Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-25 22:30:24

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: num_coasters[park_id, coaster_type] (integer)
- Operational parameters align with expected linear objective: maximize ∑(satisfaction_score[park_id, coaster_type] × num_coasters[park_id, coaster_type])
- Business configuration includes: Cost of each roller coaster type (used for Used in budget constraint calculation), Space required for each roller coaster type (used for Used in space constraint calculation)
- 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": "roller_coaster",
  "iteration": 1,
  "business_context": "A theme park chain aims to optimize the distribution of roller coasters across its parks globally to maximize visitor satisfaction, while adhering to budget, space, and maximum roller coaster constraints per park.",
  "optimization_problem_description": "The objective is to maximize the total visitor satisfaction score across all parks by determining the number of roller coasters of each type to install in each park, subject to budget, space, and maximum roller coaster constraints.",
  "optimization_formulation": {
    "objective": "maximize \u2211(satisfaction_score[park_id, coaster_type] \u00d7 num_coasters[park_id, coaster_type])",
    "decision_variables": "num_coasters[park_id, coaster_type] (integer)",
    "constraints": [
      "\u2211(coaster_cost \u00d7 num_coasters[park_id, coaster_type]) \u2264 budget[park_id] for each park_id",
      "\u2211(coaster_space \u00d7 num_coasters[park_id, coaster_type]) \u2264 space[park_id] for each park_id",
      "\u2211(num_coasters[park_id, coaster_type]) \u2264 max_coasters[park_id] for each park_id"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "satisfaction_score[park_id, coaster_type]": {
        "currently_mapped_to": "visitor_satisfaction_scores.score",
        "mapping_adequacy": "good",
        "description": "Visitor satisfaction score for each roller coaster type in each park"
      }
    },
    "constraint_bounds": {
      "budget[park_id]": {
        "currently_mapped_to": "park_budgets.budget",
        "mapping_adequacy": "good",
        "description": "Budget limit for each park"
      },
      "space[park_id]": {
        "currently_mapped_to": "park_available_space.space",
        "mapping_adequacy": "good",
        "description": "Available space for roller coasters in each park"
      },
      "max_coasters[park_id]": {
        "currently_mapped_to": "park_max_coasters.max_coasters",
        "mapping_adequacy": "good",
        "description": "Maximum number of roller coasters allowed in each park"
      }
    },
    "decision_variables": {
      "num_coasters[park_id, coaster_type]": {
        "currently_mapped_to": "visitor_satisfaction_scores.num_coasters",
        "mapping_adequacy": "good",
        "description": "Number of roller coasters of each type in each park",
        "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 visitor satisfaction scores, budget limits, available space, and maximum roller coasters per park. Configuration logic updates include scalar parameters for cost and space requirements of roller coaster types.

CREATE TABLE visitor_satisfaction_scores (
  park_id INTEGER,
  coaster_type STRING,
  score FLOAT,
  num_coasters INTEGER
);

CREATE TABLE park_budgets (
  park_id INTEGER,
  budget INTEGER
);

CREATE TABLE park_available_space (
  park_id INTEGER,
  space INTEGER
);

CREATE TABLE park_max_coasters (
  park_id INTEGER,
  max_coasters INTEGER
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on industry standards, realistic theme park budgets, and typical roller coaster costs and space requirements. The data ensures that the optimization problem is meaningful and solvable by providing a balance between visitor satisfaction, budget constraints, and space limitations.

-- Realistic data for visitor_satisfaction_scores
INSERT INTO visitor_satisfaction_scores (park_id, coaster_type, score, num_coasters) VALUES (1, 'Wooden', 8.5, 2);
INSERT INTO visitor_satisfaction_scores (park_id, coaster_type, score, num_coasters) VALUES (1, 'Steel', 9.0, 3);
INSERT INTO visitor_satisfaction_scores (park_id, coaster_type, score, num_coasters) VALUES (1, 'Inverted', 7.5, 1);
INSERT INTO visitor_satisfaction_scores (park_id, coaster_type, score, num_coasters) VALUES (2, 'Wooden', 8.0, 1);
INSERT INTO visitor_satisfaction_scores (park_id, coaster_type, score, num_coasters) VALUES (2, 'Steel', 9.5, 4);
INSERT INTO visitor_satisfaction_scores (park_id, coaster_type, score, num_coasters) VALUES (2, 'Inverted', 8.0, 2);
INSERT INTO visitor_satisfaction_scores (park_id, coaster_type, score, num_coasters) VALUES (3, 'Wooden', 7.0, 1);
INSERT INTO visitor_satisfaction_scores (park_id, coaster_type, score, num_coasters) VALUES (3, 'Steel', 9.0, 5);
INSERT INTO visitor_satisfaction_scores (park_id, coaster_type, score, num_coasters) VALUES (3, 'Inverted', 8.5, 3);

