Unit ii-3-am
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the assignment model of operations research (OR)
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Unit ii-3-am
- 3. • Suppose there are 5 machines and 5 jobs to be performed. • 1 machine can do only 1 job at a time. • How the machines should be assigned to the jobs? • Given – the cost of performing a job on a machine. • An Assignment Problem (AP) is always square. • The assignment is done on a one-to-one matching basis.
- 4. • Only one allocation is possible in a given row or column. • The AP is inherently degenerate, as the assignment is done on a one-to-one matching basis. (In the above 5 machine 5 job problem, there would be 25 cells, but allocation can be made only in 5 cells, whereas 9 cells should be allocated.) • Total number of assignments possible = n!
- 6. Hungarian Method Machine X Y Z Job A 25 31 35 B 15 20 24 C 22 19 17 • Machine opportunity and job opportunity cost to be determined. • The minimum element of a particular row or column to be subtracted from all elements of that row or column.
- 7. Machine X Y Z Job A 25 31 35 B 15 20 24 C 22 19 17
- 8. Machine X Y Z Job A 25 31 35 B 15 20 24 C 22 19 17
- 9. Machine X Y Z Job A 25-25 31-25 35-25 B 15-15 20-15 24-15 C 22-17 19-17 17-17
- 10. Machine X Y Z Job A 0 6 10 B 0 5 9 C 5 2 0 Job Opportunity Cost 1st Reduced Cost Matrix
- 11. Machine X Y Z Job A 0 6 10 B 0 5 9 C 5 2 0
- 12. Machine X Y Z Job A 0 4 10 B 0 3 9 C 5 0 0 Machine Opportunity Cost 2nd Reduced Cost (Total Opportunity Cost) Matrix
- 13. 0 4 10 0 3 9 5 0 0 • Optimal assignment is that assignment where total opportunity cost is zero. • We draw horizontal and vertical lines so as to cross all the zero elements using the minimum number of lines.
- 14. 0 4 10 0 3 9 5 0 0 • Optimal assignment is that assignment where total opportunity cost is zero. • We draw horizontal and vertical lines so as to cross all the zero elements using the minimum number of lines. • If the minimum number of lines required to do so is equal to the number of rows or columns, then optimum allocation is possible.
- 15. 0 4 10 0 3 9 5 0 0 • Optimal assignment is that assignment where total opportunity cost is zero. • We draw horizontal and vertical lines so as to cross all the zero elements using the minimum number of lines. • If the minimum number of lines required to do so is equal to the number of rows or columns, then optimum allocation is possible. • In this case it is not possible.
- 16. 0 4 10 0 3 9 5 0 0 • We identify the minimum element not covered by lines.
- 17. 0 4 10 0 3 9 5 0 0 • We identify the minimum element not covered by lines. • In this case it is 3.
- 18. 0 4 10 0 3 9 5 0 0 • We subtract it from…
- 19. Assignment Model: Applications • Assign salespeople to sales territories. • Assign vehicles to routes. • Assign accountants to client accounts. • Assign contracts to bidders through systematic evaluation of bids from competing suppliers. • Assign naval vessels to patrol sectors. • Schedule teachers to classes.
- 20. Assignment Model: Applications • Matching men to machines according to pieces produced per hour by each individual on each machine. • Matching teams to projects by the expected cost of each team to accomplish each project.