A linear programming computer package is needed.
Seastrand Oil Company produces two grades of gasoline: regular and high octane. Both gasolines are produced by blending two types of crude oil. Although both types of crude oil contain the two important ingredients required to produce both gasolines, the percentage of important ingredients in each type of crude oil differs, as does the cost per gallon. The percentage of ingredients A and B in each type of crude oil and the cost per gallon are shown.
Crude Oil Cost Ingredient A Ingredient B
1 $0.2 20% 60%
2 $0.25 50% 30%
Each gallon of regular gasoline must contain at least 40% of ingredient A, whereas each gallon of high octane can contain at most 50% of ingredient B. Daily demand for regular and high-octane gasoline is
1,000,000 and 700,000 gallons, respectively. How many gallons of each type of crude oil should be used in the two gasolines to satisfy daily demand at a minimum cost? (Round your answers to the nearest gallon.)
What is the total minimum cost (in $) for the gallons? (Round your answer to the nearest dollar.)
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This problem can be solved using linear programming. Let's denote:
- x1 = gallons of crude oil 1 used for regular gasoline
- x2 = gallons of crude oil 2 used for regular gasoline
- y1 = gallons of crude oil 1 used for high octane gasoline
- y2 = gallons of crude oil 2 used for high octane gasoline
We need to minimize the total cost, which is 0.2*(x1+y1) + 0.25*(x2+y2).
Subject to the following constraints:
1. x1 + x2 = 1,000,000 (daily demand for regular gasoline)
2. y1 + y2 = 700,000 (daily demand for high octane gasoline)
3. 0.2*x1 + 0.5*x2 >= 0.4*(x1+x2) (ingredient A in regular gasoline)
4. 0.6*y1 + 0.3*y2 <= 0.5*(y1+y2) (ingredient B in high octane gasoline)
This is a linear programming problem that can be solved using a linear programming solver. The exact solution will depend on the specific solver used and its rounding rules.
However, it's important to note that this problem may not have a feasible solution. The constraints for the percentages of ingredients A and B in the regular and high octane gasolines, respectively, may not be satisfied with the given percentages of these ingredients in the two types of crude oil. In particular, it may not be possible to produce regular gasoline with at least 40% of ingredient A using the given types of crude oil, and it may not be possible to produce high octane gasoline with at most 50% of ingredient B using the given types of crude oil.
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