## Related questions with answers

Olé Oil produces three products heating oil, gasoline, and jet fuel. The average octane levels must be at least 4.5 for heating oil, 8.5 for gas, and 7.0 for jet fuel. To produce these products Olé purchases two types of oil: crude 1 (at $12 per barrel) and crude 2 (at$10 per barrel). Each day, at most 10,000 barrels of each type of oil can be purchased. Before crude can be used to produce products for sale, it must be distilled. Each day, at most 15,000 barrels of oil can be distilled. It costs 10¢ to distill a barrel of oil. The result of distillation is as follows: (1) Each barrel of crude 1 yields 0.6 barrel of naphtha, 0.3 barrel of distilled 1, and 0.1 barrel of distilled 2. (2) Each barrel of crude 2 yields 0.4 barrel of naphtha, 0.2 barrel of distilled 1, and 0.4 barrel of distilled 2. Distilled naphtha can be used only to produce gasoline or jet fuel. Distilled oil can be used to produce heating oil or it can be sent through the catalytic cracker (at a cost of 15¢ per barrel). Each day, at most 5,000 barrels of distilled oil can be sent through the cracker. Each barrel of distilled 1 sent through the cracker yields 0.8 barrel of cracked 1 and 0.2 barrel of cracked 2. Each barrel of distilled 2 sent through the cracker yields 0.7 barrel of cracked 1 and 0.3 barrel of cracked 2. Cracked oil can be used to produce gasoline and jet fuel but not to produce heating oil. The octane level of each type of oil is as follows: naphtha, 8; distilled 1, 4; distilled 2, 5; cracked 1, 9; cracked 2, 6. All heating oil produced can be sold at $14 per barrel; all gasoline produced.$18 per barrel; and all jet fuel produced, $16 per barrel. Marketing considerations dictate that at least 3,000 barrels of each product must be produced daily. Formulate an LP to maximize Olé’s daily profit.

Solution

VerifiedHeating oil, gasoline and jet fuel are produced. Octan levels are at least $4.5,8.5$ and $7$ respectively. Crude $1$ can be purchased at $\$12$ per barrel and crude $2$ at $\$10$ per barrel. Each barrel of crude $1$ distilled yields $0.6$ barrels of nahptha , $0.3$ barrels of distilled $1$ and $0.1$ barrel of distilled $2.$ Crude $2$ yields $0.4$ barrel of nahptha, $0.2$ barrels of distilled $1$ and $0.4$ of distilled $2.$ Distilled oil can be cracked and to be more precise: distilled $1$ yields $0.8$ of cracked $1$ and $0.2$ of cracked $2.$ Similarly, each barrel of distilled $2$ yields $0.7$ of cracked $1$ and $0.3$ of cracked $2.$ Nahptha, distilled $1,$ distilled $2,$ cracked $1$ and cracked $2$ have the following octane levels:$8,4,5,9$ and $6,$ respectively. Our aim is to maximize daily profit. The objective function will be denoted by $z.$ Distilled nahptha can be used only to produce gasoline ($n_g$) and jet fuel ($n_j$). Distilled oil can only be used to produce heating oil or it can be sent through the catalytic cracker. Similarly, cracked oil can be used to produce gasoline and jet fuel. Let us $h,g$ and $j$ be the barrel of heating oil, gasoline and jet fuel sold, respectively. Also let us $c_1$ and $c_2$ denote the barrels of crude $1$ and crude $2$ bought. Crude must be distilled before being used. If we bought $c_1$ of crude $1$ then we obtain $0.6c_1$ barrels of distilled nahptha, $0.3c_1$ barrels of distilled $1$ and $0.1c_4$ barrels of distilled $2.$

Define $h_1$ and $h_2$ as barrels od distilled crude$1$ and distilled crude $2$ used for heating oil. Let us $d_1$ and $d_2$ be the barrels of distilled crude oil $1$ and crude oil $2$ sent through cracker. Also define $cr_{ij}$ as cracked oil from distilled type into two cracked types. Let us write the objective function $z$ that we mentioned:

$z=14h+16g+18j-12.1c_1-10.1c_2-0.15(d_1+d_2)$

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