EGR 549 – Example 12, page 86Page 1 of 4
April 22, 2007
By: Kristofer Olsen
Parisay’s notes in red.
Sunco Oil manufactures three types of gasoline. Each type is produced by blending three types of crude oil. The sales price per barrel of gasoline and the purchase price per barrel of crude oil are given below. Sunco must maximize its daily profits. In other words, revenues are maximized while costs are minimized.
Sales Price Purchase Price
Gasper Barrel ($)Crudeper Barrel ($)
1 70 1 45
2 60 2 35
3 50 3 25
Octane Sulfur
CrudeRatingContent (%)
1 12 0.5
2 6 2.0
3 8 3.0
Sunco must make two types of decisions: first, how much money should be spent in advertising each type of gas, and second, how to blend each type of gasoline from the three types of crude oil available.
ai = dollars spent daily on advertising gas i (i = 1, 2, 3)
xij = barrels of crude oil i used daily to produce gas j (i = 1, 2, 3; j = 1, 2, 3)
Define variables
Originally: Xij= barrels of crude oil I used daily to produce gas j, any positive value
ai = dollars spent daily on advertising gas i, any positive value
For ILP: all variables should be positive integer
Max z = 21 x11 + 11 x12 + x13 + 31 x21 + 21 x22 + 11 x23 + 41 x31 + 31 x32 + 21 x33 – a1 – a2 – a3
WinQSB Input:
WinQSB Output:
Report:
A maximum profit of $287,500 can be achieved by Sunco Oil, with the following quantities of production:3,000 barrels of gas 1 are producedfrom 2,000 barrels of crude 1 and 1,000 barrels of crude 2; 9,500 barrels of gas 2 are produced from 2,200 barrels of crude 1, 4,000 barrels of crude 2 and 3,300 barrels of crude 3; 1,000 barrels of gas 3 are produced from 800 barrels of crude 1 and 200 barrels of crude 3.
As long as the sum of $750 is spent on advertising for gas 2, the maximum profit can be achieved. Stimulating the demand for gas 2 is desirable because gas 2 is the cheapest, highest quality gas that can be produced in mass quantity from the given supply of crude. The higher the quality of gas is available to the consumers, the better the profit will be. The profit contribution from gas 2 along is $210,500, more than 73% of the total profit, while only $73,000 of profit is contributed by the most valuable gas to consumers, gas 1.
For every 10 barrels of demand more than the straight demand of 2,000 barrels for gas 2 that are stimulated, the profit increases by $1, whereas stimulating the demand for gas 1 or gas 3 is a drag on the profit, and less profit would be achieved if more money were spent on advertising for gas 1 or gas 3. This means that it would be unwise to consider stimulating the demand for gas 1 or gas 3, since the profit would be driven down by an additional barrel of demand for either of the two: by nearly $21 for an additional barrel of demand for gas 1 and by nearly $41 for an additional barrel of demand for gas 3..
In the past, fractional barrels of crude were used in gas production process, but recently the process was simplified by restricting the blending of crude to whole barrels of crude. The simplification of the gas production process is established by restricting the blending of crude to whole barrels. Mistakes are caused by the more complicated approach of using fractional barrels of crude to produce the gas. In altering its process from using fractional barrels to using only whole barrels of oil, the fact is that a cost of $250 was saved by Sunco.
Please refer to the files in Class Note web page on this problem. This problem had 4 solutions which one of them was integer values (by chance). That is why you get that same solution with ILP. Also there is extensive discussion on report writing there.