XYZ Oil Company produces four products in its refinery in unit I and in unit II; gasoline, heating oil, jet fuel, and lubricating oil. These products are made from four available crude oils. The first three crudes are processed in unit I; the fourth crude is processed in either unit I or unit II. OnIy unit II is capable of producing lube oil in addition to the other three products.
Resources (input) Availability
(in barrels per week)
Crude 1 and 3: up to 200,000 each
Crude 2: up to 150,000
Crude 4: up to 250,000
Input-output coefficients
Tablo shows the input-output relationship between crudes (inputs)
and products (outputs). these are the yields. For example, a
barrel of crude I yields 0.5 barrels of gasoline, 0.3 barrels of
heating oil, and 0.1 barrels of jet fuel.
| Unit | Crude | Gasoline | Heating oil | Lube oil | Jet fuel | Loss |
| I | 1 | 0.5 | 0.3 | 0 | 0.1 | 0.1 |
| I | 2 | 0.4 | 0.2 | 0 | 0.2 | 0.2 |
| I | 3 | 0.6 | 0.2 | 0 | 0.1 | 0.1 |
| I | 4 | 0.5 | 0.1 | 0 | 0.2 | 0.2 |
| II | 4 | 0.4 | 0.1 | 0.3 | 0.1 | 0.1 |
Marketability
Each week, Company can sell up to 250,000 barrels of gasoline;
120,000 barrels of heating oil; 30,000 barrels of lube oil; and
l00,000 barrels of jet fuel.
Profit
For each barrel of crude 1 refined, Company realizes $1.50
profit, crude 2 hrings $2.00 per barrel, crude 3 brings $1.10 per
barrel, and crude 4 brings $2.00 per barrel fL processed in unit
1 and $3.00 per barrel if processed in unit II. The objective of
Company is to maximize profits. The problem is: How much crude
oil of each available type should Company refine each week to
meet the proflt maximization?
Define the decision variables
X1 = Amount of crude 1 to be processed
X2 = Amount of crude 2 to be processed
X3 = Amount of crude 3 to be processed
X4 = Amount of crude 4 to be processed
X5 = Amount of crude 4 to be processed
(X4 + X5 = Amount of crude 4 to be processed)
The optiinal solution
X1 = 150,000 barrels
X2 = 150,000 barrels
X3 = 0
X4 = 150,000 barrels
X5 = 100,000 barrels
Total profit = $1,125,000 (each week)