Iteration final - PROBLEM_DESCRIPTION
Sequence: 5
Timestamp: 2025-07-25 22:47:08

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: x_ij[i,j]: number of aircraft i assigned to airport j (integer)
- Operational parameters align with expected linear objective: minimize sum(cost_ij * x_ij) where cost_ij is the operational cost of assigning aircraft i to airport j, and x_ij is the number of aircraft i assigned to airport j
- Business configuration includes: Maximum number of aircraft movements allowed at an airport (used for Used in movement_capacity_j[j] constraint)
- Business logic formulas to express in natural language: Total operational cost of aircraft assignments (calculation method for Used in objective function)
- 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": "aircraft",
  "iteration": 1,
  "business_context": "Optimizing the assignment of aircraft to airports to minimize total operational costs while ensuring that each airport's passenger demand is met and aircraft capacity constraints are respected.",
  "optimization_problem_description": "Minimize the total operational costs associated with assigning aircraft to airports, ensuring that the total passenger capacity of assigned aircraft meets or exceeds the airport's passenger demand, and that the number of aircraft assigned does not exceed the airport's aircraft movement capacity.",
  "optimization_formulation": {
    "objective": "minimize sum(cost_ij * x_ij) where cost_ij is the operational cost of assigning aircraft i to airport j, and x_ij is the number of aircraft i assigned to airport j",
    "decision_variables": "x_ij[i,j]: number of aircraft i assigned to airport j (integer)",
    "constraints": [
      "sum(passenger_capacity_i * x_ij[i,j]) >= passenger_demand_j[j] for each airport j",
      "sum(x_ij[i,j]) <= aircraft_movements_j[j] for each airport j"
    ]
  },
  "current_optimization_to_schema_mapping": {
    "objective_coefficients": {
      "cost_ij[i,j]": {
        "currently_mapped_to": "operational_costs.cost",
        "mapping_adequacy": "good",
        "description": "Operational cost of assigning aircraft i to airport j"
      }
    },
    "constraint_bounds": {
      "passenger_demand_j[j]": {
        "currently_mapped_to": "airport.passenger_demand",
        "mapping_adequacy": "good",
        "description": "Passenger demand at airport j"
      },
      "aircraft_movements_j[j]": {
        "currently_mapped_to": "airport.aircraft_movements",
        "mapping_adequacy": "good",
        "description": "Maximum number of aircraft movements at airport j"
      }
    },
    "decision_variables": {
      "x_ij[i,j]": {
        "currently_mapped_to": "operational_costs.aircraft_count",
        "mapping_adequacy": "good",
        "description": "Number of aircraft i assigned to airport j",
        "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 tables for operational costs and aircraft capacities, modifying the airport table to better align with constraints, and adding configuration logic for scalar parameters and formulas.

CREATE TABLE aircraft_capacity (
  aircraft_type STRING,
  passenger_capacity INTEGER
);

CREATE TABLE operational_costs (
  aircraft_type STRING,
  airport_code STRING,
  cost FLOAT,
  aircraft_count INTEGER
);

CREATE TABLE airport (
  airport_code STRING,
  passenger_demand INTEGER,
  aircraft_movements INTEGER
);


```

CURRENT STORED VALUES:
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on industry standards, realistic operational costs, and typical airport and aircraft capacities. Relationships between tables were considered to ensure data consistency and optimization solvability.

-- Realistic data for aircraft_capacity
INSERT INTO aircraft_capacity (aircraft_type, passenger_capacity) VALUES ('Boeing 737', 150);
INSERT INTO aircraft_capacity (aircraft_type, passenger_capacity) VALUES ('Airbus A320', 180);
INSERT INTO aircraft_capacity (aircraft_type, passenger_capacity) VALUES ('Boeing 777', 300);

-- Realistic data for operational_costs
INSERT INTO operational_costs (aircraft_type, airport_code, cost, aircraft_count) VALUES ('Boeing 737', 'JFK', 5000.0, 0);
INSERT INTO operational_costs (aircraft_type, airport_code, cost, aircraft_count) VALUES ('Airbus A320', 'JFK', 5500.0, 0);
INSERT INTO operational_costs (aircraft_type, airport_code, cost, aircraft_count) VALUES ('Boeing 777', 'JFK', 8000.0, 0);
INSERT INTO operational_costs (aircraft_type, airport_code, cost, aircraft_count) VALUES ('Boeing 737', 'LAX', 4800.0, 0);
INSERT INTO operational_costs (aircraft_type, airport_code, cost, aircraft_count) VALUES ('Airbus A320', 'LAX', 5200.0, 0);

-- Realistic data for airport
INSERT INTO airport (airport_code, passenger_demand, aircraft_movements) VALUES ('JFK', 15000, 50);
INSERT INTO airport (airport_code, passenger_demand, aircraft_movements) VALUES ('LAX', 20000, 60);
INSERT INTO airport (airport_code, passenger_demand, aircraft_movements) VALUES ('ORD', 18000, 55);


