Suppose that instead of measuring shortage in terms of cost per shortage per year, a cost of P dollars is incurred for each unit the firm is short. This cost does not depend on the length of time before the backlogged demand is satisfied. Determine a new expression for the annual shortage cost as a function of Q and b, and solve GMB's problem (Example 12.3) with this way of costing shortages for reasonable values of P. (What values of P do you think are reasonable?)
Example 12.3
ORDERING AUDIO CDS AT GMB WITH SHORTAGES ALLOWED
GMB is a mail-order distributor of audio CDs that sells approximately 50,000 CDs per year. Each CD is packaged in a jewel case that GMB buys from a supplier. The fixed cost of placing an order for jewel cases is $200. GMB pays $0.50 for each jewel case, and its cost of capital is 10%. The cost of storing a jewel case for one year is $0.50. GMB believes it can afford to run out of jewel cases from time to time, reasoning that this simply makes the time between customer orders and customer deliveries a bit longer. It knows that there is some cost of doing this-impatient customers can take their business elsewhere- but it is not sure what dollar amount p to attach to this cost. It decides to use a trial value of p = $52, reasoning that this value implies a $1 penalty for each extra week a customer has to wait because of a backlogged jewel case. GMB wants to develop a spreadsheet model to find the optimal order quantity, the optimal amount to backlog, and the optimal annual cost. It also wants to see how sensitive these quantities are to the unit shortage cost p.
Objective To find the order quantity and the maximum shortage allowed that minimize total annual cost, and to see how sensitive the solution is to the unit shortage cost.