solved example: assignment model of operations research

1. Assignment Model
Problems

2. A fast food chain wants to build four stores. In the past, the chain has
used six different construction companies and, having been satisfied
with each, has invited each to bid on each job. The final bids (in
thousands of rupees) are show in the following table. Since the fast food
chain wants to have each of the new stores ready as quickly as possible,
it will award at most one job to a construction company. What
assignment results in minimum total cost to the fast food chain?
Construction Companies
1 2 3 4 5 6
Store 1 85.3 88 87.5 82.4 89.1 86.7
Store 2 78.9 77.4 77.4 76.5 79.3 78.3
Store 3 82 81.3 82.4 80.6 83.5 81.7
Store 4 84.3 84.6 86.2 83.3 84.4 85.5

3. Balancing the assignment
problem

4. Construction Companies
1 2 3 4 5 6
Store 1 85.3 88 87.5 82.4 89.1 86.7
Store 2 78.9 77.4 77.4 76.5 79.3 78.3
Store 3 82 81.3 82.4 80.6 83.5 81.7
Store 4 84.3 84.6 86.2 83.3 84.4 85.5
Dummy 1
Dummy 2

5. 1 2 3 4 5 6
S1 85.3 88 87.5 82.4 89.1 86.7
S2 78.9 77.4 77.4 76.5 79.3 78.3
S3 82 81.3 82.4 80.6 83.5 81.7
S4 84.3 84.6 86.2 83.3 84.4 85.5
D1 0 0 0 0 0 0
D2 0 0 0 0 0 0

6. Applying the Hungarian method

7. 1st Reduced Cost Matrix
SUBTRACTING THE MINIMUM ELEMENT OF
EACH ROW FROM EACH ELEMENT OF THAT
ROW

8. 1 2 3 4 5 6
S1 85.3–82.4 88–82.4 87.5–82.4 82.4–82.4 89.1–82.4 86.7–82.4
S2 78.9–76.5 77.4–76.5 77.4–76.5 76.5–76.5 79.3–76.5 78.3–76.5
S3 82–80.6 81.3–80.6 82.4–80.6 80.6–80.6 83.5–80.6 81.7–80.6
S4 84.3–83.3 84.6–83.3 86.2–83.3 83.3–83.3 84.4–83.3 85.5–83.3
D1 0–0 0–0 0–0 0–0 0–0 0–0
D2 0–0 0–0 0–0 0–0 0–0 0–0

9. 1 2 3 4 5 6
S1 2.9 5.6 5.1 0 6.7 4.3
S2 2.4 0.9 0.9 0 2.8 1.8
S3 1.4 0.7 1.8 0 2.9 1.1
S4 1 1.3 2.9 0 1.1 2.2
D1 0 0 0 0 0 0
D2 0 0 0 0 0 0

10. 2nd Reduced Cost Matrix
SUBTRACTING THE MINIMUM ELEMENT OF
EACH COLUMN FROM EACH ELEMENT OF
THAT COLUMN

11. 1 2 3 4 5 6
S1 2.9–0 5.6–0 5.1–0 0–0 6.7–0 4.3–0
S2 2.4–0 0.9–0 0.9–0 0–0 2.8–0 1.8–0
S3 1.4–0 0.7–0 1.8–0 0–0 2.9–0 1.1–0
S4 1–0 1.3–0 2.9–0 0–0 1.1–0 2.2–0
D1 0–0 0–0 0–0 0–0 0–0 0–0
D2 0–0 0–0 0–0 0–0 0–0 0–0

12. 1 2 3 4 5 6
S1 2.9 5.6 5.1 0 6.7 4.3
S2 2.4 0.9 0.9 0 2.8 1.8
S3 1.4 0.7 1.8 0 2.9 1.1
S4 1 1.3 2.9 0 1.1 2.2
D1 0 0 0 0 0 0
D2 0 0 0 0 0 0

13. Striking off all 0’s using
minimum number of
vertical/horizontal lines

14. 1 2 3 4 5 6
S1 2.9 5.6 5.1 0 6.7 4.3
S2 2.4 0.9 0.9 0 2.8 1.8
S3 1.4 0.7 1.8 0 2.9 1.1
S4 1 1.3 2.9 0 1.1 2.2
D1 0 0 0 0 0 0
D2 0 0 0 0 0 0

