The demand for motorcycle tires improted byu Dizie Import- Export is 40,000/year and may be assumed to be uniform throughout the year. The cost of ordering a shipment of tires is $400, and the cost of storing each tire for a year is $2. Determine how many tires should be in each shipment if the ordering and storage costs are to be minimized. (Assume that each shipment arrives just as the previous one has been sold)
Suppose there are x shipments. Then there are 40000/x tires in a shipment, and this is the number of tires for which storage is needed throughout the year (even if the number actually in storage decreases to zero as the time for a new shipment approaches). Therefore, the storage costs are 2*40000/x 80000/x. The ordering costs are 400x for the x shipments, so the total ordering and storage costs C for the year is given by C 400x + 80000/x Even though x must be a positive integer, let's treat it as a continuous variable. Then dC/dx 400 - 80000/x? dC/dx 0 ? 400 80000/x? ? x √200 10√2 ≈ 14.14 (we discard the negative solution because x must by definition be positive) Observe that d?C/dx? 160000/x? 0 for positive x, so the critical point is a (relative) minimum. But you can show that dC/dx 0 for 0 x 10√2, and dC/dx 0 for x 10√2 so x 10√2 gives the absolute minimum for C (among all positive numbers x). Now, since x must be an integer, we need to consider integer values on each side of the continuous solution. x14: 40000/14 2857+; 2857*14 39998, so we need two shipments with an additional tire. That means we have to have storage for 2858 tires. So the true cost of 14 shipments is 2858*$2 + 14*$400 $11316. x15: 40000/15 2666.67; 2666*15 39990, so we need 10 shipments of 2667 tires and 5 shipments of 2666 tires. Again, we need storage for the larger number, so the true cost of 15 shipments is 2667*$2 + 15*$400 $11334.
