Optimal Assignment using the Hungarian Algorithm: An Example, Lecture notes of Operational Research

Download Optimal Assignment using the Hungarian Algorithm: An Example and more Lecture notes Operational Research in PDF only on Docsity! MULTIPLE SOLUTIONS Example 1 Given the following matrix, find the optimal assignment. 1 2 3 4 5 1 2 3 4 5 5 0 3 2 6 0 0 5 4 7 0 3 0 4 0 0 1 0 3 0 6 5 0 0 0 Solution: Note that all the rows and columns have at least one zero. Row 1 has a single zero in column 2. So make an assignment, delete (mark X) the second zero in column 2. This is shown in table 7. 0 denotes assignment Table 7 1 2 3 4 5 1 2 3 4 5 5 0 3 2 6 0 0 5 4 7 0x 3 0 4 0 0x 1 0 3 0 6 5 0x 0 0x Row 2 has a single zero in the first column. So make an assignment and delete the remaining zeros in column 1 as shown in table 8. Table 8 1 2 3 4 5 docsity.com 1 2 3 4 5 5 0 3 2 6 0 0x 5 4 7 0x 3 0 4 0 0x 1 0 3 0 6 5 0 0 0 Row 3, 4 and 5 have more than a single zero. So we skip these rows and examine the columns. Columns 3 has three zeros and so omit it. Column 4 has a single zero in row 5. So we make an assignment, deleting the remaining zeros in row 5. The result is as shown in table 9. Table 9 1 2 3 4 5 1 2 3 4 5 5 0 3 2 6 0 0x 5 4 7 0x 3 0 4 0 0x 1 0 3 0 6 5 0x 0 0x Now we have two zeros in rows 3 and 4 in columns 3 and they occupy the corners of a square. An arbitrary assignment has to be made. If we make an assignment in (3, 3) and delete the remaining zero in row 3 and in column 3, this leaves one zero in the position (4, 5) and an assignment is made there. Thus we have a solution to the problem as in table 10. Table 10 1 2 3 4 5 1 2 3 5 0 3 2 6 0 0x 5 4 7 0x 3 0 4 0x 0 1 0x 3 0 docsity.com