Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-25 22:28:42

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_i ∈ {0, 1} for each song i, indicating whether the song is stored locally.
- Operational parameters align with expected linear objective: minimize ∑(file_size_i × x_i), where x_i is a binary decision variable indicating whether song i is stored locally.
- Business configuration includes: Minimum number of songs to store locally. (used for Constraint bound for total songs stored.), Minimum average rating of stored songs. (used for Constraint bound for average rating.), Maximum number of songs per artist to store locally. (used for Constraint bound for songs per artist.), Minimum number of songs per genre to store locally. (used for Constraint bound for songs per genre.)
- 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": "music_1",
  "iteration": 1,
  "business_context": "A music streaming platform aims to optimize its storage and bandwidth usage by selecting a subset of songs to store locally on servers, minimizing the total file size while ensuring a diverse and high-quality music library.",
  "optimization_problem_description": "Minimize the total file size of songs stored locally, subject to constraints on the minimum number of songs per genre, the minimum average rating of songs, and the maximum number of songs per artist.",
  "optimization_formulation": {
    "objective": "minimize \u2211(file_size_i \u00d7 x_i), where x_i is a binary decision variable indicating whether song i is stored locally.",
    "decision_variables": "x_i \u2208 {0, 1} for each song i, indicating whether the song is stored locally.",
    "constraints": [
      "\u2211(x_i) \u2265 min_songs (minimum total songs stored)",
      "\u2211(rating_i \u00d7 x_i) / \u2211(x_i) \u2265 min_avg_rating (minimum average rating)",
      "\u2211(x_i) \u2264 max_songs_per_artist for each artist (maximum songs per artist)",
      "\u2211(x_i) \u2265 min_songs_per_genre for each genre (minimum songs per genre)"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "file_size_i": {
        "currently_mapped_to": "files.file_size",
        "mapping_adequacy": "good",
        "description": "File size of song i in MB."
      }
    },
    "constraint_bounds": {
      "min_songs": {
        "currently_mapped_to": "business_configuration_logic.min_songs",
        "mapping_adequacy": "good",
        "description": "Minimum number of songs to store locally."
      },
      "min_avg_rating": {
        "currently_mapped_to": "business_configuration_logic.min_avg_rating",
        "mapping_adequacy": "good",
        "description": "Minimum average rating of stored songs."
      },
      "max_songs_per_artist": {
        "currently_mapped_to": "business_configuration_logic.max_songs_per_artist",
        "mapping_adequacy": "good",
        "description": "Maximum number of songs per artist to store locally."
      },
      "min_songs_per_genre": {
        "currently_mapped_to": "business_configuration_logic.min_songs_per_genre",
        "mapping_adequacy": "good",
        "description": "Minimum number of songs per genre to store locally."
      }
    },
    "decision_variables": {
      "x_i": {
        "currently_mapped_to": "decision_variables.is_stored_locally",
        "mapping_adequacy": "good",
        "description": "Binary decision variable indicating whether song i is stored locally.",
        "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 new tables for decision variables and constraints, moving scalar parameters to business_configuration_logic.json, and updating the data dictionary to reflect optimization mappings.

CREATE TABLE files (
  file_size INTEGER,
  is_stored_locally BOOLEAN
);

CREATE TABLE decision_variables (
  song_id INTEGER,
  is_stored_locally BOOLEAN
);

CREATE TABLE constraints (
  constraint_type STRING,
  constraint_value INTEGER
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic scenarios for a music streaming platform, considering typical file sizes, song ratings, and genre/artist distributions. Parameters were set to ensure a diverse and high-quality library while optimizing storage.

-- Realistic data for files
INSERT INTO files (file_size, is_stored_locally) VALUES (8, False);
INSERT INTO files (file_size, is_stored_locally) VALUES (12, True);
INSERT INTO files (file_size, is_stored_locally) VALUES (6, False);

-- Realistic data for decision_variables
INSERT INTO decision_variables (song_id, is_stored_locally) VALUES (1, False);
INSERT INTO decision_variables (song_id, is_stored_locally) VALUES (2, True);
INSERT INTO decision_variables (song_id, is_stored_locally) VALUES (3, False);

