Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-25 22:25:55

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: Races_per_Track: integer variables representing the number of races to be held at each track
- Operational parameters align with expected linear objective: maximize ∑(Seating × Races_per_Track)
- Business configuration includes: Total available seating capacity across all tracks (used for Upper bound for total seating utilization constraint)
- 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": "race_track",
  "iteration": 1,
  "business_context": "A race track management company aims to maximize the total seating capacity utilization across all races in a given year, considering constraints on the number of races per track and the availability of seating.",
  "optimization_problem_description": "Maximize the total seating capacity utilization by deciding the number of races to be held at each track, subject to constraints on the maximum number of races per track and the total available seating capacity.",
  "optimization_formulation": {
    "objective": "maximize \u2211(Seating \u00d7 Races_per_Track)",
    "decision_variables": "Races_per_Track: integer variables representing the number of races to be held at each track",
    "constraints": "1. Races_per_Track \u2264 Max_Races for each track, 2. \u2211(Seating \u00d7 Races_per_Track) \u2264 Total_Seating_Capacity"
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "Seating[t]": {
        "currently_mapped_to": "track.Seating",
        "mapping_adequacy": "good",
        "description": "Seating capacity of each track, used as coefficients in the objective function"
      }
    },
    "constraint_bounds": {
      "Max_Races[t]": {
        "currently_mapped_to": "track_limits.Max_Races",
        "mapping_adequacy": "good",
        "description": "Maximum number of races allowed at each track"
      },
      "Total_Seating_Capacity": {
        "currently_mapped_to": "business_configuration_logic.Total_Seating_Capacity",
        "mapping_adequacy": "good",
        "description": "Total available seating capacity across all tracks"
      }
    },
    "decision_variables": {
      "Races_per_Track[t]": {
        "currently_mapped_to": "track_races.Races",
        "mapping_adequacy": "good",
        "description": "Number of races to be held at each track",
        "variable_type": "integer"
      }
    }
  },
  "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 missing optimization requirements and updating business configuration logic for scalar parameters and formulas.

CREATE TABLE track (
  Seating INTEGER
);

CREATE TABLE track_races (
  Races INTEGER
);

CREATE TABLE track_limits (
  Max_Races INTEGER
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic race track capacities, typical race schedules, and operational constraints to ensure a meaningful optimization problem.

-- Realistic data for track
INSERT INTO track (Seating) VALUES (5000);
INSERT INTO track (Seating) VALUES (10000);
INSERT INTO track (Seating) VALUES (15000);

-- Realistic data for track_races
INSERT INTO track_races (Races) VALUES (3);
INSERT INTO track_races (Races) VALUES (5);
INSERT INTO track_races (Races) VALUES (7);

-- Realistic data for track_limits
INSERT INTO track_limits (Max_Races) VALUES (10);
INSERT INTO track_limits (Max_Races) VALUES (12);
INSERT INTO track_limits (Max_Races) VALUES (15);


```

DATA DICTIONARY:
{
  "tables": {
    "track": {
      "business_purpose": "Stores information about each race track",
      "optimization_role": "objective_coefficients",
      "columns": {
        "Seating": {
          "data_type": "INTEGER",
          "business_meaning": "Seating capacity of the track",
          "optimization_purpose": "Coefficient in the objective function",
          "sample_values": [
            5000,
            10000,
            15000
          ]
        }
      }
    },
    "track_races": {
      "business_purpose": "Stores the number of races to be held at each track",
      "optimization_role": "decision_variables",
      "columns": {
        "Races": {
          "data_type": "INTEGER",
          "business_meaning": "Number of races to be held at the track",
          "optimization_purpose": "Decision variable in the optimization model",
          "sample_values": [
            3,
            5,
            7
          ]
        }
      }
    },
    "track_limits": {
      "business_purpose": "Stores the maximum number of races allowed per track",
      "optimization_role": "constraint_bounds",
      "columns": {
        "Max_Races": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of races allowed at the track",
          "optimization_purpose": "Upper bound constraint for races per track",
          "sample_values": [
            10,
            12,
            15
          ]
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "Total_Seating_Capacity": {
    "data_type": "INTEGER",
    "business_meaning": "Total available seating capacity across all tracks",
    "optimization_role": "Upper bound for total seating utilization constraint",
    "configuration_type": "scalar_parameter",
    "value": 300000,
    "business_justification": "Total seating capacity across all tracks, ensuring a realistic upper bound for the optimization constraint."
  }
}

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: race_track

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: Races_per_Track: integer variables representing the number of races to be held at each track
- Operational parameters align with expected linear objective: maximize ∑(Seating × Races_per_Track)
- Business configuration includes: Total available seating capacity across all tracks (used for Upper bound for total seating utilization constraint)
- 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 ∑(Seating × Races_per_Track)
- 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: 1. Races_per_Track ≤ Max_Races for each track, 2. ∑(Seating × Races_per_Track) ≤ Total_Seating_Capacity (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 missing optimization requirements and updating business configuration logic for scalar parameters and formulas.

CREATE TABLE track (
  Seating INTEGER
);

CREATE TABLE track_races (
  Races INTEGER
);

CREATE TABLE track_limits (
  Max_Races 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 race track capacities, typical race schedules, and operational constraints to ensure a meaningful optimization problem.

-- Realistic data for track
INSERT INTO track (Seating) VALUES (5000);
INSERT INTO track (Seating) VALUES (10000);
INSERT INTO track (Seating) VALUES (15000);

-- Realistic data for track_races
INSERT INTO track_races (Races) VALUES (3);
INSERT INTO track_races (Races) VALUES (5);
INSERT INTO track_races (Races) VALUES (7);

-- Realistic data for track_limits
INSERT INTO track_limits (Max_Races) VALUES (10);
INSERT INTO track_limits (Max_Races) VALUES (12);
INSERT INTO track_limits (Max_Races) VALUES (15);


