A machine shop manufactures custom brake disks for high performance cars.

Question

A machine shop manufactures custom brake disks for high performance cars. The shop needs to meet a demand of 2000 units per year. For each production run there is a setup cost of 1000 dollars, and there is a cost of 110 dollars per unit for materials and supplies. Storage of unsold units costs the shop 18 dollars per year. How many units should the shop produce on each production run in order to minimize inventory costs (setup plus storage)? Round your answer to the nearest integer. 

Answer 1

The shop will do a run of x units and when that runs out will run x more units and so on until all 2000 units are manufactured. On average, there will be x/2 units in storage at a cost of $18 per year. The number of runs will be 2000/x with an overhead of $1000 per run and $110 per unit. So the cost is:

C(x) = Storage_Cost + Manufacturing_Cost = 18(x/2) + (2000/x)(1000+110x)

C(x) = 9x+ 2000000/x + 2000(110)

To minimize, set the derivative to zero and solve for x.

C'(x) = 9-2000000/x²

x² = 2000000/9 = 2(1000)²/3²

x = (1000/3)√2 = 471.4045 ≈ 471 units per run

Answer 2

Unless there's more information, the answer is 1 run of 2000 units. Then there are no unsold units to store, and there's only one $1000 set up cost.