15. Subtracting the minimum uncut
element from other uncut elements,
and adding it to elements at the
intersections
SINCE MINIMUM NUMBER OF LINES
REQUIRED IS LESS THAN THE NUMBER OF
ROWS/COLUMNS

16. 1 2 3 4 5 6
S1 2.9–0.7 5.6–0.7 5.1–0.7 0 6.7–0.7 4.3–0.7
S2 2.4–0.7 0.9–0.7 0.9–0.7 0 2.8–0.7 1.8–0.7
S3 1.4–0.7 0.7–0.7 1.8–0.7 0 2.9–0.7 1.1–0.7
S4 1–0.7 1.3–0.7 2.9–0.7 0 1.1–0.7 2.2–0.7
D1 0 0 0 0+0.7 0 0
D2 0 0 0 0+0.7 0 0

17. 1 2 3 4 5 6
S1 2.2 4.9 4.4 0 6 3.6
S2 1.7 0.2 0.2 0 2.1 1.1
S3 0.7 0 1.1 0 2.2 0.4
S4 0.3 0.6 2.2 0 0.4 1.5
D1 0 0 0 0.7 0 0
D2 0 0 0 0.7 0 0

18. Striking off all 0’s using
minimum number of
vertical/horizontal lines

19. 1 2 3 4 5 6
S1 2.2 4.9 4.4 0 6 3.6
S2 1.7 0.2 0.2 0 2.1 1.1
S3 0.7 0 1.1 0 2.2 0.4
S4 0.3 0.6 2.2 0 0.4 1.5
D1 0 0 0 0.7 0 0
D2 0 0 0 0.7 0 0

20. Subtracting the minimum uncut
element from other uncut elements,
and adding it to elements at the
intersections
SINCE MINIMUM NUMBER OF LINES
REQUIRED IS LESS THAN THE NUMBER OF
ROWS/COLUMNS

21. 1 2 3 4 5 6
S1 2.2–0.2 4.9–0.2 4.4–0.2 0 6–0.2 3.6–0.2
S2 1.7–0.2 0.2–0.2 0.2–0.2 0 2.1–0.2 1.1–0.2
S3 0.7 0 1.1 0+0.2 2.2 0.4
S4 0.3–0.2 0.6–0.2 2.2–0.2 0 0.4–0.2 1.5–0.2
D1 0 0 0 0.7+0.2 0 0
D2 0 0 0 0.7+0.2 0 0

22. 1 2 3 4 5 6
S1 2 4.7 4.2 0 5.8 3.4
S2 1.5 0 0 0 1.9 0.9
S3 0.7 0 1.1 0.2 2.2 0.4
S4 0.1 0.4 2 0 0.2 1.3
D1 0 0 0 0.9 0 0
D2 0 0 0 0.9 0 0

23. Striking off all 0’s using
minimum number of
vertical/horizontal lines

24. 1 2 3 4 5 6
S1 2 4.7 4.2 0 5.8 3.4
S2 1.5 0 0 0 1.9 0.9
S3 0.7 0 1.1 0.2 2.2 0.4
S4 0.1 0.4 2 0 0.2 1.3
D1 0 0 0 0.9 0 0
D2 0 0 0 0.9 0 0

25. Subtracting the minimum uncut
element from other uncut elements,
and adding it to elements at the
intersections
SINCE MINIMUM NUMBER OF LINES
REQUIRED IS LESS THAN THE NUMBER OF
ROWS/COLUMNS

26. 1 2 3 4 5 6
S1 2 4.7 4.2 0 5.8 3.4
S2 1.5 0 0 0 1.9 0.9
S3 0.7 0 1.1 0.2 2.2 0.4
S4 0.1 0.4 2 0 0.2 1.3
D1 0 0 0 0.9 0 0
D2 0 0 0 0.9 0 0

27. 1 2 3 4 5 6
S1 2–0.1 4.7–0.1 4.2–0.1 0 5.8–0.1 3.4–0.1
S2 1.5 0 0 0+0.1 1.9 0.9
S3 0.7 0 1.1 0.2+0.1 2.2 0.4
S4 0.1–0.1 0.4–0.1 2–0.1 0 0.2–0.1 1.3–0.1
D1 0 0 0 0.9+0.1 0 0
D2 0 0 0 0.9+0.1 0 0

