Iteration final - PROBLEM_DESCRIPTION
Sequence: 7
Timestamp: 2025-07-27 21:44:00

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_ij: binary variable indicating if flight i is assigned to airport j
- Operational parameters align with expected linear objective: minimize total_cost = sum((distance * fee_rate) * x_ij) for all flights i and airports j
- Business logic formulas to express in natural language: Formula to calculate cost of assigning a flight to an airport (calculation method for Used to calculate objective coefficients 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": "flight_company",
  "iteration": 2,
  "business_context": "A flight company aims to optimize the allocation of its flights to different airports to minimize the total operational cost while ensuring that each airport can handle the assigned flights within its capacity.",
  "optimization_problem_description": "The objective is to minimize the total operational cost of flights by optimally assigning flights to airports. The cost is influenced by factors such as distance and airport fees. The company must ensure that each airport does not exceed its capacity and that all flights are assigned to an airport.",
  "optimization_formulation": {
    "objective": "minimize total_cost = sum((distance * fee_rate) * x_ij) for all flights i and airports j",
    "decision_variables": "x_ij: binary variable indicating if flight i is assigned to airport j",
    "constraints": [
      "sum(x_ij for all j) = 1 for each flight i",
      "sum(x_ij for all i) <= capacity_j for each airport j"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "cost_coefficient[i,j]": {
        "currently_mapped_to": "business_configuration_logic.cost_coefficient_formula",
        "mapping_adequacy": "good",
        "description": "Cost of assigning flight i to airport j, calculated as distance * fee_rate"
      }
    },
    "constraint_bounds": {
      "flight_assignment[i]": {
        "currently_mapped_to": "Flights.assignment",
        "mapping_adequacy": "good",
        "description": "Ensures each flight is assigned to exactly one airport"
      },
      "airport_capacity[j]": {
        "currently_mapped_to": "Airports.capacity",
        "mapping_adequacy": "good",
        "description": "Ensures no airport exceeds its capacity"
      }
    },
    "decision_variables": {
      "x_ij": {
        "currently_mapped_to": "Flights.assignment",
        "mapping_adequacy": "good",
        "description": "Binary variable indicating if flight i is assigned to airport j",
        "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: Schema changes include adding tables for distance and fee rate data, updating business configuration logic for cost coefficient formula, and ensuring all tables have at least 3 rows of data.

CREATE TABLE Flights (
  flight_id INTEGER,
  airport_id INTEGER,
  assignment BOOLEAN
);

CREATE TABLE Airports (
  airport_id INTEGER,
  capacity INTEGER
);

CREATE TABLE Distances (
  flight_id INTEGER,
  airport_id INTEGER,
  distance FLOAT
);

CREATE TABLE FeeRates (
  airport_id INTEGER,
  fee_rate FLOAT
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical distances between major airports and realistic fee rates charged by airports. Capacities were set to reflect typical airport operations, ensuring a balance between demand and supply.

-- Realistic data for Flights
INSERT INTO Flights (flight_id, airport_id, assignment) VALUES (1, 101, True);
INSERT INTO Flights (flight_id, airport_id, assignment) VALUES (2, 102, True);
INSERT INTO Flights (flight_id, airport_id, assignment) VALUES (3, 103, True);

-- Realistic data for Airports
INSERT INTO Airports (airport_id, capacity) VALUES (101, 120);
INSERT INTO Airports (airport_id, capacity) VALUES (102, 180);
INSERT INTO Airports (airport_id, capacity) VALUES (103, 150);

-- Realistic data for Distances
INSERT INTO Distances (flight_id, airport_id, distance) VALUES (1, 101, 550.0);
INSERT INTO Distances (flight_id, airport_id, distance) VALUES (2, 102, 800.0);
INSERT INTO Distances (flight_id, airport_id, distance) VALUES (3, 103, 950.0);

