Iteration final - PROBLEM_DESCRIPTION
Sequence: 7
Timestamp: 2025-07-27 22:37:51

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[concert_ID, stadium_ID] where x is a binary variable indicating if a concert is held at a stadium
- Operational parameters align with expected linear objective: maximize total_attendance = ∑(attendance_coefficient[concert_ID, stadium_ID] × x[concert_ID, stadium_ID])
- Business configuration includes: Default attendance coefficient when specific data is unavailable (used for Used as a fallback 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": "concert_singer",
  "iteration": 2,
  "business_context": "A concert organizer aims to maximize attendance across multiple concerts by selecting optimal concert-stadium pairings, considering stadium capacities and ensuring each concert has at least one singer.",
  "optimization_problem_description": "Maximize total attendance by selecting concerts to hold at specific stadiums, subject to stadium capacity constraints and ensuring each concert has at least one singer.",
  "optimization_formulation": {
    "objective": "maximize total_attendance = \u2211(attendance_coefficient[concert_ID, stadium_ID] \u00d7 x[concert_ID, stadium_ID])",
    "decision_variables": "x[concert_ID, stadium_ID] where x is a binary variable indicating if a concert is held at a stadium",
    "constraints": [
      "\u2211(x[concert_ID, stadium_ID]) >= 1 for each concert_ID",
      "\u2211(coefficient[concert_ID, stadium_ID] \u00d7 x[concert_ID, stadium_ID]) <= capacity[stadium_ID] for each stadium_ID"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "attendance_coefficient[concert_ID, stadium_ID]": {
        "currently_mapped_to": "attendance_coefficients.coefficient",
        "mapping_adequacy": "good",
        "description": "Expected attendance if the concert is held at the stadium"
      }
    },
    "constraint_bounds": {
      "capacity_constraint[stadium_ID]": {
        "currently_mapped_to": "stadium_capacity.capacity",
        "mapping_adequacy": "good",
        "description": "Maximum number of attendees a stadium can accommodate"
      }
    },
    "decision_variables": {
      "x[concert_ID, stadium_ID]": {
        "currently_mapped_to": "concert_stadium_mapping",
        "mapping_adequacy": "good",
        "description": "Binary variable indicating if a concert is held at a stadium",
        "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: Incorporated stadium capacity data into the schema, addressed mapping gaps, and updated business configuration logic for scalar parameters and formulas.

CREATE TABLE concert_stadium_mapping (
  concert_ID INTEGER,
  stadium_ID INTEGER
);

CREATE TABLE attendance_coefficients (
  concert_ID INTEGER,
  stadium_ID INTEGER,
  coefficient FLOAT
);

CREATE TABLE stadium_capacity (
  stadium_ID INTEGER,
  capacity INTEGER
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical stadium capacities and expected attendance for concerts of varying popularity, ensuring a mix of small, medium, and large events.

-- Realistic data for concert_stadium_mapping
INSERT INTO concert_stadium_mapping (concert_ID, stadium_ID) VALUES (1, 101);
INSERT INTO concert_stadium_mapping (concert_ID, stadium_ID) VALUES (2, 102);
INSERT INTO concert_stadium_mapping (concert_ID, stadium_ID) VALUES (3, 103);

-- Realistic data for attendance_coefficients
INSERT INTO attendance_coefficients (concert_ID, stadium_ID, coefficient) VALUES (1, 101, 120.0);
INSERT INTO attendance_coefficients (concert_ID, stadium_ID, coefficient) VALUES (1, 102, 180.0);
INSERT INTO attendance_coefficients (concert_ID, stadium_ID, coefficient) VALUES (2, 102, 250.0);
INSERT INTO attendance_coefficients (concert_ID, stadium_ID, coefficient) VALUES (2, 103, 300.0);
INSERT INTO attendance_coefficients (concert_ID, stadium_ID, coefficient) VALUES (3, 103, 400.0);

