MK Kuantitatif - Metode Penugasan

Summary

This video explains the Hungarian method for solving assignment problems, specifically for optimizing task allocation in a company. It covers the key steps involved, such as performing column and row reductions, scanning for zeros, and revising the table to identify optimal task assignments. The method is demonstrated through a practical example involving machines and tasks, aiming to minimize operational costs. The video also discusses special cases, such as imbalanced worker-task numbers, and how to handle them. The Hungarian method is essential for efficient resource allocation in optimization problems, ensuring cost-effective solutions.

Takeaways

• 😀 The Hungarian method is used to solve assignment problems in companies by optimally allocating workers (machines) to tasks (jobs) to minimize costs or maximize profits.
• 😀 The method assumes that the number of workers equals the number of tasks, and each worker can only perform one task, while each task can only be performed by one worker.
• 😀 The process begins by creating a table where workers (machines) are listed in rows and tasks (jobs) in columns.
• 😀 In Step 1, the smallest value in each column is subtracted from all values in that column to prepare the data for further analysis.
• 😀 In Step 2, the smallest value in each row is subtracted from all values in that row to simplify the table further.
• 😀 Step 3 involves scanning for zeroes in the matrix, followed by drawing horizontal and vertical lines to identify optimal assignment opportunities.
• 😀 The method involves revising the table by subtracting the smallest unmarked value from all unmarked entries and adding it to the intersections of the lines.
• 😀 The process continues by checking if the number of lines equals the number of tasks/ workers, and if not, the steps are repeated until an optimal solution is found.
• 😀 The final step is to select tasks based on the marked zeros in the table to ensure the minimum cost allocation of tasks to workers.
• 😀 If the number of workers and tasks is unbalanced, dummy rows or columns with zero values are added to balance the matrix before proceeding with the method.
• 😀 The Hungarian method can also be applied to maximization problems, with specific adjustments like multiplying values by negative one and looking for the largest negative values in the columns.
• 😀 To handle cases where the number of workers and tasks doesn't match, additional rows or columns filled with zero values are added, ensuring the matrix is balanced for processing.

Q & A

What is the Hungarian method used for in assignment problems?

The Hungarian method is used to solve assignment problems, specifically to allocate tasks to workers in a way that minimizes costs or maximizes profits. It is a mathematical optimization technique.

What are the key assumptions for the Hungarian method to work effectively?

The key assumptions are: the number of workers and the number of jobs must be equal, each worker can only perform one job, and each job must be assigned to exactly one worker.

What happens if the number of workers and jobs is not the same?

If the number of workers and jobs is not equal, dummy rows or columns are added to balance the matrix. These dummy rows/columns are filled with zeros to make the number of workers equal to the number of jobs.

What is the first step in the Hungarian method for solving an assignment problem?

The first step is to subtract the smallest number in each column from all the numbers in that column. This helps to normalize the values and prepare the matrix for further steps.

How do you perform the second step of the Hungarian method?

In the second step, you subtract the smallest number in each row from all the numbers in that row. This ensures that each row has at least one zero, which is important for the subsequent steps.

What is the purpose of scanning columns and rows in the Hungarian method?

The purpose of scanning columns and rows is to identify where zeros occur and to draw lines through the rows and columns that contain these zeros. These lines help in determining whether the assignment is optimal or further adjustments are needed.

How do you adjust the matrix when the number of lines does not equal the number of jobs or workers?

When the number of lines does not equal the number of jobs or workers, you adjust the matrix by finding the smallest number not covered by a line, subtracting it from all uncovered numbers, and adding it to the intersections of the lines.

What do you do if the assignment problem involves maximizing something instead of minimizing costs?

If the problem involves maximizing something, all the values in the matrix are multiplied by -1. Then, the process follows similarly to minimizing, but the goal becomes finding the largest values rather than the smallest.

What is the main difference between the Hungarian method for minimization and maximization?

The main difference is that in maximization problems, all matrix values are multiplied by -1, and the steps for selecting the smallest values in rows and columns are replaced with selecting the largest positive values.

Can the Hungarian method be used for any assignment problem?

The Hungarian method is ideal for assignment problems where the number of workers and jobs is equal. For imbalanced problems, dummy rows or columns need to be added, and if the goal is maximization, the values must be adjusted by multiplying by -1.