```

DATA DICTIONARY:
{
  "tables": {
    "aircraft_capacity": {
      "business_purpose": "Stores passenger capacity of each aircraft type",
      "optimization_role": "business_data",
      "columns": {
        "aircraft_type": {
          "data_type": "STRING",
          "business_meaning": "Type of aircraft",
          "optimization_purpose": "Identifies aircraft type",
          "sample_values": "Boeing 737, Airbus A320"
        },
        "passenger_capacity": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of passengers the aircraft can carry",
          "optimization_purpose": "Used in demand_j[j] constraint",
          "sample_values": "150, 200"
        }
      }
    },
    "operational_costs": {
      "business_purpose": "Stores operational costs of assigning aircraft to airports",
      "optimization_role": "objective_coefficients",
      "columns": {
        "aircraft_type": {
          "data_type": "STRING",
          "business_meaning": "Type of aircraft",
          "optimization_purpose": "Identifies aircraft type",
          "sample_values": "Boeing 737, Airbus A320"
        },
        "airport_code": {
          "data_type": "STRING",
          "business_meaning": "Code of the airport",
          "optimization_purpose": "Identifies airport",
          "sample_values": "JFK, LAX"
        },
        "cost": {
          "data_type": "FLOAT",
          "business_meaning": "Operational cost of assigning the aircraft to the airport",
          "optimization_purpose": "Used in objective function",
          "sample_values": "5000.0, 7500.0"
        },
        "aircraft_count": {
          "data_type": "INTEGER",
          "business_meaning": "Number of aircraft assigned",
          "optimization_purpose": "Decision variable x_ij[i,j]",
          "sample_values": "2, 3"
        }
      }
    },
    "airport": {
      "business_purpose": "Stores airport-specific data",
      "optimization_role": "constraint_bounds",
      "columns": {
        "airport_code": {
          "data_type": "STRING",
          "business_meaning": "Code of the airport",
          "optimization_purpose": "Identifies airport",
          "sample_values": "JFK, LAX"
        },
        "passenger_demand": {
          "data_type": "INTEGER",
          "business_meaning": "Total passenger demand at the airport",
          "optimization_purpose": "Used in demand_j[j] constraint",
          "sample_values": "10000, 15000"
        },
        "aircraft_movements": {
          "data_type": "INTEGER",
          "business_meaning": "Maximum number of aircraft movements at the airport",
          "optimization_purpose": "Used in movement_capacity_j[j] constraint",
          "sample_values": "50, 75"
        }
      }
    }
  }
}


BUSINESS CONFIGURATION:

BUSINESS CONFIGURATION:
{
  "max_aircraft_movements": {
    "data_type": "INTEGER",
    "business_meaning": "Maximum number of aircraft movements allowed at an airport",
    "optimization_role": "Used in movement_capacity_j[j] constraint",
    "configuration_type": "scalar_parameter",
    "value": 60,
    "business_justification": "This value is set to ensure that the airport's movement capacity is realistic and aligns with the data in the airport table."
  },
  "total_operational_cost_formula": {
    "data_type": "STRING",
    "business_meaning": "Total operational cost of aircraft assignments",
    "optimization_role": "Used in objective function",
    "configuration_type": "business_logic_formula",
    "formula_expression": "sum(cost_ij * x_ij)"
  }
}

