FuelMotion is an oil company that owns two oil processing plants that produce gasoline
from crude oil. Their Georgia Plant produces 2 thousand gallons of premium, 1 thousand
gallons of medium, and 6 thousand gallons of regular gasoline every day it operates.
Their Arizona Plant produces 3 thousand gallons of premium, 4 thousand gallons of
medium and 1 thousand gallons of regular gasoline per day of operation. Chevron has
placed an order for 110 thousand gallons of premium, 80 thousand gallons of medium,
and 90 thousand gallons of regular gasoline and has asked for the order to be filled within
30 days. If it costs FuelMotion $10,000 per day to operate the Georgia Plant and $20,000
per day to operate the Arizona Plant, then determine how it can fill Chevron's order at
minimum cost. Note: it may be necessary to produce more gas than demanded, due to the
way the plants operate.
a. Formulate a linear programming model (that is, define the variables, and write down
the objective function and all constraints mathematically).
b. Create a spreadsheet model for this problem and solve with Excel Solver.
c. What is the optimal solution? What is the optimal value?
Georgia 30
Arizona 16.66666667
To solve this problem, we can use linear programming, which is a mathematical technique for optimizing a linear objective function subject to linear constraints.
a. Formulate a linear programming model:
Let's define the variables:
- Let x1 represent the number of days the Georgia Plant operates.
- Let x2 represent the number of days the Arizona Plant operates.
Objective function:
We want to minimize the total cost, which is influenced by the operating cost of both plants. Therefore, the objective function can be defined as:
Cost = 10,000 * x1 + 20,000 * x2
Constraints:
1. The total gallons of premium gasoline produced should be at least equal to the order:
2,000 * x1 + 3,000 * x2 ≥ 110,000
2. The total gallons of medium gasoline produced should be at least equal to the order:
1,000 * x1 + 4,000 * x2 ≥ 80,000
3. The total gallons of regular gasoline produced should be at least equal to the order:
6,000 * x1 + 1,000 * x2 ≥ 90,000
4. The order needs to be filled within 30 days:
x1 + x2 ≤ 30
5. The number of days of operation cannot be negative:
x1, x2 ≥ 0
b. Create a spreadsheet model and solve with Excel Solver:
Now, we can create a spreadsheet model with the variables and constraints described above. We can use Excel Solver to find the optimal solution.
c. The optimal solution will provide the values for x1 and x2, representing the number of days each plant should operate to fulfill Chevron's order at the minimum cost. The optimal value will be the minimum cost obtained from the objective function.