-- Realistic data for constraints
INSERT INTO constraints (constraint_type, constraint_value) VALUES ('genre', 10);
INSERT INTO constraints (constraint_type, constraint_value) VALUES ('artist', 5);
INSERT INTO constraints (constraint_type, constraint_value) VALUES ('rating', 4);


```

DATA DICTIONARY:
{
  "tables": {
    "files": {
      "business_purpose": "Stores metadata about songs, including file size and storage decision.",
      "optimization_role": "objective_coefficients, decision_variables",
      "columns": {
        "file_size": {
          "data_type": "INTEGER",
          "business_meaning": "File size of the song in MB.",
          "optimization_purpose": "Coefficient in the objective function.",
          "sample_values": "5, 10, 15"
        },
        "is_stored_locally": {
          "data_type": "BOOLEAN",
          "business_meaning": "Indicates whether the song is stored locally.",
          "optimization_purpose": "Decision variable x_i.",
          "sample_values": "true, false"
        }
      }
    },
    "decision_variables": {
      "business_purpose": "Binary decision variables for song storage.",
      "optimization_role": "decision_variables",
      "columns": {
        "song_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each song.",
          "optimization_purpose": "Index for decision variable x_i.",
          "sample_values": "1, 2, 3"
        },
        "is_stored_locally": {
          "data_type": "BOOLEAN",
          "business_meaning": "Indicates whether the song is stored locally.",
          "optimization_purpose": "Decision variable x_i.",
          "sample_values": "true, false"
        }
      }
    },
    "constraints": {
      "business_purpose": "Constraints for the optimization problem.",
      "optimization_role": "constraint_bounds",
      "columns": {
        "constraint_type": {
          "data_type": "STRING",
          "business_meaning": "Type of constraint (e.g., genre, artist).",
          "optimization_purpose": "Identifies the constraint type.",
          "sample_values": "genre, artist"
        },
        "constraint_value": {
          "data_type": "INTEGER",
          "business_meaning": "Value of the constraint (e.g., minimum songs per genre).",
          "optimization_purpose": "Bound for the constraint.",
          "sample_values": "10, 5"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "min_songs": {
    "data_type": "INTEGER",
    "business_meaning": "Minimum number of songs to store locally.",
    "optimization_role": "Constraint bound for total songs stored.",
    "configuration_type": "scalar_parameter",
    "value": 100,
    "business_justification": "Ensures a substantial library for users while optimizing storage."
  },
  "min_avg_rating": {
    "data_type": "FLOAT",
    "business_meaning": "Minimum average rating of stored songs.",
    "optimization_role": "Constraint bound for average rating.",
    "configuration_type": "scalar_parameter",
    "value": 4.0,
    "business_justification": "Maintains high-quality content to enhance user satisfaction."
  },
  "max_songs_per_artist": {
    "data_type": "INTEGER",
    "business_meaning": "Maximum number of songs per artist to store locally.",
    "optimization_role": "Constraint bound for songs per artist.",
    "configuration_type": "scalar_parameter",
    "value": 5,
    "business_justification": "Prevents overrepresentation of any single artist, promoting diversity."
  },
  "min_songs_per_genre": {
    "data_type": "INTEGER",
    "business_meaning": "Minimum number of songs per genre to store locally.",
    "optimization_role": "Constraint bound for songs per genre.",
    "configuration_type": "scalar_parameter",
    "value": 10,
    "business_justification": "Ensures a diverse music library across different genres."
  }
}

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: music_1

## 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 song i, indicating whether the song is stored locally.
- Operational parameters align with expected linear objective: minimize ∑(file_size_i × x_i), where x_i is a binary decision variable indicating whether song i is stored locally.
- Business configuration includes: Minimum number of songs to store locally. (used for Constraint bound for total songs stored.), Minimum average rating of stored songs. (used for Constraint bound for average rating.), Maximum number of songs per artist to store locally. (used for Constraint bound for songs per artist.), Minimum number of songs per genre to store locally. (used for Constraint bound for songs per genre.)
- 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: minimize
- Metric to optimize: minimize ∑(file_size_i × x_i), where x_i is a binary decision variable indicating whether song i is stored locally.
- 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) ≥ min_songs (minimum total songs stored)', '∑(rating_i × x_i) / ∑(x_i) ≥ min_avg_rating (minimum average rating)', '∑(x_i) ≤ max_songs_per_artist for each artist (maximum songs per artist)', '∑(x_i) ≥ min_songs_per_genre for each genre (minimum songs per genre)'] (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 new tables for decision variables and constraints, moving scalar parameters to business_configuration_logic.json, and updating the data dictionary to reflect optimization mappings.

CREATE TABLE files (
  file_size INTEGER,
  is_stored_locally BOOLEAN
);

CREATE TABLE decision_variables (
  song_id INTEGER,
  is_stored_locally BOOLEAN
);

CREATE TABLE constraints (
  constraint_type STRING,
  constraint_value 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 realistic scenarios for a music streaming platform, considering typical file sizes, song ratings, and genre/artist distributions. Parameters were set to ensure a diverse and high-quality library while optimizing storage.

-- Realistic data for files
INSERT INTO files (file_size, is_stored_locally) VALUES (8, False);
INSERT INTO files (file_size, is_stored_locally) VALUES (12, True);
INSERT INTO files (file_size, is_stored_locally) VALUES (6, False);

-- Realistic data for decision_variables
INSERT INTO decision_variables (song_id, is_stored_locally) VALUES (1, False);
INSERT INTO decision_variables (song_id, is_stored_locally) VALUES (2, True);
INSERT INTO decision_variables (song_id, is_stored_locally) VALUES (3, False);

-- Realistic data for constraints
INSERT INTO constraints (constraint_type, constraint_value) VALUES ('genre', 10);
INSERT INTO constraints (constraint_type, constraint_value) VALUES ('artist', 5);
INSERT INTO constraints (constraint_type, constraint_value) VALUES ('rating', 4);


```

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

Response:
# Complete Optimization Problem and Solution: music_1

## 1. Problem Context and Goals

### Context  
A music streaming platform is focused on optimizing its storage and bandwidth usage by strategically selecting a subset of songs to store locally on its servers. The platform must decide which songs to store locally, represented by binary decisions for each song. The primary goal is to minimize the total file size of the songs stored locally, ensuring efficient use of storage resources.  