-- Realistic data for park_budgets
INSERT INTO park_budgets (park_id, budget) VALUES (1, 1000000);
INSERT INTO park_budgets (park_id, budget) VALUES (2, 1500000);
INSERT INTO park_budgets (park_id, budget) VALUES (3, 2000000);

-- Realistic data for park_available_space
INSERT INTO park_available_space (park_id, space) VALUES (1, 10000);
INSERT INTO park_available_space (park_id, space) VALUES (2, 15000);
INSERT INTO park_available_space (park_id, space) VALUES (3, 20000);

-- Realistic data for park_max_coasters
INSERT INTO park_max_coasters (park_id, max_coasters) VALUES (1, 5);
INSERT INTO park_max_coasters (park_id, max_coasters) VALUES (2, 7);
INSERT INTO park_max_coasters (park_id, max_coasters) VALUES (3, 10);


```

DATA DICTIONARY:
{
  "tables": {
    "visitor_satisfaction_scores": {
      "business_purpose": "Visitor satisfaction score for each roller coaster type in each park",
      "optimization_role": "objective_coefficients",
      "columns": {
        "park_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each park",
          "optimization_purpose": "Index for park in optimization model",
          "sample_values": "1, 2, 3"
        },
        "coaster_type": {
          "data_type": "STRING",
          "business_meaning": "Type of roller coaster",
          "optimization_purpose": "Index for coaster type in optimization model",
          "sample_values": "Wooden, Steel, Inverted"
        },
        "score": {
          "data_type": "FLOAT",
          "business_meaning": "Visitor satisfaction score",
          "optimization_purpose": "Coefficient in objective function",
          "sample_values": "8.5, 9.0, 7.5"
        },
        "num_coasters": {
          "data_type": "INTEGER",
          "business_meaning": "Number of roller coasters of this type in the park",
          "optimization_purpose": "Decision variable in optimization model",
          "sample_values": "2, 3, 1"
        }
      }
    },
    "park_budgets": {
      "business_purpose": "Budget limit for each park",
      "optimization_role": "constraint_bounds",
      "columns": {
        "park_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each park",
          "optimization_purpose": "Index for park in optimization model",
          "sample_values": "1, 2, 3"
        },
        "budget": {
          "data_type": "INTEGER",
          "business_meaning": "Budget limit for the park",
          "optimization_purpose": "Bound in budget constraint",
          "sample_values": "1000000, 1500000, 2000000"
        }
      }
    },
    "park_available_space": {
      "business_purpose": "Available space for roller coasters in each park",
      "optimization_role": "constraint_bounds",
      "columns": {
        "park_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each park",
          "optimization_purpose": "Index for park in optimization model",
          "sample_values": "1, 2, 3"
        },
        "space": {
          "data_type": "INTEGER",
          "business_meaning": "Available space for roller coasters",
          "optimization_purpose": "Bound in space constraint",
          "sample_values": "10000, 15000, 20000"
        }
      }
    },
    "park_max_coasters": {
      "business_purpose": "Maximum number of roller coasters allowed in each park",
      "optimization_role": "constraint_bounds",
      "columns": {
        "park_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each park",
          "optimization_purpose": "Index for park in optimization model",
          "sample_values": "1, 2, 3"
        },
        "max_coasters": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of roller coasters allowed",
          "optimization_purpose": "Bound in maximum roller coasters constraint",
          "sample_values": "5, 7, 10"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "coaster_cost": {
    "data_type": "INTEGER",
    "business_meaning": "Cost of each roller coaster type",
    "optimization_role": "Used in budget constraint calculation",
    "configuration_type": "scalar_parameter",
    "value": 500000,
    "business_justification": "This cost reflects the average investment required for a high-quality roller coaster, ensuring realistic budget constraints."