-- Realistic data for FeeRates
INSERT INTO FeeRates (airport_id, fee_rate) VALUES (101, 12.0);
INSERT INTO FeeRates (airport_id, fee_rate) VALUES (102, 18.0);
INSERT INTO FeeRates (airport_id, fee_rate) VALUES (103, 22.0);


```

DATA DICTIONARY:
{
  "tables": {
    "Flights": {
      "business_purpose": "Represents the assignment of flights to airports",
      "optimization_role": "decision_variables",
      "columns": {
        "flight_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each flight",
          "optimization_purpose": "Identifies flights in optimization model",
          "sample_values": "1, 2, 3"
        },
        "airport_id": {
          "data_type": "INTEGER",
          "business_meaning": "Identifier for the assigned airport",
          "optimization_purpose": "Links flights to airports in optimization model",
          "sample_values": "101, 102, 103"
        },
        "assignment": {
          "data_type": "BOOLEAN",
          "business_meaning": "Indicates if a flight is assigned to an airport",
          "optimization_purpose": "Decision variable in optimization model",
          "sample_values": "true, false"
        }
      }
    },
    "Airports": {
      "business_purpose": "Contains data about airports including their capacity",
      "optimization_role": "business_data",
      "columns": {
        "airport_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each airport",
          "optimization_purpose": "Identifies airports in optimization model",
          "sample_values": "101, 102, 103"
        },
        "capacity": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of flights an airport can handle",
          "optimization_purpose": "Constraint bound in optimization model",
          "sample_values": "100, 150, 200"
        }
      }
    },
    "Distances": {
      "business_purpose": "Represents the distance between each flight and airport for cost calculation",
      "optimization_role": "objective_coefficients",
      "columns": {
        "flight_id": {
          "data_type": "INTEGER",
          "business_meaning": "Unique identifier for each flight",
          "optimization_purpose": "Used in cost calculation",
          "sample_values": "1, 2, 3"
        },
        "airport_id": {
          "data_type": "INTEGER",
          "business_meaning": "Identifier for the airport",
          "optimization_purpose": "Used in cost calculation",
          "sample_values": "101, 102, 103"
        },
        "distance": {
          "data_type": "FLOAT",
          "business_meaning": "Distance between flight and airport",
          "optimization_purpose": "Used in cost calculation",
          "sample_values": "500.0, 750.0, 1000.0"
        }
      }
    },
    "FeeRates": {
      "business_purpose": "Represents the fee rate for each airport for cost calculation",
      "optimization_role": "objective_coefficients",
      "columns": {
        "airport_id": {
          "data_type": "INTEGER",
          "business_meaning": "Identifier for the airport",
          "optimization_purpose": "Used in cost calculation",
          "sample_values": "101, 102, 103"
        },
        "fee_rate": {
          "data_type": "FLOAT",
          "business_meaning": "Fee rate for using the airport",
          "optimization_purpose": "Used in cost calculation",
          "sample_values": "10.0, 15.0, 20.0"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "cost_coefficient_formula": {
    "data_type": "STRING",
    "business_meaning": "Formula to calculate cost of assigning a flight to an airport",
    "optimization_role": "Used to calculate objective coefficients in optimization model",
    "configuration_type": "business_logic_formula",
    "formula_expression": "distance * fee_rate"
  }
}

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: flight_company

## 1. Problem Context and Goals

### Context  
[Regenerate business context that naturally aligns with LINEAR optimization formulation. Ensure:]
- Business decisions match expected decision variables: x_ij: binary variable indicating if flight i is assigned to airport j
- Operational parameters align with expected linear objective: minimize total_cost = sum((distance * fee_rate) * x_ij) for all flights i and airports j
- Business logic formulas to express in natural language: Formula to calculate cost of assigning a flight to an airport (calculation method for Used to calculate objective coefficients 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: minimize
- Metric to optimize: minimize total_cost = sum((distance * fee_rate) * x_ij) for all flights i and airports j
- 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: ['sum(x_ij for all j) = 1 for each flight i', 'sum(x_ij for all i) <= capacity_j for each airport j'] (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: Schema changes include adding tables for distance and fee rate data, updating business configuration logic for cost coefficient formula, and ensuring all tables have at least 3 rows of data.

CREATE TABLE Flights (
  flight_id INTEGER,
  airport_id INTEGER,
  assignment BOOLEAN
);

CREATE TABLE Airports (
  airport_id INTEGER,
  capacity INTEGER
);

CREATE TABLE Distances (
  flight_id INTEGER,
  airport_id INTEGER,
  distance FLOAT
);

CREATE TABLE FeeRates (
  airport_id INTEGER,
  fee_rate FLOAT
);


```

### 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 distances between major airports and realistic fee rates charged by airports. Capacities were set to reflect typical airport operations, ensuring a balance between demand and supply.

-- Realistic data for Flights
INSERT INTO Flights (flight_id, airport_id, assignment) VALUES (1, 101, True);
INSERT INTO Flights (flight_id, airport_id, assignment) VALUES (2, 102, True);
INSERT INTO Flights (flight_id, airport_id, assignment) VALUES (3, 103, True);

-- Realistic data for Airports
INSERT INTO Airports (airport_id, capacity) VALUES (101, 120);
INSERT INTO Airports (airport_id, capacity) VALUES (102, 180);
INSERT INTO Airports (airport_id, capacity) VALUES (103, 150);

-- Realistic data for Distances
INSERT INTO Distances (flight_id, airport_id, distance) VALUES (1, 101, 550.0);
INSERT INTO Distances (flight_id, airport_id, distance) VALUES (2, 102, 800.0);
INSERT INTO Distances (flight_id, airport_id, distance) VALUES (3, 103, 950.0);