The platform has established several operational parameters to maintain a diverse and high-quality music library. These include:  
- A minimum number of songs that must be stored locally to ensure a substantial library for users.  
- A minimum average rating for the stored songs to maintain high-quality content and enhance user satisfaction.  
- A maximum number of songs per artist to prevent overrepresentation of any single artist and promote diversity.  
- A minimum number of songs per genre to ensure a well-rounded music library across different genres.  

These parameters are defined in the business configuration and serve as the foundation for the optimization constraints. The decision-making process is linear, focusing solely on whether each song is stored locally, without involving complex relationships such as variable products or divisions.  

### Goals  
The optimization goal is to minimize the total file size of the songs stored locally. This is achieved by selecting a subset of songs that meets all operational constraints while keeping the combined file size as small as possible. Success is measured by the efficient use of storage resources, ensuring that the platform maintains a diverse and high-quality library without exceeding storage limitations.  

## 2. Constraints  

The optimization problem is subject to the following constraints, which ensure the platform meets its operational and quality requirements:  
1. **Minimum Total Songs Stored**: The total number of songs stored locally must meet or exceed a specified minimum. This ensures a substantial library for users.  
2. **Minimum Average Rating**: The average rating of the songs stored locally must meet or exceed a specified threshold. This maintains high-quality content and enhances user satisfaction.  
3. **Maximum Songs per Artist**: The number of songs stored locally for any single artist must not exceed a specified limit. This prevents overrepresentation of any artist and promotes diversity.  
4. **Minimum Songs per Genre**: The number of songs stored locally for each genre must meet or exceed a specified minimum. This ensures a diverse music library across different genres.  

These constraints are designed to align with linear mathematical forms, ensuring the optimization problem remains straightforward and computationally efficient.  

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating new tables for decision variables and constraints, moving scalar parameters to business_configuration_logic.json, and updating the data dictionary to reflect optimization mappings.

CREATE TABLE files (
  file_size INTEGER,
  is_stored_locally BOOLEAN
);

CREATE TABLE decision_variables (
  song_id INTEGER,
  is_stored_locally BOOLEAN
);

CREATE TABLE constraints (
  constraint_type STRING,
  constraint_value INTEGER
);
```

### Data Dictionary  
The data dictionary provides a clear mapping of tables and columns to their business purposes and optimization roles:  
- **files**: Stores metadata about songs, including file size and storage decisions.  
  - `file_size`: Represents the file size of the song in MB. This is used as a coefficient in the objective function to minimize total storage.  
  - `is_stored_locally`: Indicates whether the song is stored locally. This serves as the binary decision variable in the optimization problem.  
- **decision_variables**: Contains binary decision variables for song storage.  
  - `song_id`: Unique identifier for each song, used to index the decision variables.  
  - `is_stored_locally`: Indicates whether the song is stored locally, aligning with the decision variable in the optimization problem.  
- **constraints**: Defines the constraints for the optimization problem.  
  - `constraint_type`: Specifies the type of constraint (e.g., genre, artist).  
  - `constraint_value`: Represents the value of the constraint (e.g., minimum songs per genre), serving as the bound for the constraint in the optimization problem.  

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic scenarios for a music streaming platform, considering typical file sizes, song ratings, and genre/artist distributions. Parameters were set to ensure a diverse and high-quality library while optimizing storage.

-- Realistic data for files
INSERT INTO files (file_size, is_stored_locally) VALUES (8, False);
INSERT INTO files (file_size, is_stored_locally) VALUES (12, True);
INSERT INTO files (file_size, is_stored_locally) VALUES (6, False);

-- Realistic data for decision_variables
INSERT INTO decision_variables (song_id, is_stored_locally) VALUES (1, False);
INSERT INTO decision_variables (song_id, is_stored_locally) VALUES (2, True);
INSERT INTO decision_variables (song_id, is_stored_locally) VALUES (3, False);

-- Realistic data for constraints
INSERT INTO constraints (constraint_type, constraint_value) VALUES ('genre', 10);
INSERT INTO constraints (constraint_type, constraint_value) VALUES ('artist', 5);
INSERT INTO constraints (constraint_type, constraint_value) VALUES ('rating', 4);
```
