Riverside oil company in eastern Kentucky produces 3 different grades of gasoline. They are regular, premium, and supreme grades. Each barrel of regular grade sells for $77 while premium grade sells for $82 per barrel and supreme grade sells for $89 per barrel. Petroleum component A, B and C are purchased to be blended together for the production of three different grades of gasoline. The availability and costs of petroleum components A, B, and C are: A: 8000 barrels available at $32 cost per barrel; B: 7000 barrels available at $45 cost per barrel; C: 9000 barrels available at $39 cost per barrel.
The following constraints/blending specifications apply: Regular: at 30% of A; no more than 30% of B; no restriction on C
Premium: at least 25% of B; no restriction on A and C
Supreme: no more than 25% of A; no restriction on B; at least 40% on C.
QUESTIONS:
a. List the linear program for this problem.
b. Report optimal way to blend these three petroleum components for production.
c. What is the maximized profit?
a. To list the linear program for this problem, we need to define the decision variables, objective function, and constraints.
Decision Variables:
Let x1 be the number of barrels of component A used.
Let x2 be the number of barrels of component B used.
Let x3 be the number of barrels of component C used.
Objective Function:
Maximize Profit = 77x1 + 82x2 + 89x3
Constraints:
Regular:
0.3x1 ≤ x2 (30% of A cannot exceed 30% of B)
Premium:
x2 ≥ 0.25(x2 + x3) (at least 25% of B)
or
0.75x2 - 0.25x3 ≥ 0
Supreme:
0.25x1 ≤ 0.4x3 (no more than 25% of A)
or
0.25x1 - 0.4x3 ≤ 0
Availability Constraints:
x1 ≤ 8000 (limit of component A)
x2 ≤ 7000 (limit of component B)
x3 ≤ 9000 (limit of component C)
Non-negativity Constraints:
x1, x2, x3 ≥ 0
b. To find the optimal way to blend these three petroleum components, you can solve the linear program using methods such as graphical method, simplex method, or software like Excel Solver or Linear Programming software.
c. The maximized profit can be determined by evaluating the objective function at the optimal solution obtained from solving the linear program.
a. The linear program for this problem can be formulated as follows:
Let x1, x2, and x3 represent the number of barrels of petroleum components A, B, and C respectively to be used in the blend.
Objective function:
Maximize Profit = 77x1 + 82x2 + 89x3
Subject to the following constraints:
Regular grade:
0.30x1 <= x2
x1 + x2 + x3 <= 8000
Premium grade:
0.25x2 <= x3
Supreme grade:
x1 <= 0.25(x1 + x2 + x3)
x3 >= 0.40(x1 + x2 + x3)
Non-negativity constraints:
x1 >= 0
x2 >= 0
x3 >= 0
b. To find the optimal way to blend the petroleum components for production, solve the linear program to obtain the values of x1, x2, and x3 that satisfy the constraints and maximize the profit.
c. The maximized profit can be determined by substituting the values of x1, x2, and x3 obtained from the solution of the linear program into the objective function.