Question:

(a) Shale Oil, located in the island of Aruba, has a capacity of 600,000 barrels of crude oil per day.  The final products from the refinery include two types of unleaded gasoline: regular  and premium.  The refining process encompasses three stages:

(1) a distillation tower that produces a feedstock, 

(2) a cracker unit that produces gasoline stock by using a portion of the feedstock produced from the distillation tower, and

(3) a blender unit that blends the gasoline stock from the cracker unit and the feedstock from the distillation tower.

Both the regular and the premium gasoline can be produced from either the feedstock or the gasoline stock during the blending process, although at different production costs.  The company estimates that the net profit per barrel of regular gasoline is $7.70 and $5.20, depending on whether it is blended from feedstock or from gasoline stock.  The corresponding profit values for the premium grade are $12.30 and $10.40.

According to design specifications, it takes five barrels of crude oil to produce one barrel of feedstock.  The cracker units cannot use more than 40,000 barrels of feedstock a day.  All remaining feedstock is used directly in the blender unit to produce the end-product gasoline.  The demand limits for regular and premium gasoline are 80,000 and 50,000 barrels per day, respectively.

Formulate the above problem as a Linear Programming Problem in order to determine the optimum production schedule for the refinery.

(b) Solve the following linear programming problem using the Simplex Method:

Minimize

z = 4x₁+ x2

subject to the following constraints:

3x₁ + x₂ = 3

4x₁ + 3x₂ ≥6

x₁ + 2x₂ ≤4

x₁ + x₂ ≥ 0