Neilsen cookie company sells its assorted butter cookies in containers that have a net content of 1 lb. the estimated demand for the cookies is 500,000 1 lb containers. the setup cost for each production run is $507, and the manufacturing cost is $0.53 for each container of cookies. the cost of storing each container of cookies over the year is $0.36. assuming uniformity of demand throughout the year and instantaneous production, how many containers of cookies should neilsen produce per production run in order to minimize the production cost? hint: following the method of example 5, show that the total production cost is given by the function below. then minimize the function c on the interval (0, 500,000). (round your answer to the nearest whole number.) c(x) = 253,500,000 x + 0.18x + 265,000.

  37,528 containers of cookies

Step-by-step explanation:

a) Cost Function

For the fixed annual demand of 500,000 containers, the manufacturing cost of $0.53 each will total ...

  manufacturing cost = 500,000×$0.53 = $265,000 annually.

For a production batch size of x containers, those x containers will be put into storage, and withdrawn at the uniform rate of 500,000 containers per year. On average, (1/2)x containers will be in storage, so storage costs will be ...

  storage cost = (1/2)($0.36x) = $0.18x . . . . annually

For annual demand of 500,000 containers, and a production batch size of x, there will be 500,000/x production batches each year. The setup cost is $507 for each of those, so the annual setup cost is ...

  setup cost = (500,000/x)($507) = $253,500,000/x . . . . annually

The total cost of producing 500,000 containers annually will be ...

  C = setup cost + storage cost + manufacturing cost

  C(x) = 253,500,000/x + 0.18x + 265,000

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b) Minimal-Cost Batch Size

Cost will be minimized when its derivative with respect to x is zero.

  dC/dx = -253,500,000/x^2 +0.18 = 0

  x^2 = 253,500,000/0.18 . . . . . . . solve for x^2

  x ≈ 37,527.8 ≈ 37,528 . . . . . . . . . take the square root

The size of the production run that will minimize production cost is 37,528 boxes.

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Comment on the solution

You may notice that the equation we finally solve for batch size is equivalent to one that sets setup cost equal to storage cost:

  253,500,000/x = 0.18x

This relation (setup cost = storage cost) is the general solution for this sort of problem regarding batch size.

6hryt%4578855

1. 3.4/4

2. 0.8/7

3. 0.5 x 3

4. 0.9 x 6

5. 10.8

6. 2.35

7. 0.567

8. 9.35

9. 0.02

10. 0.06

11. 0.16

12. 0.42

13. 0.63

14. 5.6

15. 8.1

16. 9.9

17. 0.28

18. 0.48

19. 0.27

20. 0.49

21. 0.45

22. 0.72

23. 0.9

24. 0.88

25. 0.75

26. 0.16

27. 0.328

28. 0.495

29. 0.72

30. 1.068