A manager intends to order a new machine and must now decide on the number of spare parts to order along with the machine. The parts cost $453 each and have no salvage value. The manager has compiled the frequency distribution for the probable usage of spare parts:4 spare parts with frequency 0.13;5 spare parts with frequency 0.22;6 spare parts with frequency 0.12;7 spare parts with frequency 0.05; and 8 spare parts with frequency 0.48. For what range of shortage costs would stocking 6 spare parts constitute an optimal decision ?

A Quality Analyst wants to construct a control chart for determining whether three machines, all producing the same product, are under control with regard to a particular quality variable. Accordingly, he sampled four units of output from each machine, with the following results : Machine #1 measurements [14, 13, 18, 17]; Machine #2 measurements [ 14, 13, 13, 19]; Machine #3 measurements [ 19, 17, 12, 21]. Using the factors for three sigma control limits, what are x-bar chart upper and lower control limits?