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: aircraft

## 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[i,j]: number of aircraft i assigned to airport j (integer)
- Operational parameters align with expected linear objective: minimize sum(cost_ij * x_ij) where cost_ij is the operational cost of assigning aircraft i to airport j, and x_ij is the number of aircraft i assigned to airport j
- Business configuration includes: Maximum number of aircraft movements allowed at an airport (used for Used in movement_capacity_j[j] constraint)
- Business logic formulas to express in natural language: Total operational cost of aircraft assignments (calculation method for Used in objective function)
- 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 sum(cost_ij * x_ij) where cost_ij is the operational cost of assigning aircraft i to airport j, and x_ij is the number of aircraft i assigned to airport 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(passenger_capacity_i * x_ij[i,j]) >= passenger_demand_j[j] for each airport j', 'sum(x_ij[i,j]) <= aircraft_movements_j[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 1 Database Schema
-- Objective: Schema changes include creating tables for operational costs and aircraft capacities, modifying the airport table to better align with constraints, and adding configuration logic for scalar parameters and formulas.

CREATE TABLE aircraft_capacity (
  aircraft_type STRING,
  passenger_capacity INTEGER
);

CREATE TABLE operational_costs (
  aircraft_type STRING,
  airport_code STRING,
  cost FLOAT,
  aircraft_count INTEGER
);

CREATE TABLE airport (
  airport_code STRING,
  passenger_demand INTEGER,
  aircraft_movements 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 operational costs, and typical airport and aircraft capacities. Relationships between tables were considered to ensure data consistency and optimization solvability.

-- Realistic data for aircraft_capacity
INSERT INTO aircraft_capacity (aircraft_type, passenger_capacity) VALUES ('Boeing 737', 150);
INSERT INTO aircraft_capacity (aircraft_type, passenger_capacity) VALUES ('Airbus A320', 180);
INSERT INTO aircraft_capacity (aircraft_type, passenger_capacity) VALUES ('Boeing 777', 300);

-- Realistic data for operational_costs
INSERT INTO operational_costs (aircraft_type, airport_code, cost, aircraft_count) VALUES ('Boeing 737', 'JFK', 5000.0, 0);
INSERT INTO operational_costs (aircraft_type, airport_code, cost, aircraft_count) VALUES ('Airbus A320', 'JFK', 5500.0, 0);
INSERT INTO operational_costs (aircraft_type, airport_code, cost, aircraft_count) VALUES ('Boeing 777', 'JFK', 8000.0, 0);
INSERT INTO operational_costs (aircraft_type, airport_code, cost, aircraft_count) VALUES ('Boeing 737', 'LAX', 4800.0, 0);
INSERT INTO operational_costs (aircraft_type, airport_code, cost, aircraft_count) VALUES ('Airbus A320', 'LAX', 5200.0, 0);

