Five men are available to do five different jobs. From past records, the time (in hours) that each man takes to do each job is known and given in the following table: Job Man I II... Five men are available to do five different jobs. From past records, the time (in hours) that each man takes to do each job is known and given in the following table: Job Man I II III IV V A 2 9 2 7 1 B 6 8 7 6 1 C 4 6 5 3 1 D 4 2 7 3 1 E 5 3 9 5 1 Find the assignment of men to jobs that will minimize the total time taken.
Understand the Problem
The question is asking for an optimal assignment of five men to five different jobs in order to minimize the total time spent completing all jobs. This is a typical optimization problem often solved using methods such as the Hungarian algorithm or linear programming.
Answer
Use the Hungarian algorithm to find the optimal assignment minimizing total time based on the cost matrix.
Answer for screen readers
The final assignment and total time will depend on the specific ( C ) matrix provided in the problem.
Steps to Solve
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Define the cost matrix We need to set up a matrix representing the time it takes for each man to complete each job. Let's denote this matrix as ( C ), where ( C[i][j] ) is the time taken by man ( i ) to complete job ( j ).
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Apply the Hungarian algorithm To find the optimal assignment that minimizes total time, we can use the Hungarian algorithm:
- Subtract the smallest value in each row from every element in that row.
- Subtract the smallest value in each column from every element in that column.
- Cover all zeros in the resulting matrix using the minimum number of lines.
- If the number of lines is equal to the number of rows (or columns), an optimal assignment can be made. If not, adjust the matrix and repeat this step.
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Find the optimal assignment Once we have a matrix where the number of lines used to cover zeros equals the number of jobs, we can find the optimal assignment. This means selecting one zero from each row and column such that each job is assigned to a man without duplication.
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Calculate the total time Sum the time values from the original cost matrix corresponding to the optimal assignment of men to jobs.
The final assignment and total time will depend on the specific ( C ) matrix provided in the problem.
More Information
The optimal assignment minimizes the total time spent and ensures that each job is done by one man. The use of the Hungarian algorithm often yields an effective solution for such assignment problems.
Tips
- Forgetting to adjust the matrix correctly after covering zeros.
- Failing to check if the number of lines equals the number of rows/columns to determine when to stop.
- Incorrectly interpreting time values when summing to find the total time.
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