Iteration final - PROBLEM_DESCRIPTION
Sequence: 7
Timestamp: 2025-07-27 21:59:36

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] is a binary variable indicating if artist i is selected (1 if selected, 0 otherwise)
- Operational parameters align with expected linear objective: maximize total_weeks_on_top = ∑(artist_weeks_on_top.Weeks_on_Top[i] * x[i])
- 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_4",
  "iteration": 2,
  "business_context": "A music festival organizer wants to maximize the exposure of artists by selecting the optimal set of artists to perform at a festival, considering their popularity and availability.",
  "optimization_problem_description": "Maximize the total weeks on top of the charts for selected artists while ensuring the number of artists does not exceed festival capacity and each artist is available.",
  "optimization_formulation": {
    "objective": "maximize total_weeks_on_top = \u2211(artist_weeks_on_top.Weeks_on_Top[i] * x[i])",
    "decision_variables": "x[i] is a binary variable indicating if artist i is selected (1 if selected, 0 otherwise)",
    "constraints": [
      "\u2211(x[i]) \u2264 festival_capacity.Capacity",
      "x[i] \u2264 artist.Availability_Status[i] for all i"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "Weeks_on_Top[i]": {
        "currently_mapped_to": "artist_weeks_on_top.Weeks_on_Top",
        "mapping_adequacy": "good",
        "description": "Number of weeks artist i was on top of the charts"
      }
    },
    "constraint_bounds": {
      "Capacity": {
        "currently_mapped_to": "festival_capacity.Capacity",
        "mapping_adequacy": "good",
        "description": "Maximum number of artists that can be selected"
      }
    },
    "decision_variables": {
      "x[i]": {
        "currently_mapped_to": "artist.Artist_ID",
        "mapping_adequacy": "good",
        "description": "Binary variable indicating if artist 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 Weeks_on_Top table to address missing data for objective coefficients, updated data dictionary, and adjusted business configuration logic for scalar parameters.

CREATE TABLE artist (
  Artist_ID INTEGER,
  Availability_Status BOOLEAN
);

CREATE TABLE artist_weeks_on_top (
  Artist_ID INTEGER,
  Weeks_on_Top INTEGER
);

CREATE TABLE festival_capacity (
  Capacity INTEGER
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical festival capacities, artist popularity metrics, and availability patterns observed in the music industry.

-- Realistic data for artist
INSERT INTO artist (Artist_ID, Availability_Status) VALUES (1, True);
INSERT INTO artist (Artist_ID, Availability_Status) VALUES (2, False);
INSERT INTO artist (Artist_ID, Availability_Status) VALUES (3, True);

-- Realistic data for artist_weeks_on_top
INSERT INTO artist_weeks_on_top (Artist_ID, Weeks_on_Top) VALUES (1, 12);
INSERT INTO artist_weeks_on_top (Artist_ID, Weeks_on_Top) VALUES (2, 8);
INSERT INTO artist_weeks_on_top (Artist_ID, Weeks_on_Top) VALUES (3, 15);

-- Realistic data for festival_capacity
INSERT INTO festival_capacity (Capacity) VALUES (2);


```

DATA DICTIONARY:
{
  "tables": {
    "artist": {
      "business_purpose": "Stores information about artists",
      "optimization_role": "decision_variables",
      "columns": {
        "Artist_ID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each artist",
          "optimization_purpose": "Used as a decision variable in the optimization model",
          "sample_values": "1, 2, 3"
        },
        "Availability_Status": {
          "data_type": "BOOLEAN",
          "business_meaning": "Indicates if the artist is available on the festival date",
          "optimization_purpose": "Used to enforce availability constraint",
          "sample_values": "true, false"
        }
      }
    },
    "artist_weeks_on_top": {
      "business_purpose": "Stores the number of weeks each artist was on top of the charts",
      "optimization_role": "objective_coefficients",
      "columns": {
        "Artist_ID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each artist",
          "optimization_purpose": "Links to artist table for objective coefficient",
          "sample_values": "1, 2, 3"
        },
        "Weeks_on_Top": {
          "data_type": "INTEGER",
          "business_meaning": "Number of weeks the artist was on top of the charts",
          "optimization_purpose": "Used as an objective coefficient in the optimization model",
          "sample_values": "5, 10, 15"
        }
      }
    },
    "festival_capacity": {
      "business_purpose": "Stores the maximum number of artists for the festival",
      "optimization_role": "constraint_bounds",
      "columns": {
        "Capacity": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of artists that can be selected",
          "optimization_purpose": "Used as a constraint bound in the optimization model",
          "sample_values": "10"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:


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

EXACT MARKDOWN STRUCTURE TO FOLLOW:

# Complete Optimization Problem and Solution: music_4

## 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 if artist i is selected (1 if selected, 0 otherwise)
- Operational parameters align with expected linear objective: maximize total_weeks_on_top = ∑(artist_weeks_on_top.Weeks_on_Top[i] * x[i])
- 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_weeks_on_top = ∑(artist_weeks_on_top.Weeks_on_Top[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: ['∑(x[i]) ≤ festival_capacity.Capacity', 'x[i] ≤ artist.Availability_Status[i] for all 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 Weeks_on_Top table to address missing data for objective coefficients, updated data dictionary, and adjusted business configuration logic for scalar parameters.

CREATE TABLE artist (
  Artist_ID INTEGER,
  Availability_Status BOOLEAN
);

CREATE TABLE artist_weeks_on_top (
  Artist_ID INTEGER,
  Weeks_on_Top INTEGER
);

CREATE TABLE festival_capacity (
  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 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical festival capacities, artist popularity metrics, and availability patterns observed in the music industry.

-- Realistic data for artist
INSERT INTO artist (Artist_ID, Availability_Status) VALUES (1, True);
INSERT INTO artist (Artist_ID, Availability_Status) VALUES (2, False);
INSERT INTO artist (Artist_ID, Availability_Status) VALUES (3, True);

-- Realistic data for artist_weeks_on_top
INSERT INTO artist_weeks_on_top (Artist_ID, Weeks_on_Top) VALUES (1, 12);
INSERT INTO artist_weeks_on_top (Artist_ID, Weeks_on_Top) VALUES (2, 8);
INSERT INTO artist_weeks_on_top (Artist_ID, Weeks_on_Top) VALUES (3, 15);

-- Realistic data for festival_capacity
INSERT INTO festival_capacity (Capacity) VALUES (2);


