Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-27 22:12:03

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] is a binary variable indicating whether song i is stored (1) or not (0)
- Operational parameters align with expected linear objective: maximize total_rating = ∑(rating[i] * x[i])
- Business configuration includes: The total storage capacity available for storing songs (used for Used as a constraint bound in optimization model), The minimum total resolution quality required for stored songs (used for Used as a constraint bound in 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": "music_1",
  "iteration": 1,
  "business_context": "A music streaming service aims to optimize its storage allocation to maximize the total rating of songs available on the platform, considering the file size and resolution of each song, while adhering to storage capacity and resolution quality constraints.",
  "optimization_problem_description": "Maximize the total rating of songs stored on the platform, subject to storage capacity and resolution quality constraints. Each song has a file size and resolution, and the platform has a limited storage capacity.",
  "optimization_formulation": {
    "objective": "maximize total_rating = \u2211(rating[i] * x[i])",
    "decision_variables": "x[i] is a binary variable indicating whether song i is stored (1) or not (0)",
    "constraints": [
      "\u2211(file_size[i] * x[i]) <= total_storage_capacity",
      "\u2211(resolution[i] * x[i]) >= minimum_total_resolution"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "rating[i]": {
        "currently_mapped_to": "song.rating",
        "mapping_adequacy": "good",
        "description": "The rating of each song used to calculate the total rating of stored songs"
      }
    },
    "constraint_bounds": {
      "total_storage_capacity": {
        "currently_mapped_to": "business_configuration_logic.total_storage_capacity",
        "mapping_adequacy": "good",
        "description": "The total storage capacity available for storing songs"
      },
      "minimum_total_resolution": {
        "currently_mapped_to": "business_configuration_logic.minimum_total_resolution",
        "mapping_adequacy": "good",
        "description": "The minimum total resolution quality required for stored songs"
      }
    },
    "decision_variables": {
      "x[i]": {
        "currently_mapped_to": "song.f_id",
        "mapping_adequacy": "good",
        "description": "Binary variable indicating whether song i is stored",
        "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 adding tables for missing data requirements and updating configuration logic for scalar parameters and formulas.

CREATE TABLE song (
  f_id INTEGER,
  rating FLOAT,
  file_size INTEGER
);

CREATE TABLE song_file_data (
  song_id INTEGER,
  resolution INTEGER
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical file sizes and resolutions for music streaming services, ensuring a mix of high and low ratings to reflect a realistic catalog.

-- Realistic data for song
INSERT INTO song (f_id, rating, file_size) VALUES (1, 4.5, 5000);
INSERT INTO song (f_id, rating, file_size) VALUES (2, 3.8, 3000);
INSERT INTO song (f_id, rating, file_size) VALUES (3, 5.0, 7000);

-- Realistic data for song_file_data
INSERT INTO song_file_data (song_id, resolution) VALUES (1, 1080);
INSERT INTO song_file_data (song_id, resolution) VALUES (2, 720);
INSERT INTO song_file_data (song_id, resolution) VALUES (3, 480);


```

DATA DICTIONARY:
{
  "tables": {
    "song": {
      "business_purpose": "Stores metadata about each song",
      "optimization_role": "objective_coefficients",
      "columns": {
        "f_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each song",
          "optimization_purpose": "Used as a decision variable in optimization",
          "sample_values": "1, 2, 3"
        },
        "rating": {
          "data_type": "FLOAT",
          "business_meaning": "The rating of each song",
          "optimization_purpose": "Used to calculate the total rating of stored songs",
          "sample_values": "4.5, 3.8, 5.0"
        },
        "file_size": {
          "data_type": "INTEGER",
          "business_meaning": "The file size of each song",
          "optimization_purpose": "Used in storage capacity constraint",
          "sample_values": "5000, 3000, 7000"
        }
      }
    },
    "song_file_data": {
      "business_purpose": "Stores file size and resolution data for each song",
      "optimization_role": "business_data",
      "columns": {
        "song_id": {
          "data_type": "INTEGER",
          "business_meaning": "References the unique identifier of a song",
          "optimization_purpose": "Links file data to songs",
          "sample_values": "1, 2, 3"
        },
        "resolution": {
          "data_type": "INTEGER",
          "business_meaning": "The resolution quality of each song",
          "optimization_purpose": "Used in resolution quality constraint",
          "sample_values": "1080, 720, 480"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "total_storage_capacity": {
    "data_type": "INTEGER",
    "business_meaning": "The total storage capacity available for storing songs",
    "optimization_role": "Used as a constraint bound in optimization model",
    "configuration_type": "scalar_parameter",
    "value": 15000,
    "business_justification": "Reflects a realistic storage limit for a small to medium-sized music streaming service."
  },
  "minimum_total_resolution": {
    "data_type": "INTEGER",
    "business_meaning": "The minimum total resolution quality required for stored songs",
    "optimization_role": "Used as a constraint bound in optimization model",
    "configuration_type": "scalar_parameter",
    "value": 2000,
    "business_justification": "Ensures a minimum quality threshold across the catalog, aligning with industry standards."
  }
}

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] is a binary variable indicating whether song i is stored (1) or not (0)
- Operational parameters align with expected linear objective: maximize total_rating = ∑(rating[i] * x[i])
- Business configuration includes: The total storage capacity available for storing songs (used for Used as a constraint bound in optimization model), The minimum total resolution quality required for stored songs (used for Used as a constraint bound in 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_rating = ∑(rating[i] * x[i])
- 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: ['∑(file_size[i] * x[i]) <= total_storage_capacity', '∑(resolution[i] * x[i]) >= minimum_total_resolution'] (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 adding tables for missing data requirements and updating configuration logic for scalar parameters and formulas.

CREATE TABLE song (
  f_id INTEGER,
  rating FLOAT,
  file_size INTEGER
);

CREATE TABLE song_file_data (
  song_id INTEGER,
  resolution 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 file sizes and resolutions for music streaming services, ensuring a mix of high and low ratings to reflect a realistic catalog.

-- Realistic data for song
INSERT INTO song (f_id, rating, file_size) VALUES (1, 4.5, 5000);
INSERT INTO song (f_id, rating, file_size) VALUES (2, 3.8, 3000);
INSERT INTO song (f_id, rating, file_size) VALUES (3, 5.0, 7000);

-- Realistic data for song_file_data
INSERT INTO song_file_data (song_id, resolution) VALUES (1, 1080);
INSERT INTO song_file_data (song_id, resolution) VALUES (2, 720);
INSERT INTO song_file_data (song_id, resolution) VALUES (3, 480);


