Describe the algorithm for assignment problem.
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Describe the algorithm for assignment problem.
The Assignment Problem is a mathematical optimization problem that seeks to determine the most efficient assignment of tasks to resources, minimizing the total cost or time required to complete all tasks. The algorithm for solving the Assignment Problem is known as the Hungarian Algorithm, also referred to as the Munkres Algorithm. Here's a brief description of the algorithm:
Step 1: Initialization: Start with a cost matrix representing the costs associated with assigning each task to each resource. If necessary, convert the cost matrix into a square matrix by adding dummy rows or columns to ensure an equal number of tasks and resources.
Step 2: Row Reduction: Reduce each row of the cost matrix by subtracting the minimum cost in that row from all elements in the row. This ensures that at least one zero is present in each row.
Step 3: Column Reduction: Reduce each column of the cost matrix by subtracting the minimum cost in that column from all elements in the column. This ensures that at least one zero is present in each column.
Step 4: Assignment: Starting from the top-left corner of the cost matrix, find the smallest uncovered element (i.e., not crossed out) and mark it. Then, check for other uncovered elements in the same row or column and mark them as well. Repeat this process until all rows and columns have at least one marked element.
Step 5: Test for Optimality: Check whether the number of marked elements equals the number of rows or columns. If it does, the current assignment is optimal. If not, proceed to the next step.
Step 6: Adjust Matrix: Determine the smallest uncovered element (let it be (m)) and subtract (m) from all uncovered elements. Add (m) to all elements covered by two lines (rows and columns) and remove one line (row or column) from each covered zero.
Step 7: Repeat: Repeat Steps 4 to 6 until an optimal assignment is achieved, where each row and column has exactly one marked element.
Step 8: Interpretation: Interpret the assignment by identifying which tasks are assigned to which resources based on the marked elements in the cost matrix.
The Hungarian Algorithm efficiently solves the Assignment Problem in polynomial time complexity, making it suitable for applications in logistics, scheduling, and resource allocation.