Question

The next three questions are based on the following information. The Low Knock Oil Company produces two grades of cut-rate gasoline for industrial distribution. The grades, regular and economy, are produced by refining a blend of two types of crude oil, type X100 and type X220. Each crude oil differs not only in cost per barrel, but in composition as well. The following indicates the percentage of crucial ingredients found in each of the crude oils and the cost per barrel for each:

CRUDE OIL TYPE INGREDIENT A (%) INGREDIENT B (%) COST/BARREL ($) X100 35 55 30.00 X220 60 25 34.80

Weekly demand for the regular grade of Low Knock gasoline is at least 25,000 barrels, and demand for the economy is at least 32,000 barrels per week. At least 45% of each barrel of regular must be ingredient A. At most 50% of each barrel of economy should contain ingredient B. While the gasoline yield from one barrel of crude depends on the type of crude and the type of processing used, we will assume for the sake of this example that one barrel of crude oil will yield 0.46 barrel of gasoline. Hint: You may refer to the Low Knock Oil Company example analyzed on page 326-327 in the textbook (Program 8.9), and simply adjust your constraints for the demands accordingly.

Question 1

To minimize the production cost, the optimal amount of X100 crude oil used in producing regular gasoline is  barrels, and the optimal amount of X220 crude oil used in producing regular gasoline is  barrel. (Please round to the closest integer and include no units.)

Question 2

To minimize the production cost, the optimal amount of X100 crude oil used in producing economy gasoline is  barrels, and the optimal amount of X220 crude oil used in producing economy gasoline is  barrel. (Please round to the closest integer and include no units.)

Answer 1

Let

X1= barrels of crude X100 blended to produce refined regular

X2=barrels of crude X100 blended r efined economy

X3=barrels of X220 blended in refined regular

X4=barrels of X220 blended in refined economy

minimize cost=$30X1+$30X2+$34.80X3+$34.80X4

(X1+X3)=total amount in refined regular 0.45(X1+X3)=ingredient A required

0.35X1+0.60X3≥0.45X1+0.45X3

0.35X1+0.60X3=ingredient A in refined regular gas

-0.10X1+0.15X3≥0

Minimize cost = 30X1 + 30X2 + 34.80X3 + 34.80X4

subject to

X1+X3≥25000

X2+X4≥32,000

–0.10X1+0.15X3≥0

0.05X2–0.25X4≤0

X1,X2,X3,X4≥0

We get X1 =15000

X2 = 26666.67

X3= 10000

X4 = 5333.33

Total cost =1783600