A manufacturer of Bluray Discs BDs estimates the yearly dema

A manufacturer of Blu-ray Discs (BDs) estimates the yearly demand for a BD to be 10,000. It costs $400 to set up the machinery to burn the the BDs, plus $3 for each one produced. If it costs the company $2 to store a Bd for a year, how many should be burned at a time and how many production runs will be needed to minimize costs?

Solution

Let x be the number of BDs burned at a time. Then, an average of x/2 BDs are stored throughout the year. At a sorage cost of $ 2 per BD per year, the annual storage costs are (x/2)*2 = x. The cost per run is 400 + x*3 = 3x + 400. Now 10000 BDs @ x BDs per run will require 10000/x runs. Therefore, the production costs are (3x + 400)( 10000/x).Thus, the total cost = production cost + storage cost C (x) = (3x + 400)( 10000/x) + x = x + 30000 + 4000000 x^-1 . Then dC/dx = 1 - 4000000 x^-2 . Thus dC/dx = 0 when x = 2000. Also d²C/dx² = 4000000x^-3 which is positive when x = 2000. Thus, the total cost is minimum when 2000 BDs are burned in a run and when there are 10000/2000 = 5 production runs in an year.