-- Realistic data for airport
INSERT INTO airport (airport_code, passenger_demand, aircraft_movements) VALUES ('JFK', 15000, 50);
INSERT INTO airport (airport_code, passenger_demand, aircraft_movements) VALUES ('LAX', 20000, 60);
INSERT INTO airport (airport_code, passenger_demand, aircraft_movements) VALUES ('ORD', 18000, 55);


```

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

Response:
# Complete Optimization Problem and Solution: aircraft

## 1. Problem Context and Goals

### Context  
The business problem involves optimizing the assignment of aircraft to airports to minimize total operational costs while ensuring that each airport's passenger demand is met and aircraft capacity constraints are respected. The decision to be made is the number of each type of aircraft assigned to each airport, which directly impacts operational costs. Each aircraft type has a specific passenger capacity, and each airport has a defined passenger demand and a maximum number of aircraft movements it can handle. The operational cost of assigning a specific aircraft type to a particular airport is known and varies based on the aircraft and airport combination. The business configuration includes a maximum number of aircraft movements allowed at each airport, which ensures that the airport's movement capacity is not exceeded. The total operational cost is calculated as the sum of the costs of all aircraft assignments, where each cost is the product of the operational cost for a specific aircraft-airport pair and the number of aircraft assigned to that pair.

### Goals  
The primary goal of this optimization is to minimize the total operational costs associated with assigning aircraft to airports. Success is measured by achieving the lowest possible sum of operational costs across all aircraft assignments while ensuring that all airports meet their passenger demand and do not exceed their aircraft movement capacity. The optimization focuses on making precise operational decisions that lead to a linear formulation, avoiding any scenarios that would require nonlinear relationships such as variable products or divisions.

## 2. Constraints    

The optimization must adhere to the following constraints:  
1. **Passenger Demand Constraint**: For each airport, the total passenger capacity of all assigned aircraft must meet or exceed the airport's passenger demand. This ensures that the airport can accommodate all passengers.  
2. **Aircraft Movement Constraint**: For each airport, the total number of assigned aircraft must not exceed the airport's maximum number of aircraft movements. This ensures that the airport's movement capacity is not exceeded.  

These constraints are designed to ensure that the optimization respects both passenger demand and operational capacity limitations, leading to a feasible and realistic solution.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating tables for operational costs and aircraft capacities, modifying the airport table to better align with constraints, and adding configuration logic for scalar parameters and formulas.

CREATE TABLE aircraft_capacity (
  aircraft_type STRING,
  passenger_capacity INTEGER
);

CREATE TABLE operational_costs (
  aircraft_type STRING,
  airport_code STRING,
  cost FLOAT,
  aircraft_count INTEGER
);

CREATE TABLE airport (
  airport_code STRING,
  passenger_demand INTEGER,
  aircraft_movements INTEGER
);
```

### Data Dictionary  
- **aircraft_capacity**:  
  - **aircraft_type**: Identifies the type of aircraft.  
  - **passenger_capacity**: Represents the maximum number of passengers the aircraft can carry, used to ensure passenger demand is met.  

- **operational_costs**:  
  - **aircraft_type**: Identifies the type of aircraft.  
  - **airport_code**: Identifies the airport.  
  - **cost**: Represents the operational cost of assigning the aircraft to the airport, used in the objective function.  
  - **aircraft_count**: Represents the number of aircraft assigned, which is the decision variable in the optimization.  

- **airport**:  
  - **airport_code**: Identifies the airport.  
  - **passenger_demand**: Represents the total passenger demand at the airport, used in the passenger demand constraint.  
  - **aircraft_movements**: Represents the maximum number of aircraft movements at the airport, used in the aircraft movement constraint.  

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on industry standards, realistic operational costs, and typical airport and aircraft capacities. Relationships between tables were considered to ensure data consistency and optimization solvability.

-- Realistic data for aircraft_capacity
INSERT INTO aircraft_capacity (aircraft_type, passenger_capacity) VALUES ('Boeing 737', 150);
INSERT INTO aircraft_capacity (aircraft_type, passenger_capacity) VALUES ('Airbus A320', 180);
INSERT INTO aircraft_capacity (aircraft_type, passenger_capacity) VALUES ('Boeing 777', 300);

-- Realistic data for operational_costs
INSERT INTO operational_costs (aircraft_type, airport_code, cost, aircraft_count) VALUES ('Boeing 737', 'JFK', 5000.0, 0);
INSERT INTO operational_costs (aircraft_type, airport_code, cost, aircraft_count) VALUES ('Airbus A320', 'JFK', 5500.0, 0);
INSERT INTO operational_costs (aircraft_type, airport_code, cost, aircraft_count) VALUES ('Boeing 777', 'JFK', 8000.0, 0);
INSERT INTO operational_costs (aircraft_type, airport_code, cost, aircraft_count) VALUES ('Boeing 737', 'LAX', 4800.0, 0);
INSERT INTO operational_costs (aircraft_type, airport_code, cost, aircraft_count) VALUES ('Airbus A320', 'LAX', 5200.0, 0);

-- Realistic data for airport
INSERT INTO airport (airport_code, passenger_demand, aircraft_movements) VALUES ('JFK', 15000, 50);
INSERT INTO airport (airport_code, passenger_demand, aircraft_movements) VALUES ('LAX', 20000, 60);
INSERT INTO airport (airport_code, passenger_demand, aircraft_movements) VALUES ('ORD', 18000, 55);
```
