Assignment Problems

Assignment Model: Hungarian Algorithm and Its Applications

The Assignment Problem is a specific type of optimization problem where resources or tasks need to be allocated to activities on a one-to-one basis. The goal is to minimize the total cost or maximize the total profit. The Hungarian Algorithm is a popular method for solving such problems efficiently.

Mathematical Formulation

Variables:

Let $x_{ij}$ be a binary variable where:

$$x_{ij}=\{\begin{array}{ll}1& \text{if job }i\text{ is assigned to facility }j\\ 0& \text{otherwise<span style="font-size: x-large; text-align: left;"></span>}\end{array}$$

Objective Function:

Minimize total assignment cost:

$$\text{Minimize }Z=\sum _{i=1}^n\sum _{j=1}^n{C}_{ij}{x}_{i\mathrm{j<span\; style="font-size:\; x-large;"></span>}}$$

where $C_{ij}$ is the cost of assigning job $i$ to facility $j$.

Constraints:

1. Each job is assigned to exactly one facility:

$$\sum _{j=1}^n{x}_{ij}=1\text{for all }i$$

2. Each facility handles exactly one job:

$$\sum _{i=1}^n{x}_{ij}=1\text{for all }j$$

3. Binary constraint:

$$x_{ij}\in \{0,1\}$$

Hungarian Algorithm Steps

The Hungarian Algorithm is used for solving assignment problems efficiently. It involves the following steps:

• Row Reduction:
  • For each row, subtract the smallest element in that row from all elements of that row. This ensures that each row has at least one zero.
• Column Reduction:
  • For each column, subtract the smallest element in that column from all elements of that column. This ensures that each column has at least one zero.
• Assignment:
  • Mark Zeros:
    • In the reduced matrix, look for rows and columns with exactly one zero and make assignments to these zeros.
    • Cross out all other zeros in the same row and column as the assigned zero to prevent multiple assignments in the same row or column.
  • Repeat:
    • Continue marking and assigning zeros in rows and columns until all rows and columns have exactly one assignment.
• Optimality Check:
  • Cover Zeros:
    • Draw the minimum number of lines (horizontal and vertical) required to cover all zeros in the matrix.
    • Steps to Draw Lines:
      • Mark all rows without assignments.
      • Mark all columns containing zeros in these rows.
      • Mark all rows that have assignments in the marked columns.
      • Repeat until no more rows or columns can be marked.
    • If the number of lines is equal to the number of rows or columns, an optimal assignment has been found. Otherwise, proceed to the next step.
• Adjust the Matrix:
  • Adjust Values:
    • Find the smallest element not covered by any line.
    • Subtract this smallest element from all uncovered elements.
    • Add this smallest element to the elements at the intersections of lines.
  • Repeat:
    • Return to the assignment step with the adjusted matrix.

Applications of the Hungarian Algorithm

• Job Assignment:
  • Assign workers to tasks or jobs to minimize total cost or maximize efficiency.
• Resource Allocation:
  • Allocate resources or machines to jobs to minimize the total operational cost.
• Scheduling:
  • Optimal scheduling of tasks in production processes or project management.
• Transportation and Logistics:
  • Assign shipments or routes to minimize transportation costs.

Important Considerations

• Complexity:
  • The Hungarian Algorithm operates in polynomial time, making it efficient for large assignment problems.
• Degeneracy:
  • The algorithm effectively handles degeneracy by ensuring that the number of assignments matches the number of rows or columns.
• Versatility:
  • While the Hungarian Algorithm is tailored for square assignment problems (equal number of rows and columns), it can be adapted for rectangular problems by adding dummy rows or columns.

By following these steps, the Hungarian Algorithm provides an efficient and effective solution to assignment problems, ensuring optimal allocation of resources or tasks.