Retail Chain Order

Question

An electronic retail chain wishes to minimize costs for a particular popular model of laptop computer. This retail chain has a line of credit to finance its inventory and the current holding rate is 5.5%. The chain estimates it can sell 670,000 units per year and it pays $375 dollars per unit. It costs $320 dollars to place each order. How many units should it order each time? Round answer to nearest laptop unit.

Answer 1

let x orders be placed in the year.  Then, the average holding time is 1/2x.  (There are orders every 1/x, and the average item is held for half the time.)

Thus, the total cost is C(x) = 320x + 675000 * 375 * 1/2x * .055

Then, we minimize by taking the first derivative and setting it equal to 0

C'(x) = 320 - 675000 * 375 * .055/2x^2

Then, 320 = 675000 * 375 * .055/2x^2

x^2 = 675000 * 375 * .055/640

x = sqrt(675000 * 375 * .055/640) = 147.49, which rounds to 147.

Note that the second derivative is 675000 * 375 * .055/x^3 is positive for all x > 0, so this is a minimum.