```

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

Response:
# Complete Optimization Problem and Solution: music_4

## 1. Problem Context and Goals

### Context  
The music festival organizer is tasked with selecting a set of artists to perform at an upcoming festival. The primary decision involves choosing which artists to invite, where each artist's selection is represented by a binary decision variable. The goal is to maximize the festival's exposure by selecting artists who have spent the most weeks at the top of the music charts. This decision-making process is constrained by the festival's capacity, which limits the number of artists that can be invited, and the availability of each artist on the festival date.

The operational parameters are straightforward: the objective is to maximize the total number of weeks the selected artists have been on top of the charts. This is achieved by summing the product of each artist's weeks on top and their selection status. The festival's capacity acts as a constraint, ensuring that the number of selected artists does not exceed the venue's limit. Additionally, only artists who are available on the festival date can be selected.

### Goals  
The optimization goal is to maximize the festival's exposure by selecting artists who have the highest cumulative weeks on top of the charts. The metric for optimization is the total weeks on top, calculated by summing the weeks on top for each selected artist. Success is measured by the total exposure achieved, aligning with the objective of maximizing the festival's impact through artist selection.

## 2. Constraints    

The selection of artists is subject to two primary constraints. First, the total number of artists selected must not exceed the festival's capacity. This ensures that the festival remains within logistical and operational limits. Second, an artist can only be selected if they are available on the festival date. These constraints are designed to ensure that the selection process is both feasible and aligned with the festival's operational requirements.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 2 Database Schema
-- Objective: Added Weeks_on_Top table to address missing data for objective coefficients, updated data dictionary, and adjusted business configuration logic for scalar parameters.

CREATE TABLE artist (
  Artist_ID INTEGER,
  Availability_Status BOOLEAN
);

CREATE TABLE artist_weeks_on_top (
  Artist_ID INTEGER,
  Weeks_on_Top INTEGER
);

CREATE TABLE festival_capacity (
  Capacity INTEGER
);
```

### Data Dictionary  
- **Artist Table**: This table contains information about each artist, including a unique identifier and their availability status. The availability status indicates whether an artist is available to perform on the festival date, which is crucial for ensuring that only available artists are considered for selection.

- **Artist Weeks on Top Table**: This table records the number of weeks each artist has been at the top of the music charts. This data serves as the basis for the optimization objective, as it reflects the potential exposure each artist can bring to the festival.

- **Festival Capacity Table**: This table defines the maximum number of artists that can be selected for the festival. It serves as a constraint in the optimization model, ensuring that the number of selected artists does not exceed the festival's logistical capacity.

### Current Stored Values  
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical festival capacities, artist popularity metrics, and availability patterns observed in the music industry.

-- Realistic data for artist
INSERT INTO artist (Artist_ID, Availability_Status) VALUES (1, True);
INSERT INTO artist (Artist_ID, Availability_Status) VALUES (2, False);
INSERT INTO artist (Artist_ID, Availability_Status) VALUES (3, True);

-- Realistic data for artist_weeks_on_top
INSERT INTO artist_weeks_on_top (Artist_ID, Weeks_on_Top) VALUES (1, 12);
INSERT INTO artist_weeks_on_top (Artist_ID, Weeks_on_Top) VALUES (2, 8);
INSERT INTO artist_weeks_on_top (Artist_ID, Weeks_on_Top) VALUES (3, 15);

-- Realistic data for festival_capacity
INSERT INTO festival_capacity (Capacity) VALUES (2);
```
