=== Problem Context ===
# Complete Optimization Problem and Solution: world_1

## 1. Problem Context and Goals

### Context  
A government agency is tasked with allocating a fixed budget across multiple cities to maximize the total population served. The allocation must ensure equitable distribution across districts while adhering to budget constraints. Each city has a specific population size, and the budget allocated to a city directly impacts the number of people served. The agency has established operational parameters to guide the allocation process:  
- **Total Budget Available for Allocation**: The agency has a fixed total budget of 1,000,000 units, which cannot be exceeded.  
- **Minimum Budget Allocation per City**: Each city must receive at least 50,000 units to ensure even smaller cities receive a fair share.  
- **Minimum Budget Allocation per District**: Each district must receive a minimum of 200,000 units to ensure equitable distribution across regions.  

The decision variables in this problem are the budget allocations to each city, represented as continuous values. The objective is to maximize the total population served by allocating the budget in a way that respects these constraints.  

### Goals  
The primary goal of this optimization problem is to maximize the total population served by the budget allocation. This is achieved by prioritizing cities with larger populations while ensuring that all cities and districts receive their minimum required allocations. Success is measured by the total population served, which is directly proportional to the budget allocated to each city. The optimization process ensures that the allocation is both efficient and equitable, aligning with the agency’s operational and policy objectives.  

## 2. Constraints  

The budget allocation process is subject to the following constraints:  
1. **Total Budget Limit**: The sum of all budget allocations across cities must not exceed the total available budget of 1,000,000 units.  
2. **Minimum Allocation per City**: Each city must receive at least 50,000 units of the budget to ensure a fair distribution.  
3. **Minimum Allocation per District**: The total budget allocated to all cities within a district must be at least 200,000 units to maintain equity across regions.  

These constraints ensure that the budget allocation is both feasible and aligned with the agency’s operational requirements.  

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating tables for budget allocation, districts, and cities. Configuration logic updates include adding scalar parameters for total budget, minimum allocation per city, and minimum allocation per district.

CREATE TABLE city (
  CityID INTEGER,
  Population INTEGER,
  DistrictID INTEGER
);

CREATE TABLE district (
  DistrictID INTEGER,
  DistrictName STRING
);

CREATE TABLE budget_allocation (
  CityID INTEGER,
  Allocation FLOAT
);
```

### Data Dictionary  
- **city**: Stores city-specific data, including population and district affiliation.  
  - **CityID**: Unique identifier for each city.  
  - **Population**: Population of the city, used as the coefficient in the objective function.  
  - **DistrictID**: District to which the city belongs, used for grouping in district-level constraints.  
- **district**: Stores district-specific data, including district names.  
  - **DistrictID**: Unique identifier for each district.  
  - **DistrictName**: Name of the district, used for business context.  
- **budget_allocation**: Stores the budget allocated to each city.  
  - **CityID**: City to which the budget is allocated.  
  - **Allocation**: Budget allocated to the city, representing the decision variable in the optimization problem.  


=== Schema ===
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating tables for budget allocation, districts, and cities. Configuration logic updates include adding scalar parameters for total budget, minimum allocation per city, and minimum allocation per district.

CREATE TABLE city (
  CityID INTEGER,
  Population INTEGER,
  DistrictID INTEGER
);

CREATE TABLE district (
  DistrictID INTEGER,
  DistrictName STRING
);

CREATE TABLE budget_allocation (
  CityID INTEGER,
  Allocation FLOAT
);



=== LLM Response ===
```sql
-- Query Description: Retrieve the population of each city along with its district ID.
-- This data is crucial for the objective function, as the goal is to maximize the total population served.
-- The population acts as the coefficient in the optimization problem.
SELECT CityID, Population, DistrictID
FROM city;

-- Query Description: Retrieve the total population per district.
-- This data is useful for understanding the distribution of population across districts, which can influence the allocation strategy.
SELECT DistrictID, SUM(Population) AS TotalPopulation
FROM city
GROUP BY DistrictID;

-- Query Description: Retrieve the current budget allocations for each city.
-- This data represents the decision variables in the optimization problem and is necessary to ensure that the allocations meet the constraints.
SELECT CityID, Allocation
FROM budget_allocation;

-- Query Description: Retrieve the total budget allocated per district.
-- This data is important for enforcing the district-level minimum allocation constraint.
SELECT DistrictID, SUM(Allocation) AS TotalAllocation
FROM budget_allocation
JOIN city ON budget_allocation.CityID = city.CityID
GROUP BY DistrictID;

-- Query Description: Retrieve the total budget allocated across all cities.
-- This data is necessary to ensure that the total budget constraint is not exceeded.
SELECT SUM(Allocation) AS TotalBudgetAllocated
FROM budget_allocation;

-- Query Description: Retrieve the list of cities with their current allocations and populations.
-- This data is useful for verifying that each city meets the minimum allocation requirement and for understanding the impact of allocations on population served.
SELECT city.CityID, city.Population, budget_allocation.Allocation
FROM city
LEFT JOIN budget_allocation ON city.CityID = budget_allocation.CityID;

-- Query Description: Retrieve the list of districts with their names and IDs.
-- This data is useful for providing context and ensuring that district-level constraints are applied correctly.
SELECT DistrictID, DistrictName
FROM district;

-- Query Description: Retrieve the number of cities per district.
-- This data is useful for understanding the distribution of cities across districts, which can influence the allocation strategy.
SELECT DistrictID, COUNT(CityID) AS NumberOfCities
FROM city
GROUP BY DistrictID;

-- Query Description: Retrieve the cities that currently have allocations below the minimum required (50,000 units).
-- This data is important for identifying cities that need adjustments to meet the minimum allocation constraint.
SELECT city.CityID, city.Population, budget_allocation.Allocation
FROM city
LEFT JOIN budget_allocation ON city.CityID = budget_allocation.CityID
WHERE budget_allocation.Allocation < 50000 OR budget_allocation.Allocation IS NULL;

-- Query Description: Retrieve the districts that currently have total allocations below the minimum required (200,000 units).
-- This data is important for identifying districts that need adjustments to meet the minimum allocation constraint.
SELECT city.DistrictID, SUM(budget_allocation.Allocation) AS TotalAllocation
FROM city
LEFT JOIN budget_allocation ON city.CityID = budget_allocation.CityID
GROUP BY city.DistrictID
HAVING SUM(budget_allocation.Allocation) < 200000 OR SUM(budget_allocation.Allocation) IS NULL;
```

These queries are designed to retrieve the most relevant data for solving the optimization problem, including data for the objective function, decision variables, and constraints. They also provide aggregated and summary data that can help in understanding the current state of allocations and identifying areas that need adjustment.