```

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

Response:
# Complete Optimization Problem and Solution: race_track

## 1. Problem Context and Goals

### Context  
A race track management company is tasked with optimizing the utilization of seating capacity across all its tracks for the upcoming year. The company operates multiple tracks, each with a specific seating capacity and a maximum allowable number of races. The primary business decision involves determining the number of races to be held at each track to maximize the total seating capacity utilization. This decision must respect two key operational constraints: the maximum number of races allowed at each track and the total available seating capacity across all tracks. The total seating capacity is a fixed value determined by the company’s business configuration, ensuring that the overall seating utilization does not exceed the company’s operational limits. The problem is designed to align with linear optimization principles, focusing on straightforward relationships between seating capacities, race counts, and operational constraints without introducing nonlinear complexities.

### Goals  
The optimization goal is to maximize the total seating capacity utilization across all races held at the company’s tracks. This is achieved by strategically determining the number of races at each track, weighted by their respective seating capacities. Success is measured by the total seating utilization, which is the sum of the seating capacity of each track multiplied by the number of races held there. The objective is to ensure that the company’s resources are used efficiently while adhering to operational constraints.

## 2. Constraints  

The optimization problem is subject to two primary constraints:  
1. **Race Limit per Track**: The number of races held at each track cannot exceed the maximum number of races allowed for that track. This ensures that each track operates within its capacity for hosting events.  
2. **Total Seating Capacity**: The total seating utilization across all tracks must not exceed the company’s total available seating capacity. This ensures that the company’s overall seating resources are not overutilized.  

These constraints are designed to align with linear optimization principles, ensuring that the relationships between variables and constraints remain straightforward and avoid nonlinear complexities.

## 3. Available Data  

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

CREATE TABLE track (
  Seating INTEGER
);

CREATE TABLE track_races (
  Races INTEGER
);

CREATE TABLE track_limits (
  Max_Races INTEGER
);
```

### Data Dictionary  
- **track**: Stores information about each race track.  
  - **Seating**: Represents the seating capacity of the track. This value is used as a coefficient in the optimization objective to calculate total seating utilization.  
- **track_races**: Stores the number of races to be held at each track.  
  - **Races**: Represents the decision variable in the optimization model, determining the number of races at each track.  
- **track_limits**: Stores the maximum number of races allowed per track.  
  - **Max_Races**: Represents the upper bound constraint for the number of races at each track.  

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic race track capacities, typical race schedules, and operational constraints to ensure a meaningful optimization problem.

-- Realistic data for track
INSERT INTO track (Seating) VALUES (5000);
INSERT INTO track (Seating) VALUES (10000);
INSERT INTO track (Seating) VALUES (15000);

-- Realistic data for track_races
INSERT INTO track_races (Races) VALUES (3);
INSERT INTO track_races (Races) VALUES (5);
INSERT INTO track_races (Races) VALUES (7);

-- Realistic data for track_limits
INSERT INTO track_limits (Max_Races) VALUES (10);
INSERT INTO track_limits (Max_Races) VALUES (12);
INSERT INTO track_limits (Max_Races) VALUES (15);
```
