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 ABC and 1 ton of ingredient DEF and yields 2 tons of an output product. The second process costs $30,000 per batch, requires 2 and 7 tons of the ingredients ABC and DEF, respectively, and yields 5 tons of the product. The third process costs $11,000 per batch, requires 9 and 2 tons of the ingredients, respectively, and yields 1 ton of the product. Orchid 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 ABC and 60 tons of ingredient DEF at hand. The following LP model solves the problem: min 14x, +30x + 11x, Subject to: 2x + 5x2 + x3 50 3x + 2x2 +9x 75 x+7x+2x, 60 20 a) What is the marginal cost of production per ton of output? [i.e. By how much is the total cost going to increase if one more ton of the product is produced?] b) How much would it cost to produce 70 tons of the new pharmaceutical products? c) How much would it cost to produce 100 tons of the new pharmaceutical products? d) How much should Orchid be willing to pay to obtain additional 20 tons of ingredient ABC? What about 20 tons of ingredient DEF? e) How cheap would the third process have to become before it might be used in an optimal solution? f) How much would the cost of the 50 tons of product increase if process 2 actually costs $32,000 per batch? g) How much would the cost of the 50 tons of product decrease if process 1 costs $13,000 per batch? h) If additional 10 tons of ingredient ABC and 5 tons of ingredient DEF become available, what will happen to the total cost? i) Suppose that the engineering department is thinking about a new process that produces 6 tons of the product using 3 tons of each of the two original ingredients. At what cost would this new process be economic to use? Suppose that the three processes actually use 0.1, 0.3, and 0.2 ton per batch of a third ingredient but we do not know exactly how much of it is available. Determine the minimum amount needed if the optimal solution is not to change (the current optimal quantities used of each ingredient remain unchanged).