An order for 45 castings has been received. Each casting is produced independently, with a 0.85 probability of an individual casting being good. The customer will accept as few as 40 and as many as 50 castings at a unit sales price of $2000; each casting costs $800 to produce. (Hint: The optimum lot size is at least 55 castings.)

a. Determine the lot size that maximizes expected profit.

b. Determine the economic lot size assuming a 0.98 probability of a good unit.