-- Realistic data for FeeRates
INSERT INTO FeeRates (airport_id, fee_rate) VALUES (101, 12.0);
INSERT INTO FeeRates (airport_id, fee_rate) VALUES (102, 18.0);
INSERT INTO FeeRates (airport_id, fee_rate) VALUES (103, 22.0);


```

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

Response:
# Complete Optimization Problem and Solution: flight_company

## 1. Problem Context and Goals

### Context  
The flight company is focused on optimizing the allocation of its flights to various airports to minimize the overall operational costs. The decision-making process involves determining whether a flight should be assigned to a specific airport, represented by a binary decision variable. The operational costs are influenced by the distance between the flight's origin and the airport, as well as the fee rate charged by the airport. The company aims to ensure that each flight is assigned to exactly one airport, and that no airport exceeds its capacity to handle flights. The cost of assigning a flight to an airport is calculated by multiplying the distance by the fee rate, which serves as the basis for the objective coefficients in the optimization model. This approach ensures that the problem remains linear, avoiding any nonlinear relationships such as variable products or divisions. The business configuration includes both scalar parameters and business logic formulas, ensuring consistency and flexibility in parameter tuning.

### Goals  
The primary goal of the optimization is to minimize the total operational cost associated with assigning flights to airports. This involves reducing the sum of costs calculated by multiplying the distance and fee rate for each flight-airport assignment. Success in this optimization is measured by achieving the lowest possible total cost while adhering to the constraints of flight assignments and airport capacities. The objective is clearly defined in linear terms, focusing on minimizing the total cost without involving any complex mathematical expressions.

## 2. Constraints    

The optimization problem is subject to several constraints that ensure the feasibility and practicality of the solution:

- Each flight must be assigned to exactly one airport. This ensures that all flights are accounted for and properly managed within the system.
- The total number of flights assigned to an airport must not exceed its capacity. This constraint ensures that airports operate within their limits, maintaining efficient and safe operations.

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

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 2 Database Schema
-- Objective: Schema changes include adding tables for distance and fee rate data, updating business configuration logic for cost coefficient formula, and ensuring all tables have at least 3 rows of data.

CREATE TABLE Flights (
  flight_id INTEGER,
  airport_id INTEGER,
  assignment BOOLEAN
);

CREATE TABLE Airports (
  airport_id INTEGER,
  capacity INTEGER
);

CREATE TABLE Distances (
  flight_id INTEGER,
  airport_id INTEGER,
  distance FLOAT
);

CREATE TABLE FeeRates (
  airport_id INTEGER,
  fee_rate FLOAT
);


```

### Data Dictionary  
The data dictionary provides a comprehensive overview of the tables and columns, explaining their business purposes and roles in the optimization process:

- **Flights Table**: Represents the assignment of flights to airports. It includes:
  - `flight_id`: A unique identifier for each flight, used to identify flights in the optimization model.
  - `airport_id`: An identifier for the assigned airport, linking flights to airports in the optimization model.
  - `assignment`: A boolean indicating if a flight is assigned to an airport, serving as a decision variable in the optimization model.

- **Airports Table**: Contains data about airports, including their capacity. It includes:
  - `airport_id`: A unique identifier for each airport, used to identify airports in the optimization model.
  - `capacity`: The maximum number of flights an airport can handle, serving as a constraint bound in the optimization model.

- **Distances Table**: Represents the distance between each flight and airport for cost calculation. It includes:
  - `flight_id`: A unique identifier for each flight, used in cost calculation.
  - `airport_id`: An identifier for the airport, used in cost calculation.
  - `distance`: The distance between the flight and airport, used in cost calculation.

- **FeeRates Table**: Represents the fee rate for each airport for cost calculation. It includes:
  - `airport_id`: An identifier for the airport, used in cost calculation.
  - `fee_rate`: The fee rate for using the airport, used in cost calculation.

### Current Stored Values  
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical distances between major airports and realistic fee rates charged by airports. Capacities were set to reflect typical airport operations, ensuring a balance between demand and supply.

-- Realistic data for Flights
INSERT INTO Flights (flight_id, airport_id, assignment) VALUES (1, 101, True);
INSERT INTO Flights (flight_id, airport_id, assignment) VALUES (2, 102, True);
INSERT INTO Flights (flight_id, airport_id, assignment) VALUES (3, 103, True);

-- Realistic data for Airports
INSERT INTO Airports (airport_id, capacity) VALUES (101, 120);
INSERT INTO Airports (airport_id, capacity) VALUES (102, 180);
INSERT INTO Airports (airport_id, capacity) VALUES (103, 150);

-- Realistic data for Distances
INSERT INTO Distances (flight_id, airport_id, distance) VALUES (1, 101, 550.0);
INSERT INTO Distances (flight_id, airport_id, distance) VALUES (2, 102, 800.0);
INSERT INTO Distances (flight_id, airport_id, distance) VALUES (3, 103, 950.0);

-- Realistic data for FeeRates
INSERT INTO FeeRates (airport_id, fee_rate) VALUES (101, 12.0);
INSERT INTO FeeRates (airport_id, fee_rate) VALUES (102, 18.0);
INSERT INTO FeeRates (airport_id, fee_rate) VALUES (103, 22.0);


```
