canada, UK"
        }
      }
    },
    "player_selection": {
      "business_purpose": "Binary decision variables indicating whether each player is selected",
      "optimization_role": "decision_variables",
      "columns": {
        "Player_ID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each player",
          "optimization_purpose": "Links to poker_player table",
          "sample_values": "1, 2, 3"
        },
        "Selected": {
          "data_type": "BOOLEAN",
          "business_meaning": "Binary decision variable indicating whether player is selected",
          "optimization_purpose": "Decision variable in the optimization model",
          "sample_values": "true, false"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "nationality_diversity_formula": {
    "data_type": "STRING",
    "business_meaning": "Ensures selected players represent at least three different nationalities",
    "optimization_role": "Constraint to enforce diversity in nationalities",
    "configuration_type": "business_logic_formula",
    "formula_expression": "COUNT(DISTINCT Nationality) >= 3"
  }
}

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: poker_player

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: x_i ∈ {0,1} for each player i, indicating whether the player is selected
- Operational parameters align with expected linear objective: maximize ∑(Earnings_i × x_i) where x_i is a binary decision variable indicating whether player i is selected
- Business logic formulas to express in natural language: Ensures selected players represent at least three different nationalities (calculation method for Constraint to enforce diversity in nationalities)
- 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 ∑(Earnings_i × x_i) where x_i is a binary decision variable indicating whether player i is selected
- 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: ['∑(x_i) ≥ 3 for each distinct nationality', '∑(x_i for Money_Rank_i < 50) ≤ 0.2 × ∑(x_i)'] (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: Added Nationality column to poker_player table to enforce diversity constraint. Updated business configuration logic to include nationality diversity formula.

CREATE TABLE poker_player (
  Earnings FLOAT,
  Money_Rank INTEGER,
  Selected BOOLEAN,
  Nationality STRING
);

CREATE TABLE player_selection (
  Player_ID INTEGER,
  Selected BOOLEAN
);


```

### 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 realistic poker tournament scenarios, ensuring diversity in nationalities and a mix of money ranks to make the optimization problem meaningful.

-- Realistic data for poker_player
INSERT INTO poker_player (Earnings, Money_Rank, Selected, Nationality) VALUES (1200.0, 45, False, 'USA');
INSERT INTO poker_player (Earnings, Money_Rank, Selected, Nationality) VALUES (1500.0, 55, False, 'Canada');
INSERT INTO poker_player (Earnings, Money_Rank, Selected, Nationality) VALUES (2000.0, 60, False, 'UK');
INSERT INTO poker_player (Earnings, Money_Rank, Selected, Nationality) VALUES (1000.0, 50, False, 'Australia');
INSERT INTO poker_player (Earnings, Money_Rank, Selected, Nationality) VALUES (1800.0, 58, False, 'Germany');

-- Realistic data for player_selection
INSERT INTO player_selection (Player_ID, Selected) VALUES (1, False);
INSERT INTO player_selection (Player_ID, Selected) VALUES (2, False);
INSERT INTO player_selection (Player_ID, Selected) VALUES (3, False);
INSERT INTO player_selection (Player_ID, Selected) VALUES (4, False);
INSERT INTO player_selection (Player_ID, Selected) VALUES (5, False);


```

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

Response:
# Complete Optimization Problem and Solution: poker_player

## 1. Problem Context and Goals

### Context  
A poker tournament organizer is tasked with selecting a subset of poker players to maximize the total earnings of the selected players. The selection process must ensure diversity in nationalities, requiring that the chosen players represent at least three different countries. Additionally, the organizer aims to limit the number of selected players with low money ranks, ensuring that no more than 20% of the selected players have a money rank below 50.  

The decision to select a player is represented by a binary choice: each player is either selected (1) or not selected (0). The total earnings of the selected players are calculated by summing the earnings of each chosen player. The organizer uses the players' earnings as the primary metric to optimize, while the constraints on nationality diversity and money rank are enforced to maintain a balanced and competitive tournament lineup.  

