1. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India

2. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India ASSIGNMENT MODEL 5 CHAPTER

3. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 3 Learning Objectives Solve assignment problems with the Hungarian (matrix reduction) method Solve minimisation as well as maximisation problems.

4. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 4 Assignment Model - Characteristics Assignment model deals with one to one assignment of workers to machines, jobs to machines, men to tasks and so on. It is a special case of transportation model. Each allocation has a cost just as each route has a cost in the transportation model. Number of jobs and machines or men and tasks and so on must be equal. Only one assignment is made in each row or column.

5. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 5 Assignment Problem 1 2 3 4 A D C B 36 20 31 17 24 32 40 12 22 40 38 18 16 39 35 36

6. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 6 Assignment Problem Jobs Machines 1234 A20363117 B24324012 C22403818 D36393516

7. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 7 Calculate row opportunity cost by subtracting the smallest number in each row from every number in that row Jobs Machines 1234 A20363117 B24324012 C22403818 D36393516 Jobs Machines 1234 A319140 B1220280 C422200 D 23190 Step 1 – Determine the Opportunity Cost Table

8. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 8 Calculate total opportunity cost by subtracting the smallest number in each column from every number in that column Jobs Machines 1234 A319140 B1220280 C422200 D 23190 Jobs Machines 1234 A0000 B91140 C1360 D17450

9. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 9 Draw the minimum number of vertical and horizontal straight lines necessary to cover zeros in the table Jobs Machines 1234 A0000 B91140 C1360 D17450 1 2 Step 2 – Determine if an optimal assignment can be made or not

10. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 10 If the number of lines equals the number of rows or columns, then one can make an optimal assignment If the number of lines does not equal the number of rows or columns –subtract the smallest number not covered by a line from every other uncovered number –add the same number to any number lying at the intersection of any two lines (twice covered number) –return to step 2

11. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 11 The least uncovered number is 1. Subtract from every other uncovered number and add to twice covered numbers. 05417D 0631C 01419B 0000A 4321 Machines Jobs 1 2 Machines 1234 A0001 B80130 C0250 D16340 Step 3 – Revise Total Opportunity Cost

12. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 12 Jobs Machines 1234 A0001 B80130 C0250 D16340 Check if optimal assignment can be made Draw the minimum number of vertical and horizontal straight lines necessary to cover zeros in the table

13. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 13 As number of lines is equal to the number of jobs i.e. 4, an optimal assignment can be made Start with a row or column with only one zero. Assign that cell. Delete all other cells with zero in that row or column and repeat procedure 04316D 0520C 01308B 1000A 4321 Machines Jobs

14. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 14 Solution Assign Job A to Machine 3 Rs 31 Assign Job B to Machine 2 Rs 32 Assign Job C to Machine 1 Rs 22 Assign Job D to Machine 4 Rs 16 Total Rs 101

15. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 15 Unbalanced Problem In case the number of rows and columns are not equal, add a dummy row or column as required. The cost of assigning a dummy or assigning to a dummy is zero.

16. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 16 Maximisation Problem For a maximisation problem, gains or profits are converted into opportunity loss by subtracting all values from the highest value. Since the problem is to maximise profits, it can be solved by minimising the opportunity loss.

17. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 17 Summary of Steps Step 1 – Determine the Opportunity Cost Table –Calculate row opportunity cost by subtracting the smallest number in each row from every number in that row –Calculate total opportunity cost by subtracting the smallest number in each column from every number in that column

18. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 18 Summary of Steps Step 2 – Determine if an optimal assignment can be made or not –Draw the minimum number of vertical and horizontal straight lines necessary to cover zeros in the table –If the number of lines equals the number of rows or columns, then one can make an optimal assignment. If not, go to Step 3.

19. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 19 Summary of Steps Step 3 – Revise Total Opportunity Cost –subtract the smallest number not covered by a line from every other uncovered number –add the same number to any number lying at the intersection of any two lines (twice covered number) –return to step 2.

20. Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 20 Summary of Steps In case an optimal assignment can be made, start with a row or column with only one zero. Assign that cell. Delete all other cells with zero in that row or column and repeat this procedure till all assignments are made.