  },
  "coaster_space": {
    "data_type": "INTEGER",
    "business_meaning": "Space required for each roller coaster type",
    "optimization_role": "Used in space constraint calculation",
    "configuration_type": "scalar_parameter",
    "value": 2000,
    "business_justification": "This space requirement is based on typical land usage for roller coasters, ensuring realistic space constraints."
  }
}

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: roller_coaster

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: num_coasters[park_id, coaster_type] (integer)
- Operational parameters align with expected linear objective: maximize ∑(satisfaction_score[park_id, coaster_type] × num_coasters[park_id, coaster_type])
- Business configuration includes: Cost of each roller coaster type (used for Used in budget constraint calculation), Space required for each roller coaster type (used for Used in space constraint calculation)
- 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 ∑(satisfaction_score[park_id, coaster_type] × num_coasters[park_id, coaster_type])
- 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: ['∑(coaster_cost × num_coasters[park_id, coaster_type]) ≤ budget[park_id] for each park_id', '∑(coaster_space × num_coasters[park_id, coaster_type]) ≤ space[park_id] for each park_id', '∑(num_coasters[park_id, coaster_type]) ≤ max_coasters[park_id] for each park_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 1 Database Schema
-- Objective: Schema changes include creating new tables for visitor satisfaction scores, budget limits, available space, and maximum roller coasters per park. Configuration logic updates include scalar parameters for cost and space requirements of roller coaster types.

CREATE TABLE visitor_satisfaction_scores (
  park_id INTEGER,
  coaster_type STRING,
  score FLOAT,
  num_coasters INTEGER
);

CREATE TABLE park_budgets (
  park_id INTEGER,
  budget INTEGER
);

CREATE TABLE park_available_space (
  park_id INTEGER,
  space INTEGER
);

CREATE TABLE park_max_coasters (
  park_id INTEGER,
  max_coasters 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 industry standards, realistic theme park budgets, and typical roller coaster costs and space requirements. The data ensures that the optimization problem is meaningful and solvable by providing a balance between visitor satisfaction, budget constraints, and space limitations.

-- Realistic data for visitor_satisfaction_scores
INSERT INTO visitor_satisfaction_scores (park_id, coaster_type, score, num_coasters) VALUES (1, 'Wooden', 8.5, 2);
INSERT INTO visitor_satisfaction_scores (park_id, coaster_type, score, num_coasters) VALUES (1, 'Steel', 9.0, 3);
INSERT INTO visitor_satisfaction_scores (park_id, coaster_type, score, num_coasters) VALUES (1, 'Inverted', 7.5, 1);
INSERT INTO visitor_satisfaction_scores (park_id, coaster_type, score, num_coasters) VALUES (2, 'Wooden', 8.0, 1);
INSERT INTO visitor_satisfaction_scores (park_id, coaster_type, score, num_coasters) VALUES (2, 'Steel', 9.5, 4);
INSERT INTO visitor_satisfaction_scores (park_id, coaster_type, score, num_coasters) VALUES (2, 'Inverted', 8.0, 2);
INSERT INTO visitor_satisfaction_scores (park_id, coaster_type, score, num_coasters) VALUES (3, 'Wooden', 7.0, 1);
INSERT INTO visitor_satisfaction_scores (park_id, coaster_type, score, num_coasters) VALUES (3, 'Steel', 9.0, 5);
INSERT INTO visitor_satisfaction_scores (park_id, coaster_type, score, num_coasters) VALUES (3, 'Inverted', 8.5, 3);

-- Realistic data for park_budgets
INSERT INTO park_budgets (park_id, budget) VALUES (1, 1000000);
INSERT INTO park_budgets (park_id, budget) VALUES (2, 1500000);
INSERT INTO park_budgets (park_id, budget) VALUES (3, 2000000);

-- Realistic data for park_available_space
INSERT INTO park_available_space (park_id, space) VALUES (1, 10000);
INSERT INTO park_available_space (park_id, space) VALUES (2, 15000);
INSERT INTO park_available_space (park_id, space) VALUES (3, 20000);

-- Realistic data for park_max_coasters
INSERT INTO park_max_coasters (park_id, max_coasters) VALUES (1, 5);
INSERT INTO park_max_coasters (park_id, max_coasters) VALUES (2, 7);
INSERT INTO park_max_coasters (park_id, max_coasters) VALUES (3, 10);


