Introduction
The assignment model is a specialized optimization technique derived from the transportation model, focused on efficiently allocating resources such as employees to tasks, machines to jobs, or teachers to classes while minimizing total assignment costs.
Overview of the Assignment Model
The assignment model involves determining whether to allocate a specific source (e.g., worker) to a specific destination (e.g., task), with each decision represented as a binary variable (0 or 1). For example, if there are three employees and three tasks, each unique pairing is a potential assignment, resulting in 3!=6 possible ways to assign all employees to all tasks.
Key Characteristics
- One-to-One Assignment: Each task receives only one assignee, and each assignee is assigned only one task.
- Square Cost Matrix: The problem is represented by a square matrix, reflecting equal numbers of sources and destinations.
- Unique Assignment per Row/Column: The optimal solution guarantees that each row and column of the cost matrix contains only one assignment.
- Unit Capacity and Demand: Both the supply and demand at each source and destination are exactly one unit, distinguishing the assignment problem from general transportation problems.
Problem Constraints
- One Assignment per Task and Worker: Each assignment must have at most one assignee, and each assignee receives at most one assignment.
- Non-negativity: All assignment values must be non-negative.
- Integer Programming: Assignments are restricted to integers (specifically, 0 or 1), ensuring binary decisions.
Applications
Assignment models are widely used in various domains, including:
- Assigning machines to specific factory orders.
- Allocating salespeople to territories.
- Matching teachers to classes.
- Assigning accountants to client accounts.
- Contract allocation to bidders during bid evaluations.
Solution Methods
Several approaches can be used to solve assignment problems:
Enumeration Method
- List all possible assignments and choose the one with minimal cost, time, or distance.
- Limitation: Feasible only for small problems due to rapid growth in the number of possible combinations.
Simplex Method
- Utilizes the linear programming simplex algorithm to solve the assignment problem, particularly suitable for problems too large for graphical methods.
- May be less efficient for highly degenerate assignment problems.
Transportation Method
- Treats the assignment model as a special case of the transportation problem and applies transportation algorithms accordingly.
Hungarian Method
- Developed by Hungarian mathematician D. König and popularized by Kuhn, the Hungarian algorithm efficiently solves assignment problems using a systematic, matrix-based procedure.
- Recognized as the fastest and most efficient way to solve assignment problems.
Assignment Problem Table Format
Below is an example of an assignment problem table as described:
|
Task 1 |
Task 2 |
Task 3 |
|
|
Worker 1 |
11 |
9 |
17 |
|
Worker 2 |
13 |
7 |
15 |
|
Worker 3 |
16 |
10 |
12 |
The assignment problem is typically represented as a matrix, where:
- Rows: Represent objects (e.g., workers) to be assigned.
- Columns: Represent targets (e.g., tasks).
- Entries: Indicate the cost (or profit) of assigning a specific worker to a particular task.
Conclusion
The assignment model is a pivotal tool in operations research, allowing for the optimal allocation of resources in a variety of practical settings. While it shares similarities with the transportation model, its distinguishing features such as binary decision variables, a square cost matrix, and unique assignments make specialized solution methods like the Hungarian algorithm especially effective.
