An assignment problem is a special form of transportation problem where all supply and demand values equal 1.

1. Introduction to the Assignment Problem

The assignment problem focuses on assigning resources to tasks, where each resource must be assigned to a task in such a way that the total cost is minimized, and each resource is assigned to exactly one task, with all supply and demand values equal to 1.

2. Formulation of the Problem

In mathematical terms, the assignment problem can be represented as a cost matrix $C$, where $C_{ij}$ represents the cost of assigning resource $i$ to task $j$. The goal is to find a matrix $X$ such that:

$X_{ij} = \begin{cases} 1 & \text{if resource } i \text{ is assigned to task } j \\ 0 & \text{otherwise} \end{cases}$

subject to the constraints:

$\sum_{j=1}^{n} X_{ij} = 1 \quad \forall i$ $\sum_{i=1}^{m} X_{ij} = 1 \quad \forall j$

Here, $m$ is the number of resources (or supply) and $n$ is the number of tasks (or demand).

3. Solving the Assignment Problem

Various algorithms exist for solving the assignment problem, including the Hungarian method, which is commonly used due to its efficiency. The process involves the following steps:

1. Construct the Cost Matrix: Formulate the cost matrix from the given data.
2. Row and Column Reduction: Subtract the smallest entry in each row from all entries of that row, and then do the same for each column.
3. Cover Zeros: Use the minimum number of lines to cover all zeros in the matrix.
4. Adjust the Matrix: If the number of lines is less than the size of the matrix, adjust the matrix by finding the smallest uncovered element and adding it to uncovered elements and subtracting it from elements that are covered twice.
5. Repeat: Continue the above steps until an optimal assignment is found.

4. Verification and Results

To verify the solution, check that each resource is assigned to exactly one task, and vice versa. The total cost can also be calculated by summing the respective costs from the original cost matrix according to the final assignment.

Final Answer

The assignment problem effectively allows for optimized resources allocation with minimal costs while ensuring efficiency. The specifics will depend on the particular cost matrix given in a practical scenario, but the outlined approach provides a systematic method to achieve an optimal solution.