# Complete Optimization Problem and Solution: department_management

## 1. Problem Context and Goals

### Context  
The company is focused on strategically distributing its total budget across various departments to enhance their overall performance rankings. Each department is allocated a portion of the total budget, represented by the decision variable fraction_allocated[i], which indicates the fraction of the total budget assigned to department i. The primary objective is to maximize the sum of the product of each department's ranking and its allocated budget fraction. The total budget available for allocation is a fixed amount, serving as a constraint in the optimization model. This approach ensures that the allocation process is linear, with clear operational parameters that align with the company's strategic goals. The business configuration includes a total budget of 3 billion, which is realistic for a mid-sized company aiming to optimize resource distribution across its departments.

### Goals  
The optimization goal is to maximize the overall effectiveness of the budget allocation by enhancing the total ranking of all departments. This is achieved by focusing on maximizing the sum of the ranking of each department multiplied by the fraction of the budget allocated to it. Success is measured by how well the allocation aligns with the expected ranking improvements, ensuring that the optimization process is linear and straightforward.

## 2. Constraints    

The budget allocation process is subject to specific constraints to ensure feasibility and effectiveness. The total budget allocated across all departments must not exceed the available budget of 3 billion. Additionally, each department's budget allocation fraction must be between 0 and 1, inclusive. These constraints are designed to maintain a linear relationship in the optimization model, ensuring that the allocation process is both practical and efficient.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating a new table for decision variables, adding a configuration parameter for total budget, and updating existing tables to align with optimization requirements.

CREATE TABLE department (
  Department_ID INTEGER,
  Ranking FLOAT,
  Budget_in_Billions FLOAT
);

CREATE TABLE decision_variables (
  Department_ID INTEGER,
  fraction_allocated FLOAT
);
```

### Data Dictionary  
The database schema is designed to support the optimization process by clearly defining the roles of each table and column in business terms:

- **Department Table**: This table captures essential information about each department, including its unique identifier, ranking, and budget requirements. The ranking serves as an objective coefficient in the optimization model, while the budget requirement acts as a constraint bound.

  - **Department_ID**: A unique identifier for each department, used to link data across tables.
  - **Ranking**: Represents the potential performance ranking of the department, serving as a key factor in determining budget allocation.
  - **Budget_in_Billions**: Indicates the budget requirement for each department, used to ensure that allocations do not exceed available resources.

- **Decision Variables Table**: This table records the fraction of the total budget allocated to each department, which is the primary decision variable in the optimization model.

  - **Department_ID**: Links the decision variable to the corresponding department.
  - **fraction_allocated**: Represents the fraction of the total budget allocated to the department, guiding the optimization process.

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical departmental budget allocations and rankings in a mid-sized company, ensuring that the total budget constraint is respected while allowing for meaningful optimization.

-- Realistic data for department
INSERT INTO department (Department_ID, Ranking, Budget_in_Billions) VALUES (1, 1.8, 0.6);
INSERT INTO department (Department_ID, Ranking, Budget_in_Billions) VALUES (2, 2.5, 1.2);
INSERT INTO department (Department_ID, Ranking, Budget_in_Billions) VALUES (3, 3.2, 1.8);

-- Realistic data for decision_variables
INSERT INTO decision_variables (Department_ID, fraction_allocated) VALUES (1, 0.25);
INSERT INTO decision_variables (Department_ID, fraction_allocated) VALUES (2, 0.35);
INSERT INTO decision_variables (Department_ID, fraction_allocated) VALUES (3, 0.4);
```