The business logic for nationality diversity is defined by a configuration that ensures the selected players come from at least three distinct nationalities. This logic is applied as a constraint in the optimization process. Similarly, the money rank constraint is derived from the players' current money ranks, ensuring that the selection adheres to the 20% limit for low-ranked players.  

### Goals  
The primary goal of this optimization problem is to maximize the total earnings of the selected poker players. This is achieved by selecting a subset of players whose combined earnings are as high as possible, while adhering to the constraints on nationality diversity and money rank.  

Success is measured by the total earnings of the selected players, which directly corresponds to the sum of the earnings of each chosen player. The optimization process ensures that the selection meets the predefined constraints, resulting in a lineup that is both financially optimal and operationally feasible.  

## 2. Constraints  

1. **Nationality Diversity Constraint**: The selected players must represent at least three different nationalities. This ensures a diverse lineup that reflects the global nature of the tournament.  

2. **Money Rank Constraint**: No more than 20% of the selected players can have a money rank below 50. This ensures that the majority of the selected players are competitively ranked, maintaining the tournament's prestige and competitiveness.  

These constraints are designed to align with the business requirements and are expressed in a way that naturally leads to linear mathematical forms, ensuring the optimization problem remains linear and tractable.  

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 2 Database Schema
-- Objective: Added Nationality column to poker_player table to enforce diversity constraint. Updated business configuration logic to include nationality diversity formula.

CREATE TABLE poker_player (
  Earnings FLOAT,
  Money_Rank INTEGER,
  Selected BOOLEAN,
  Nationality STRING
);

CREATE TABLE player_selection (
  Player_ID INTEGER,
  Selected BOOLEAN
);
```

### Data Dictionary  
- **poker_player**: Stores information about poker players, including their earnings, money rank, selection status, and nationality.  
  - **Earnings**: The earnings of each poker player, used as coefficients in the objective function to maximize total earnings.  
  - **Money_Rank**: The money rank of each player, used to enforce the constraint limiting the number of low-ranked players.  
  - **Selected**: A binary indicator (true/false) representing whether the player is selected, serving as the decision variable in the optimization model.  
  - **Nationality**: The nationality of each player, used to enforce the diversity constraint requiring at least three distinct nationalities.  

- **player_selection**: Tracks the selection status of each player, linking to the poker_player table.  
  - **Player_ID**: A unique identifier for each player, linking to the poker_player table.  
  - **Selected**: A binary indicator (true/false) representing whether the player is selected, serving as the decision variable in the optimization model.  

### Current Stored Values  
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic poker tournament scenarios, ensuring diversity in nationalities and a mix of money ranks to make the optimization problem meaningful.

-- Realistic data for poker_player
INSERT INTO poker_player (Earnings, Money_Rank, Selected, Nationality) VALUES (1200.0, 45, False, 'USA');
INSERT INTO poker_player (Earnings, Money_Rank, Selected, Nationality) VALUES (1500.0, 55, False, 'Canada');
INSERT INTO poker_player (Earnings, Money_Rank, Selected, Nationality) VALUES (2000.0, 60, False, 'UK');
INSERT INTO poker_player (Earnings, Money_Rank, Selected, Nationality) VALUES (1000.0, 50, False, 'Australia');
INSERT INTO poker_player (Earnings, Money_Rank, Selected, Nationality) VALUES (1800.0, 58, False, 'Germany');

-- Realistic data for player_selection
INSERT INTO player_selection (Player_ID, Selected) VALUES (1, False);
INSERT INTO player_selection (Player_ID, Selected) VALUES (2, False);
INSERT INTO player_selection (Player_ID, Selected) VALUES (3, False);
INSERT INTO player_selection (Player_ID, Selected) VALUES (4, False);
INSERT INTO player_selection (Player_ID, Selected) VALUES (5, False);
```
