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 700,000 1 lb containers. The setup cost for each production run is $546, and the manufacturing cost is $0.47 for each container of cookies. The cost of storing each container of cookies over the year is $0.35. 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

Answer 1

46,734 units per run

Step-by-step explanation:

total estimated demand = 700,000 containers

setup costs per production run = $546

manufacturing cost = $0.47 per container

holding cost = $0.35 per container

r = 700,000 / x

total setup costs = 546r = 546 (700,000/x) = 382,200,000/x

production costs = 0.47 x 700,000 = 329,000

storage cost per unit= 1/2r x 0.35 = 0.35/2(700,000/x) = 0.35x/1,400,000

total storage costs = 700,000 x 0.35x/1,400,000 = 0.175x

C(x) = 382,200,000/x + 0.175 x + 329,000

now we find the derivative:

C'(x) = -382,200,000/x² + 0.175

382,200,000/x² = 0.175

382,200,000 = 0.175x²

x² = 382,200,000 / 0.175 = 2,184,000,000

x = √2,184,000,000 = 46,733.28 ≈ 46,734 units per run

this answer is based on a continuous production process, there are 14.98 runs per year