$2.$ RCI Case: Read the RCI case and answer the following questions:

a$.$ What are the costs of producing each of the three product in house?

b$.$ Develop an LP model to find the optimal production schedule and outsourcing plan that minimizes total cost.

c$.$ Which resources are completely used up$?$ What are the corresponding shadow prices? If RCI could increase labor capacity in one stage$/$shift$,$ which one should they chose?

d$.$ What is the optimal production schedule if the hand cream can be outsourced at a cost of $$50$ per carton? What is the maximum price that would make outsourcing hand cream optimal?

e$.$ What is the optimal production schedule if demand for each product increases by $10\%?($assume hand cream can be outsourced at $$50$ per carton$)$