28. 1 2 3 4 5 6
S1 1.9 4.6 4.1 0 5.7 3.3
S2 1.5 0 0 0.1 1.9 0.9
S3 0.7 0 1.1 0.3 2.2 0.4
S4 0 0.3 1.9 0 0.1 1.1
D1 0 0 0 1 0 0
D2 0 0 0 1 0 0

29. Striking off all 0’s using
minimum number of
vertical/horizontal lines

30. 1 2 3 4 5 6
S1 1.9 4.6 4.1 0 5.7 3.3
S2 1.5 0 0 0.1 1.9 0.9
S3 0.7 0 1.1 0.3 2.2 0.4
S4 0 0.3 1.9 0 0.1 1.1
D1 0 0 0 1 0 0
D2 0 0 0 1 0 0

31. Making assignments
SINCE MINIMUM NUMBER OF LINES
REQUIRED IS EQUAL TO THE NUMBER OF
ROWS/COLUMNS

32. 1 2 3 4 5 6
S1 1.9 4.6 4.1 0 5.7 3.3
S2 1.5 0 0 0.1 1.9 0.9
S3 0.7 0 1.1 0.3 2.2 0.4
S4 0 0.3 1.9 0 0.1 1.1
D1 0 0 0 1 0 0
D2 0 0 0 1 0 0

33. 1 2 3 4 5 6
S1 1.9 4.6 4.1 0 5.7 3.3
S2 1.5 0 0 0.1 1.9 0.9
S3 0.7 0 1.1 0.3 2.2 0.4
S4 0 0.3 1.9 0 0.1 1.1
D1 0 0 0 1 0 0
D2 0 0 0 1 0 0
×

34. 1 2 3 4 5 6
S1 1.9 4.6 4.1 0 5.7 3.3
S2 1.5 0 0 0.1 1.9 0.9
S3 0.7 0 1.1 0.3 2.2 0.4
S4 0 0.3 1.9 0 0.1 1.1
D1 0 0 0 1 0 0
D2 0 0 0 1 0 0
×

35. 1 2 3 4 5 6
S1 1.9 4.6 4.1 0 5.7 3.3
S2 1.5 0 0 0.1 1.9 0.9
S3 0.7 0 1.1 0.3 2.2 0.4
S4 0 0.3 1.9 0 0.1 1.1
D1 0 0 0 1 0 0
D2 0 0 0 1 0 0
×
×
×
×

36. 1 2 3 4 5 6
S1 1.9 4.6 4.1 0 5.7 3.3
S2 1.5 0 0 0.1 1.9 0.9
S3 0.7 0 1.1 0.3 2.2 0.4
S4 0 0.3 1.9 0 0.1 1.1
D1 0 0 0 1 0 0
D2 0 0 0 1 0 0
×
×
×
×

37. 1 2 3 4 5 6
S1 1.9 4.6 4.1 0 5.7 3.3
S2 1.5 0 0 0.1 1.9 0.9
S3 0.7 0 1.1 0.3 2.2 0.4
S4 0 0.3 1.9 0 0.1 1.1
D1 0 0 0 1 0 0
D2 0 0 0 1 0 0
×
×
×
×
×
×

38. 1 2 3 4 5 6
S1 1.9 4.6 4.1 0 5.7 3.3
S2 1.5 0 0 0.1 1.9 0.9
S3 0.7 0 1.1 0.3 2.2 0.4
S4 0 0.3 1.9 0 0.1 1.1
D1 0 0 0 1 0 0
D2 0 0 0 1 0 0
×
×
×
×
×
×

39. 1 2 3 4 5 6
S1 1.9 4.6 4.1 0 5.7 3.3
S2 1.5 0 0 0.1 1.9 0.9
S3 0.7 0 1.1 0.3 2.2 0.4
S4 0 0.3 1.9 0 0.1 1.1
D1 0 0 0 1 0 0
D2 0 0 0 1 0 0
×
×
×
×
×
×
×
×

40. 1 2 3 4 5 6
S1 1.9 4.6 4.1 0 5.7 3.3
S2 1.5 0 0 0.1 1.9 0.9
S3 0.7 0 1.1 0.3 2.2 0.4
S4 0 0.3 1.9 0 0.1 1.1
D1 0 0 0 1 0 0
D2 0 0 0 1 0 0
×
×
×
×
×
×
×
×

