Question $5$: A foundry has received an order of $20$ custom$-$designed castings. The casting process costs $$700$ per unit scheduled. If a casting is good then it is machined to specifications at an added cost of $$400$ per unit $.$If a casting is not sold, it has a recycle value of $$200.$ The customer has indicated a willingness to pay $$2000$ per casting for $20$ acceptable castings. The customer also agreed to pay $$1500$ each for one or two additional castings. However the customer is not willing to purchase fewer than $20$ castings and not willing to pay more than $22$ castings. Based on the historical data the following probability distribution have been estimated.

$\backslash$table$[[\backslash$table$[[-$ Grood$],[$Castings$]],$Number of Castings Scheduled$],[20,21,22,23,24,25,26,27,28,29,30],[12,0\mathrm{.}10,0\mathrm{.}05,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00],[13,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}05,0\mathrm{.}05,0\mathrm{.}05,0\mathrm{.}05,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00],[14,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}05,0\mathrm{.}05,0\mathrm{.}05,0\mathrm{.}05,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00],[15,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}05,0\mathrm{.}05,0\mathrm{.}05,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00],[16,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}05,0\mathrm{.}05,0\mathrm{.}05,0\mathrm{.}05,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00],[17,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}05,0\mathrm{.}05,0\mathrm{.}05,0\mathrm{.}05,0\mathrm{.}00,0\mathrm{.}00],[18,0\mathrm{.}20,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}05,0\mathrm{.}05,0\mathrm{.}10,0\mathrm{.}05,0\mathrm{.}05,0\mathrm{.}05],[19,0\mathrm{.}10,0\mathrm{.}15,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}05,0\mathrm{.}05],[20,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}15,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}05],[21,0,00,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0,10,0\mathrm{.}10,0\mathrm{.}10],[22,0\mathrm{.}00,0,00,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10],[23,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10],[24,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10],[25,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10],[26,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0,05,0\mathrm{.}05,0\mathrm{.}10,0\mathrm{.}10,0\mathrm{.}10],[27,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}05,0\mathrm{.}05,0\mathrm{.}10,0\mathrm{.}10],[28,0\mathrm{.}00,0\mathrm{.}000,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0,00,0,00,0\mathrm{.}00,0\mathrm{.}05,0\mathrm{.}05,0\mathrm{.}05],[29,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}05,0,05],[30,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}00,0\mathrm{.}05]]$

If $27$ casting is scheduled for production what would be expected profit?