Question

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.

Answer 1

37,528 containers of cookies

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For a production batch size of x containers, those x containers will be put into storage, and withdrawn at the uniform rate of 500,000 containers per year.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.