=== Problem Context ===
# Complete Optimization Problem and Solution: school_bus

## 1. Problem Context and Goals

### Context  
The business problem involves optimizing the assignment of drivers to schools to minimize the total travel distance while ensuring operational efficiency. Each driver resides in a specific home city, and each school is located in a distinct location. The travel distance between a driver's home city and a school's location is calculated using the Euclidean distance formula, which measures the straight-line distance between two points based on their coordinates. 

The decision to assign a driver to a school is represented as a binary choice: a driver is either assigned to a school (1) or not (0). The business configuration includes two key parameters:  
1. **Maximum number of drivers per school**: Each school must be assigned exactly one driver to ensure efficient operations.  
2. **Maximum number of schools per driver**: Each driver can be assigned to at most one school to avoid overburdening drivers.  

The goal is to make these assignments in a way that minimizes the total travel distance across all driver-school pairs, ensuring that the constraints are satisfied. This problem is naturally suited for a linear optimization formulation, as it involves straightforward relationships between decision variables, coefficients, and constraints without requiring nonlinear operations like multiplication or division of variables.

### Goals  
The primary optimization goal is to minimize the total travel distance incurred by assigning drivers to schools. This is achieved by summing the distances for all driver-school pairs where an assignment is made. Success is measured by the total distance value, which is directly derived from the distances stored in the distance matrix. The objective is to find the optimal set of assignments that satisfies the constraints while achieving the lowest possible total travel distance.

## 2. Constraints  

The problem is subject to the following constraints:  
1. **Each school must be assigned exactly one driver**: This ensures that every school has a dedicated driver for its operations.  
2. **Each driver can be assigned to at most one school**: This prevents drivers from being overburdened by multiple assignments.  

These constraints are expressed in terms of the binary assignment decisions, ensuring that the solution adheres to the operational requirements of the business. The constraints are linear in nature, as they involve simple sums of binary variables without any nonlinear relationships.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating tables for drivers, schools, and distance matrix. Configuration logic updates include scalar parameters for distance calculation and business logic formulas for assignment constraints.

CREATE TABLE drivers (
  driver_id INTEGER,
  home_city STRING
);

CREATE TABLE schools (
  school_id INTEGER,
  location STRING
);

CREATE TABLE distance_matrix (
  driver_id INTEGER,
  school_id INTEGER,
  distance FLOAT,
  assign BOOLEAN
);
```

### Data Dictionary  
- **Drivers Table**: Contains information about drivers, including their unique identifiers and home cities.  
  - `driver_id`: Unique identifier for each driver, used to identify drivers in the optimization model.  
  - `home_city`: City where the driver resides, used to calculate the distance to schools.  

- **Schools Table**: Contains information about schools, including their unique identifiers and locations.  
  - `school_id`: Unique identifier for each school, used to identify schools in the optimization model.  
  - `location`: Location of the school, used to calculate the distance from drivers' home cities.  

- **Distance Matrix Table**: Contains the travel distances between drivers' home cities and schools' locations, as well as the binary assignment decisions.  
  - `driver_id`: Unique identifier for each driver, used to identify drivers in the optimization model.  
  - `school_id`: Unique identifier for each school, used to identify schools in the optimization model.  
  - `distance`: Travel distance between a driver's home city and a school's location, used as a coefficient in the objective function.  
  - `assign`: Binary decision variable indicating whether a driver is assigned to a school, used as a decision variable in the optimization model.  


=== Schema ===
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating tables for drivers, schools, and distance matrix. Configuration logic updates include scalar parameters for distance calculation and business logic formulas for assignment constraints.

CREATE TABLE drivers (
  driver_id INTEGER,
  home_city STRING
);

CREATE TABLE schools (
  school_id INTEGER,
  location STRING
);

CREATE TABLE distance_matrix (
  driver_id INTEGER,
  school_id INTEGER,
  distance FLOAT,
  assign BOOLEAN
);



=== LLM Response ===
```sql
-- Query Description: Retrieve all driver-school pairs with their corresponding distances.
-- This is essential for the objective function, which aims to minimize the total travel distance.
SELECT driver_id, school_id, distance
FROM distance_matrix;

-- Query Description: Retrieve all drivers and their home cities.
-- This information is necessary to understand the starting points for distance calculations.
SELECT driver_id, home_city
FROM drivers;

-- Query Description: Retrieve all schools and their locations.
-- This information is necessary to understand the destinations for distance calculations.
SELECT school_id, location
FROM schools;

-- Query Description: Retrieve the count of drivers and schools to ensure the problem is balanced.
-- This helps in understanding if there are enough drivers for each school and vice versa.
SELECT 
  (SELECT COUNT(*) FROM drivers) AS total_drivers,
  (SELECT COUNT(*) FROM schools) AS total_schools;

-- Query Description: Retrieve the minimum and maximum distances in the distance matrix.
-- This helps in understanding the range of distances, which can be useful for setting bounds in the optimization model.
SELECT MIN(distance) AS min_distance, MAX(distance) AS max_distance
FROM distance_matrix;

-- Query Description: Retrieve the average distance per driver.
-- This can be useful for understanding the typical travel distance a driver might incur.
SELECT driver_id, AVG(distance) AS avg_distance
FROM distance_matrix
GROUP BY driver_id;

-- Query Description: Retrieve the average distance per school.
-- This can be useful for understanding the typical travel distance to a school.
SELECT school_id, AVG(distance) AS avg_distance
FROM distance_matrix
GROUP BY school_id;

-- Query Description: Retrieve all driver-school pairs where the assignment is currently set to true.
-- This helps in understanding the current assignments and can be used to validate constraints.
SELECT driver_id, school_id, distance
FROM distance_matrix
WHERE assign = TRUE;

-- Query Description: Retrieve the number of schools each driver is currently assigned to.
-- This helps in ensuring that the constraint of at most one school per driver is not violated.
SELECT driver_id, COUNT(school_id) AS num_schools_assigned
FROM distance_matrix
WHERE assign = TRUE
GROUP BY driver_id;

-- Query Description: Retrieve the number of drivers each school is currently assigned to.
-- This helps in ensuring that the constraint of exactly one driver per school is not violated.
SELECT school_id, COUNT(driver_id) AS num_drivers_assigned
FROM distance_matrix
WHERE assign = TRUE
GROUP BY school_id;

-- Query Description: Retrieve the total distance for all current assignments.
-- This provides the current total travel distance, which is the value to be minimized.
SELECT SUM(distance) AS total_distance
FROM distance_matrix
WHERE assign = TRUE;

-- Query Description: Retrieve the driver-school pairs with the smallest distances.
-- This can be useful for identifying potential optimal assignments.
SELECT driver_id, school_id, distance
FROM distance_matrix
ORDER BY distance ASC
LIMIT 10;

-- Query Description: Retrieve the driver-school pairs with the largest distances.
-- This can be useful for identifying assignments that should be avoided.
SELECT driver_id, school_id, distance
FROM distance_matrix
ORDER BY distance DESC
LIMIT 10;
```

These queries are designed to retrieve the most relevant data for solving the optimization problem. They provide insights into the distances, current assignments, and constraints, which are crucial for formulating and solving the optimization model.