41. 1 2 3 4 5 6
S1 1.9 4.6 4.1 0 5.7 3.3
S2 1.5 0 0 0.1 1.9 0.9
S3 0.7 0 1.1 0.3 2.2 0.4
S4 0 0.3 1.9 0 0.1 1.1
D1 0 0 0 1 0 0
D2 0 0 0 1 0 0
×
×
×
×
×
×
×
× ×
×

42. 1 2 3 4 5 6
S1 1.9 4.6 4.1 0 5.7 3.3
S2 1.5 0 0 0.1 1.9 0.9
S3 0.7 0 1.1 0.3 2.2 0.4
S4 0 0.3 1.9 0 0.1 1.1
D1 0 0 0 1 0 0
D2 0 0 0 1 0 0
×
×
×
×
×
×
×
× ×
×

43. Computing total cost

44. 1 2 3 4 5 6
S1 1.9 4.6 4.1 0 5.7 3.3
S2 1.5 0 0 0.1 1.9 0.9
S3 0.7 0 1.1 0.3 2.2 0.4
S4 0 0.3 1.9 0 0.1 1.1
D1 0 0 0 1 0 0
D2 0 0 0 1 0 0
×
×
×
×
×
×
×
× ×
×
1 2 3 4 5 6
S1 85.3 88 87.5 82.4 89.1 86.7
S2 78.9 77.4 77.4 76.5 79.3 78.3
S3 82 81.3 82.4 80.6 83.5 81.7
S4 84.3 84.6 86.2 83.3 84.4 85.5
D1 0 0 0 0 0 0
D2 0 0 0 0 0 0

45. 1 2 3 4 5 6
S1 1.9 4.6 4.1 0 5.7 3.3
S2 1.5 0 0 0.1 1.9 0.9
S3 0.7 0 1.1 0.3 2.2 0.4
S4 0 0.3 1.9 0 0.1 1.1
D1 0 0 0 1 0 0
D2 0 0 0 1 0 0
×
×
×
×
×
×
×
× ×
×
1 2 3 4 5 6
S1 85.3 88 87.5 82.4 89.1 86.7
S2 78.9 77.4 77.4 76.5 79.3 78.3
S3 82 81.3 82.4 80.6 83.5 81.7
S4 84.3 84.6 86.2 83.3 84.4 85.5
D1 0 0 0 0 0 0
D2 0 0 0 0 0 0

46. 1 2 3 4 5 6
S1 85.3 88 87.5 82.4 89.1 86.7
S2 78.9 77.4 77.4 76.5 79.3 78.3
S3 82 81.3 82.4 80.6 83.5 81.7
S4 84.3 84.6 86.2 83.3 84.4 85.5
D1 0 0 0 0 0 0
D2 0 0 0 0 0 0 Optimum Assignment and Total Cost
Construction Company Store Cost
1 4 Rs.84300
2 3 Rs.81300
3 2 Rs.77400
4 1 Rs.82400
5 - -
6 - -
Total Cost Rs.325400

47. Ans: Optimum Assignment & Total Cost
Construction Company Store Cost
1 4 Rs.84300
2 3 Rs.81300
3 2 Rs.77400
4 1 Rs.82400
5 - -
6 - -
Total Cost Rs.325400

48. A firm wants to purchase three different types of equipment and five
manufacturers have come forward to supply these machines. However,
the firm’s policy is not to accept more than one machine from any of the
manufacturers. The data relating to the price (in lakhs of rupees) quoted
by the different manufacturers are given below. Determine how best the
firm can purchase the three machines.
Machines
1 2 3
Manufacturers
A 2.99 3.11 2.68
B 2.78 2.87 2.57
C 2.92 3.05 2.80
D 2.82 3.10 2.74
E 3.11 2.90 2.64

49. A management consulting firm has a backlog of 4 contracts.
Work on these contracts must be started immediately. 3 project
leaders are available for assignment to the contracts. Because
of varying work experience of the project leaders the profit of
the consulting firm would vary based on the assignments as
shown below. The unassigned contract can be completed by
subcontracting it to an outside consultant. The profit from the
subcontract is zero. Determine the assignment of project
leaders to the contracts so as to maximise profits.
Contracts
1 2 3 4
Project
Leaders
A 13 10 9 11
B 15 17 13 20
C 6 8 11 7