```

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

Response:
# Complete Optimization Problem and Solution: roller_coaster

## 1. Problem Context and Goals

### Context  
A global theme park chain is focused on enhancing visitor satisfaction by strategically distributing roller coasters across its parks. The decision involves determining the number of roller coasters of each type to install in each park, considering visitor satisfaction scores, budget limits, available space, and maximum roller coaster capacity per park. Each roller coaster type has a fixed cost and space requirement, which are critical for ensuring that the installation plans are financially viable and physically feasible. The goal is to make operational decisions that maximize visitor satisfaction while adhering to these constraints.  

The business configuration includes the cost of each roller coaster type, set at $500,000, and the space required for each roller coaster type, set at 2,000 square units. These parameters are used to calculate budget and space constraints, ensuring that the optimization problem remains linear and aligned with real-world operational limitations.  

### Goals  
The primary goal of this optimization problem is to maximize the total visitor satisfaction across all parks. This is achieved by determining the optimal number of roller coasters of each type to install in each park, weighted by their respective visitor satisfaction scores. Success is measured by the overall increase in visitor satisfaction, which is directly influenced by the number and type of roller coasters installed. The optimization ensures that the decisions are made within the bounds of budget, space, and maximum roller coaster capacity for each park.  

## 2. Constraints  

The optimization problem is subject to the following constraints:  
1. **Budget Constraint**: The total cost of installing roller coasters in each park must not exceed the park’s allocated budget. The cost of each roller coaster type is fixed at $500,000, and the total cost is calculated by multiplying this cost by the number of roller coasters of each type installed in the park.  
2. **Space Constraint**: The total space occupied by roller coasters in each park must not exceed the park’s available space. Each roller coaster type requires 2,000 square units of space, and the total space is calculated by multiplying this requirement by the number of roller coasters of each type installed in the park.  
3. **Maximum Roller Coasters Constraint**: The total number of roller coasters installed in each park must not exceed the park’s maximum allowed capacity. This ensures that the park’s infrastructure and operational capabilities are not overwhelmed.  

These constraints ensure that the optimization problem remains linear and aligned with the operational realities of the theme parks.  

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating new tables for visitor satisfaction scores, budget limits, available space, and maximum roller coasters per park. Configuration logic updates include scalar parameters for cost and space requirements of roller coaster types.

CREATE TABLE visitor_satisfaction_scores (
  park_id INTEGER,
  coaster_type STRING,
  score FLOAT,
  num_coasters INTEGER
);

CREATE TABLE park_budgets (
  park_id INTEGER,
  budget INTEGER
);

CREATE TABLE park_available_space (
  park_id INTEGER,
  space INTEGER
);

CREATE TABLE park_max_coasters (
  park_id INTEGER,
  max_coasters INTEGER
);
```

### Data Dictionary  
- **visitor_satisfaction_scores**:  
  - **park_id**: Unique identifier for each park.  
  - **coaster_type**: Type of roller coaster (e.g., Wooden, Steel, Inverted).  
  - **score**: Visitor satisfaction score for the roller coaster type in the park.  
  - **num_coasters**: Number of roller coasters of this type in the park.  

- **park_budgets**:  
  - **park_id**: Unique identifier for each park.  
  - **budget**: Budget limit for the park.  

- **park_available_space**:  
  - **park_id**: Unique identifier for each park.  
  - **space**: Available space for roller coasters in the park.  

- **park_max_coasters**:  
  - **park_id**: Unique identifier for each park.  
  - **max_coasters**: Maximum number of roller coasters allowed in the park.  

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on industry standards, realistic theme park budgets, and typical roller coaster costs and space requirements. The data ensures that the optimization problem is meaningful and solvable by providing a balance between visitor satisfaction, budget constraints, and space limitations.

-- Realistic data for visitor_satisfaction_scores
INSERT INTO visitor_satisfaction_scores (park_id, coaster_type, score, num_coasters) VALUES (1, 'Wooden', 8.5, 2);
INSERT INTO visitor_satisfaction_scores (park_id, coaster_type, score, num_coasters) VALUES (1, 'Steel', 9.0, 3);
INSERT INTO visitor_satisfaction_scores (park_id, coaster_type, score, num_coasters) VALUES (1, 'Inverted', 7.5, 1);
INSERT INTO visitor_satisfaction_scores (park_id, coaster_type, score, num_coasters) VALUES (2, 'Wooden', 8.0, 1);
INSERT INTO visitor_satisfaction_scores (park_id, coaster_type, score, num_coasters) VALUES (2, 'Steel', 9.5, 4);
INSERT INTO visitor_satisfaction_scores (park_id, coaster_type, score, num_coasters) VALUES (2, 'Inverted', 8.0, 2);
INSERT INTO visitor_satisfaction_scores (park_id, coaster_type, score, num_coasters) VALUES (3, 'Wooden', 7.0, 1);
INSERT INTO visitor_satisfaction_scores (park_id, coaster_type, score, num_coasters) VALUES (3, 'Steel', 9.0, 5);
INSERT INTO visitor_satisfaction_scores (park_id, coaster_type, score, num_coasters) VALUES (3, 'Inverted', 8.5, 3);

-- Realistic data for park_budgets
INSERT INTO park_budgets (park_id, budget) VALUES (1, 1000000);
INSERT INTO park_budgets (park_id, budget) VALUES (2, 1500000);
INSERT INTO park_budgets (park_id, budget) VALUES (3, 2000000);

-- Realistic data for park_available_space
INSERT INTO park_available_space (park_id, space) VALUES (1, 10000);
INSERT INTO park_available_space (park_id, space) VALUES (2, 15000);
INSERT INTO park_available_space (park_id, space) VALUES (3, 20000);

-- Realistic data for park_max_coasters
INSERT INTO park_max_coasters (park_id, max_coasters) VALUES (1, 5);
INSERT INTO park_max_coasters (park_id, max_coasters) VALUES (2, 7);
INSERT INTO park_max_coasters (park_id, max_coasters) VALUES (3, 10);
```
