Eli Orchid can manufacture its newest pharmaceutical product in any of three processes. One costs $14,000 per batch, requires 3 tons of one major ingredient and 1 tons of the other, and yields 2 tons of output product. The second process costs $30,000 per batch, requires 2 and 7 tons of product. The third process costs $11,000 per batch, requires 9 and 2 tons of the ingredients, respectively, and yields 1 ton of product. Orhid wants to find the least costly way to produce at least 50 tons of the new product given that there are 75 tons of ingredient 1 and 60 tons of ingredient 2 on hand.

min 14x1 30x2 + 11x3

s.t. 2x1 + 5x2 + 1x3 >= 50

      3x1 + 2x2 + 9x3 <= 75

      1x1 + 7x2 + 2x3 <= 60

x1,x2,x3 >= 0

a. What is the marginal cost of production (Per ton of output)? 

b. How much would it cost to produce 70 tons of the new pharmaceutical product? To produce 100 tons? 

c. How much should Orchid be willing to pay to obtain 20 more tons of ingredient 1? How about ingredient 2? 

d. How cheap would the third process have to become before it might be used in an optimal solution? 

e. How much would the cost of the 50 tons of product increase if process 2 actually cost $32,000 per batch? If it cost $39,000?

f. How much would the cost of the 50 tons of product decrease if process 1 cost $13,000 per batch? If it cost $10,0000 per batch? 

g. Suppose that the engineering department is thinking about a new process that produces 6 tons of product using 3 tons of each of the two original ingredients. At what cost would this new process be economic to use?