# Complete Optimization Problem and Solution: department_management

## 1. Problem Context and Goals

### Context  
A government agency is tasked with allocating its budget across various departments to maximize overall efficiency. The efficiency of each department is determined by two key factors: its ranking within the agency and the number of employees it has. The agency has established weights to prioritize these factors, with department ranking being more influential than the number of employees. The total budget available for allocation is fixed, and each department must receive a minimum budget to function effectively, while also not exceeding a maximum budget limit. The goal is to distribute the budget in a way that maximizes the overall efficiency score across all departments, ensuring that the allocation respects both the minimum and maximum budget constraints for each department.

The agency uses specific weights to calculate the efficiency score: a weight of 0.7 is assigned to department ranking, and a weight of 0.3 is assigned to the number of employees. These weights reflect the agency's prioritization of department ranking over employee count in determining efficiency. The total budget available for allocation is set at $1,000,000, which must be distributed across all departments while adhering to their individual budget constraints.

### Goals  
The primary goal of this optimization problem is to maximize the overall efficiency score of all departments. This efficiency score is calculated as a weighted sum of each department's ranking and the number of employees, multiplied by the budget allocated to that department. The agency aims to allocate the budget in such a way that the total efficiency score is as high as possible, while ensuring that each department receives at least its minimum required budget and does not exceed its maximum budget limit. Success will be measured by achieving the highest possible efficiency score within the constraints of the total budget and the individual department budget limits.

## 2. Constraints    

The budget allocation must adhere to the following constraints:

1. **Total Budget Constraint**: The sum of the budgets allocated to all departments must not exceed the total budget available, which is $1,000,000. This ensures that the agency does not overspend its available resources.

2. **Minimum Budget Constraint**: Each department must receive at least its minimum required budget to function effectively. This ensures that no department is underfunded to the point where it cannot perform its essential functions.

3. **Maximum Budget Constraint**: Each department cannot receive more than its maximum budget limit. This ensures that no single department consumes an excessive portion of the total budget, allowing for a more equitable distribution of resources.

These constraints ensure that the budget allocation is both feasible and fair, respecting the operational needs of each department while staying within the overall budget limits.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating tables for department budgets and efficiency weights, and moving scalar parameters like total budget and weights to business_configuration_logic.json.

CREATE TABLE department_budgets (
  department_id INTEGER,
  minimum_budget FLOAT,
  maximum_budget FLOAT,
  allocated_budget FLOAT
);
```

### Data Dictionary  
The `department_budgets` table contains the following columns, each with a specific business purpose and optimization role:

- **department_id**: A unique identifier for each department. This column is used to index the budget constraints for each department.
  
- **minimum_budget**: The minimum budget required for each department to function effectively. This value serves as the lower bound for the budget allocation decision.

- **maximum_budget**: The maximum budget that can be allocated to each department. This value serves as the upper bound for the budget allocation decision.

- **allocated_budget**: The budget that is currently allocated to each department. This column represents the decision variable in the optimization problem, as it is the amount that will be adjusted to maximize the overall efficiency score.

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical government department budgets, ensuring a realistic distribution of minimum and maximum budgets across departments. The weights for efficiency calculation were set to prioritize department ranking slightly over the number of employees, reflecting common government efficiency metrics.

-- Realistic data for department_budgets
INSERT INTO department_budgets (department_id, minimum_budget, maximum_budget, allocated_budget) VALUES (1, 100000, 300000, 200000);
INSERT INTO department_budgets (department_id, minimum_budget, maximum_budget, allocated_budget) VALUES (2, 80000, 250000, 150000);
INSERT INTO department_budgets (department_id, minimum_budget, maximum_budget, allocated_budget) VALUES (3, 50000, 200000, 100000);
```