-- Realistic data for stadium_capacity
INSERT INTO stadium_capacity (stadium_ID, capacity) VALUES (101, 5000);
INSERT INTO stadium_capacity (stadium_ID, capacity) VALUES (102, 10000);
INSERT INTO stadium_capacity (stadium_ID, capacity) VALUES (103, 15000);


```

DATA DICTIONARY:
{
  "tables": {
    "concert_stadium_mapping": {
      "business_purpose": "Maps concerts to specific stadiums for planning",
      "optimization_role": "business_data",
      "columns": {
        "concert_ID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each concert",
          "optimization_purpose": "Links concerts to stadiums",
          "sample_values": "1, 2, 3"
        },
        "stadium_ID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each stadium",
          "optimization_purpose": "Links stadiums to concerts",
          "sample_values": "101, 102, 103"
        }
      }
    },
    "attendance_coefficients": {
      "business_purpose": "Stores attendance coefficients for concert-stadium combinations",
      "optimization_role": "objective_coefficients",
      "columns": {
        "concert_ID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each concert",
          "optimization_purpose": "Part of composite key for attendance coefficients",
          "sample_values": "1, 2, 3"
        },
        "stadium_ID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each stadium",
          "optimization_purpose": "Part of composite key for attendance coefficients",
          "sample_values": "101, 102, 103"
        },
        "coefficient": {
          "data_type": "FLOAT",
          "business_meaning": "Expected attendance if the concert is held at the stadium",
          "optimization_purpose": "Objective coefficient in optimization model",
          "sample_values": "150.0, 200.0, 250.0"
        }
      }
    },
    "stadium_capacity": {
      "business_purpose": "Stores maximum capacity for each stadium",
      "optimization_role": "constraint_bounds",
      "columns": {
        "stadium_ID": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each stadium",
          "optimization_purpose": "Links capacity to stadiums",
          "sample_values": "101, 102, 103"
        },
        "capacity": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of attendees a stadium can accommodate",
          "optimization_purpose": "Constraint bound in optimization model",
          "sample_values": "5000, 10000, 15000"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "default_attendance_coefficient": {
    "data_type": "INTEGER",
    "business_meaning": "Default attendance coefficient when specific data is unavailable",
    "optimization_role": "Used as a fallback in optimization model",
    "configuration_type": "scalar_parameter",
    "value": 150,
    "business_justification": "A default value of 150 is realistic for fallback scenarios where specific attendance data is unavailable, representing a moderate draw."
  }
}

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: concert_singer

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: x[concert_ID, stadium_ID] where x is a binary variable indicating if a concert is held at a stadium
- Operational parameters align with expected linear objective: maximize total_attendance = ∑(attendance_coefficient[concert_ID, stadium_ID] × x[concert_ID, stadium_ID])
- Business configuration includes: Default attendance coefficient when specific data is unavailable (used for Used as a fallback 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_attendance = ∑(attendance_coefficient[concert_ID, stadium_ID] × x[concert_ID, stadium_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: ['∑(x[concert_ID, stadium_ID]) >= 1 for each concert_ID', '∑(coefficient[concert_ID, stadium_ID] × x[concert_ID, stadium_ID]) <= capacity[stadium_ID] for each stadium_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: Incorporated stadium capacity data into the schema, addressed mapping gaps, and updated business configuration logic for scalar parameters and formulas.

CREATE TABLE concert_stadium_mapping (
  concert_ID INTEGER,
  stadium_ID INTEGER
);

CREATE TABLE attendance_coefficients (
  concert_ID INTEGER,
  stadium_ID INTEGER,
  coefficient FLOAT
);

CREATE TABLE stadium_capacity (
  stadium_ID INTEGER,
  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 stadium capacities and expected attendance for concerts of varying popularity, ensuring a mix of small, medium, and large events.

-- Realistic data for concert_stadium_mapping
INSERT INTO concert_stadium_mapping (concert_ID, stadium_ID) VALUES (1, 101);
INSERT INTO concert_stadium_mapping (concert_ID, stadium_ID) VALUES (2, 102);
INSERT INTO concert_stadium_mapping (concert_ID, stadium_ID) VALUES (3, 103);

-- Realistic data for attendance_coefficients
INSERT INTO attendance_coefficients (concert_ID, stadium_ID, coefficient) VALUES (1, 101, 120.0);
INSERT INTO attendance_coefficients (concert_ID, stadium_ID, coefficient) VALUES (1, 102, 180.0);
INSERT INTO attendance_coefficients (concert_ID, stadium_ID, coefficient) VALUES (2, 102, 250.0);
INSERT INTO attendance_coefficients (concert_ID, stadium_ID, coefficient) VALUES (2, 103, 300.0);
INSERT INTO attendance_coefficients (concert_ID, stadium_ID, coefficient) VALUES (3, 103, 400.0);

-- Realistic data for stadium_capacity
INSERT INTO stadium_capacity (stadium_ID, capacity) VALUES (101, 5000);
INSERT INTO stadium_capacity (stadium_ID, capacity) VALUES (102, 10000);
INSERT INTO stadium_capacity (stadium_ID, capacity) VALUES (103, 15000);


```

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

Response:
# Complete Optimization Problem and Solution: concert_singer

## 1. Problem Context and Goals

### Context  
The concert organizer is tasked with maximizing attendance across various concerts by strategically selecting which concerts to hold at specific stadiums. Each decision involves determining whether a concert should be held at a particular stadium, represented by a binary decision variable. The primary objective is to maximize the total expected attendance, which is calculated by summing the expected attendance for each concert-stadium pairing. This expected attendance is derived from predefined coefficients that estimate the number of attendees if a concert is held at a specific stadium. In cases where specific attendance data is unavailable, a default attendance coefficient is used as a fallback to ensure continuity in the optimization model.