```

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

Response:
# Complete Optimization Problem and Solution: music_1

## 1. Problem Context and Goals

### Context  
The music streaming service is focused on optimizing its storage allocation to enhance the overall user experience by maximizing the total rating of songs available on the platform. Each song is evaluated based on its rating, file size, and resolution quality. The decision to store a song is represented by a binary choice, where storing a song is indicated by a value of 1, and not storing it is indicated by a value of 0. The platform operates under specific constraints: a limited total storage capacity and a requirement for a minimum total resolution quality across all stored songs. These constraints ensure that the platform maintains a balance between quantity and quality, adhering to industry standards and operational limitations. The business configuration includes parameters such as the total storage capacity available for storing songs and the minimum total resolution quality required for stored songs, which are critical in guiding the optimization process.

### Goals  
The primary goal of the optimization is to maximize the total rating of the songs stored on the platform. This involves selecting a combination of songs that collectively offer the highest possible rating while respecting the constraints of storage capacity and resolution quality. The success of this optimization is measured by the total rating achieved, which is directly influenced by the ratings of the individual songs selected for storage. The objective is to ensure that the platform offers a high-quality music catalog that meets user expectations and operational requirements.

## 2. Constraints    

The optimization process is governed by two main constraints. First, the total file size of the stored songs must not exceed the available storage capacity. This ensures that the platform operates within its physical storage limits. Second, the combined resolution quality of the stored songs must meet or exceed a specified minimum threshold. This constraint guarantees that the platform maintains a certain level of audio quality, which is essential for user satisfaction. These constraints are linear in nature, focusing on the sum of individual song attributes to guide the decision-making process.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include adding tables for missing data requirements and updating configuration logic for scalar parameters and formulas.

CREATE TABLE song (
  f_id INTEGER,
  rating FLOAT,
  file_size INTEGER
);

CREATE TABLE song_file_data (
  song_id INTEGER,
  resolution INTEGER
);
```

### Data Dictionary  
The data structure is designed to support the optimization process by providing essential information about each song. The "song" table contains metadata for each song, including a unique identifier, the song's rating, and its file size. This information is crucial for determining the song's contribution to the overall rating and its impact on storage capacity. The "song_file_data" table links each song to its resolution quality, which is used to ensure that the platform meets the minimum resolution quality constraint. Each table and column is aligned with the optimization roles, ensuring that the data supports the linear formulation of the problem.

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical file sizes and resolutions for music streaming services, ensuring a mix of high and low ratings to reflect a realistic catalog.

-- Realistic data for song
INSERT INTO song (f_id, rating, file_size) VALUES (1, 4.5, 5000);
INSERT INTO song (f_id, rating, file_size) VALUES (2, 3.8, 3000);
INSERT INTO song (f_id, rating, file_size) VALUES (3, 5.0, 7000);

-- Realistic data for song_file_data
INSERT INTO song_file_data (song_id, resolution) VALUES (1, 1080);
INSERT INTO song_file_data (song_id, resolution) VALUES (2, 720);
INSERT INTO song_file_data (song_id, resolution) VALUES (3, 480);
```
