A Bookstore Chain Often Has to Place Orders for a Wide

Essay

A bookstore chain often has to place orders for a wide variety of books. The setup cost for placing an order for copies of a particular hardcover book is $100, regardless of the size of the order. The unit cost per copy is $40. The head of the purchasing department of the bookstore estimates that the cost of holding a copy of this book in inventory for one week is $6. The text's inventory position at the beginning of any week is the number of copies in inventory plus any copies that have already been ordered but have not yet arrived. The reorder policy specifies that if the inventory (x) at the beginning of the week is less than or equal to R, exactly enough copies will be ordered to bring the inventory up to a set amount Q. Thus, the bookstore will order Q − x copies. Otherwise, if the inventory is greater than R, no order will be placed that week. If an order is placed, it will arrive after a lead time of 1, 2, or 3 weeks with probabilities 0.65, 0.25, and 0.10, respectively. The weekly demand for this book is uncertain, but it can be described by a normal distribution with mean 600 and standard deviation 150. The bookstore's policy is to satisfy all demand in the week it occurs. If weekly demand cannot be satisfied completely from on-hand inventory, then an emergency order will be placed at the end of the week for the shortage. This order will arrive virtually instantaneously (via express mail delivery), but at a much higher cost of $70. It is currently the beginning of week 1, and the current inventory of this hardcover book, including any copies that might have just arrived, is 1200. There are no other orders on the way. Simulate a range of (R,Q) ordering policies, from 400<R<1400 and 1000<Q<2500, and determine which one minimizes total cost over the next 52 weeks.