Assignment model

The assignment problem is a special case of transportation problem in which the objective is to assign ‘m’ jobs or workers to ‘n’ machines such that the cost incurred is minimized.

1. Topic: ASSIGNMENT MODEL OPERATION RESEARCH

2. Assignment model

3. Assignment model Assigning of jobs to factors (men or machine) to get most optimum output or get least cost. Hungarian method is the mostly used method of solving assignment problems. Types of Assignment problems: Balanced Unbalanced

4. Objectives No workers is more than one job. No job is allotted to more than one worker. Total time taken to complete a job is minimum. The work done is cost effective.

5. Balanced assignment An assignment is called balanced assignment problem if the number of persons (factors) is same as the number of jobs. J 1 J 2 J 3 _ _ _ _ _ _ _ _ _ _ _ _ J n P 1 T 11 T 12 T 13 _ _ _ _ _ _ _ _ _ _ _ _ T 1n P 2 T 21 T 22 T 23 _ _ _ _ _ _ _ _ _ _ _ _ T 2n P 3 T 31 T 32 T 33 _ _ _ _ _ _ _ _ _ _ _ _ T 3n I I I I _ _ _ _ _ _ _ _ _ _ _ _ I I I I I _ _ _ _ _ _ _ _ _ _ _ _ I P n T n1 T n2 T n3 _ _ _ _ _ _ _ _ _ _ _ _ T nn

6. Example of balanced problem 1 2 3 4 A 1 8 4 1 B 5 7 6 5 C 3 5 4 12 D 3 1 6 3 PROFITS (IN 1000’s) PLANTS Solve the following assignment problem to get maximum profit:

7. METHOD STEP 1 Subtract all the element of each row from the largest elements of respective rows. STEP 2 Subtract the least element of each column from other elements of respective column and allot jobs. 1 2 3 4 A 7 4 7 B 2 1 2 C 9 7 8 D 3 5 3 PROFITS PLANTS PROFITS PLANTS 1 2 3 4 A 5 4 7 B 1 2 C 7 7 8 D 1 5 3

8. METHOD Hence the optimum solution is: A2 B1 C4 D3 Hence the total profit will be 8000+ 5000+ 12000 +6000 = 31000 8000 8000 12000 6000

9. Unbalanced assignment An assignment is called unbalanced assignment problem if the number of persons (factors) is not same as the number of jobs. J 1 J 2 J 3 _ _ _ _ _ _ _ _ _ _ _ _ J n-1 P 1 T 11 T 12 T 13 _ _ _ _ _ _ _ _ _ _ _ _ T 1(n-1) P 2 T 21 T 22 T 23 _ _ _ _ _ _ _ _ _ _ _ _ T 2(n-1) P 3 T 31 T 32 T 33 _ _ _ _ _ _ _ _ _ _ _ _ T 3(n-1) I I I I _ _ _ _ _ _ _ _ _ _ _ _ I I I I I _ _ _ _ _ _ _ _ _ _ _ _ I P n T n1 T n2 T n3 _ _ _ _ _ _ _ _ _ _ _ _ T n (n-1)

10. Example of Unbalanced problem Solve the following assignment problem to get it completed in least time: A B C D E 1 62 78 50 101 82 2 71 84 61 73 59 3 87 92 111 71 81 4 48 64 87 77 80 JOBS MACHINES 5

11. METHOD Subtract all the element of each row from the largest elements of respective rows. STEP 1 A B C D E 1 39 23 51 19 2 13 23 11 25 3 24 19 40 30 4 39 23 10 7 5 JOBS MACHINES

12. METHOD Subtract the least element of each column from other elements of respective column and allot jobs. STEP 2 V A B C D E 1 39 23 51 19 2 13 23 11 25 3 24 19 40 30 4 39 23 10 7 5 JOBS MACHINES

13. METHOD Cancel all zeros by drawing lines less than the number of rows or columns. STEP 3 A B C D E 1 39 23 51 19 2 13 23 11 25 3 24 19 40 30 4 39 23 10 7 5 JOBS MACHINES

14. METHOD Choose the least uncovered element and add it to all intersection points and subtract it from all uncovered elements. STEP 4 A B C D E 1 32 23 51 12 2 6 23 11 18 3 17 19 40 23 4 32 23 10 5 7 7 7 JOBS MACHINES

15. METHOD Hence the optimum solution is: A5 0 B2 84 C3 111 D1 101 E4 80 Hence the total profit will be 0+ 84+ 111 +101+ 80 = 376

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