# 3. ASSIGNMENT MODEL1.pptx

29. May 2023
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### 3. ASSIGNMENT MODEL1.pptx

• 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 shown 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 SUB TR AC TI NG THE M INI M UM ELE MENT OF E AC H R O W FR O M E AC H E LE M E NT O F 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 SUB TRACTI NG THE MINI MUM ELE MENT OF EACH COLUM N FR OM EACH ELE MENT OF THAT COLUM N
• 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 SI NC E MINI MUM NUM BER OF LI NE S REQUI RED IS LE SS THA N THE NUM BER 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 SI NC E MINI MUM NUM BER OF LI NE S REQUI RED IS LE SS THA N THE NUM BER 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 2 0. 2.2 0.4 S4 0.1 0.4 2 0 0.2 1.3 D1 0 0 0 9 0. 0 0 D2 0 0 0 9 0. 0 0
• 25. Subtracting the minimum uncut element from other uncut elements, and adding it to elements at the intersections SI NC E MINI MUM NUM BER OF LI NE S REQUI RED IS LE SS THA N THE NUM BER 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 2 0. 2.2 0.4 S4 0.1 0.4 2 0 0.2 1.3 D1 0 0 0 9 0. 0 0 D2 0 0 0 9 0. 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 1 0. 1.9 0.9 S3 0.7 0 1.1 3 0. 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 SI NC E MINI MUM NUM BER OF LI NE S REQUI RED IS EQUAL TO THE NUM BER 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 TotalCost 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 & TotalCost 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
• 57. 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
• 58. 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