The operational parameters include the stadium capacities, which act as constraints to ensure that the number of attendees does not exceed the maximum capacity of each stadium. Additionally, it is essential that each concert is paired with at least one stadium to ensure that every concert has a venue. The business configuration includes a default attendance coefficient, which is set at a realistic value of 150, representing a moderate draw for fallback scenarios.

### Goals  
The primary goal of the optimization is to maximize the total attendance across all concerts. This is achieved by selecting the optimal concert-stadium pairings that yield the highest expected attendance, based on the attendance coefficients. The success of this optimization is measured by the total expected attendance, which is calculated by summing the expected attendance for each selected concert-stadium pairing. The optimization process focuses on maximizing this metric while adhering to the constraints of stadium capacities and ensuring each concert is held at a venue.

## 2. Constraints    

The optimization problem is subject to several constraints that ensure feasibility and alignment with business requirements:

- Each concert must be held at least once, meaning that for every concert, there should be at least one stadium selected to host it. This ensures that no concert is left without a venue.
  
- The total expected attendance for a concert at a particular stadium must not exceed the stadium's capacity. This constraint ensures that the number of attendees does not surpass the maximum number that the stadium can accommodate, maintaining safety and logistical feasibility.

These constraints are expressed in business terms that naturally lead to linear mathematical forms, ensuring that the optimization problem remains linear and solvable using standard linear programming techniques.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 2 Database Schema
-- Objective: Incorporated stadium capacity data into the schema, addressed mapping gaps, and updated business configuration logic for scalar parameters and formulas.

CREATE TABLE concert_stadium_mapping (
  concert_ID INTEGER,
  stadium_ID INTEGER
);

CREATE TABLE attendance_coefficients (
  concert_ID INTEGER,
  stadium_ID INTEGER,
  coefficient FLOAT
);

CREATE TABLE stadium_capacity (
  stadium_ID INTEGER,
  capacity INTEGER
);
```

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

- **Concert-Stadium Mapping Table**: This table maps concerts to specific stadiums for planning purposes. It plays a crucial role in linking concerts to stadiums, allowing the optimization model to determine which concerts are held at which venues.

  - **concert_ID**: A unique identifier for each concert, used to link concerts to stadiums.
  - **stadium_ID**: A unique identifier for each stadium, used to link stadiums to concerts.

- **Attendance Coefficients Table**: This table stores the expected attendance coefficients for each concert-stadium combination. These coefficients are used as objective coefficients in the optimization model to calculate the expected attendance.

  - **concert_ID**: Part of the composite key for attendance coefficients, linking to the concert.
  - **stadium_ID**: Part of the composite key for attendance coefficients, linking to the stadium.
  - **coefficient**: Represents the expected attendance if the concert is held at the stadium, serving as the objective coefficient in the optimization model.

- **Stadium Capacity Table**: This table stores the maximum capacity for each stadium, serving as constraint bounds in the optimization model to ensure that attendance does not exceed stadium capacities.

  - **stadium_ID**: Links capacity to stadiums, ensuring that each stadium's capacity is correctly applied in the optimization model.
  - **capacity**: Represents the maximum number of attendees a stadium can accommodate, serving as a constraint bound in the optimization model.

### Current Stored Values  
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical stadium capacities and expected attendance for concerts of varying popularity, ensuring a mix of small, medium, and large events.

-- Realistic data for concert_stadium_mapping
INSERT INTO concert_stadium_mapping (concert_ID, stadium_ID) VALUES (1, 101);
INSERT INTO concert_stadium_mapping (concert_ID, stadium_ID) VALUES (2, 102);
INSERT INTO concert_stadium_mapping (concert_ID, stadium_ID) VALUES (3, 103);

-- Realistic data for attendance_coefficients
INSERT INTO attendance_coefficients (concert_ID, stadium_ID, coefficient) VALUES (1, 101, 120.0);
INSERT INTO attendance_coefficients (concert_ID, stadium_ID, coefficient) VALUES (1, 102, 180.0);
INSERT INTO attendance_coefficients (concert_ID, stadium_ID, coefficient) VALUES (2, 102, 250.0);
INSERT INTO attendance_coefficients (concert_ID, stadium_ID, coefficient) VALUES (2, 103, 300.0);
INSERT INTO attendance_coefficients (concert_ID, stadium_ID, coefficient) VALUES (3, 103, 400.0);

-- Realistic data for stadium_capacity
INSERT INTO stadium_capacity (stadium_ID, capacity) VALUES (101, 5000);
INSERT INTO stadium_capacity (stadium_ID, capacity) VALUES (102, 10000);
INSERT INTO stadium_capacity (stadium_ID, capacity) VALUES (103, 15000);
```
