Exercise 1.16 (Oil refinery problem)

A manager of an oil refinery has 8 million barrels of crude oil A and 5 million barrels of crude oil B allocated for production during the coming month. These resources can be used to make either gasoline, which sells for $38 per barrel, or home heating oil, which sells for $33 per barrel. There are three production processes with the following characteristics:

PIC

All quantities are in barrels. For example, with the first process, 3 barrels of crude A and 5 barrels of crude B are used to produce 4 barrels of gasoline and 3 barrels of heating oil. The costs in this table refer to variable and allocated overhead costs, and there are no separate cost items for the cost of the crudes.
Formulate a linear programming problem that would help the manager maximize net revenue over the next month.

Answers

To find the optimal allocations between the given processes, we denote their outputs by x1,x2 and x3 respectively. Our aim is to maximize the net revenue given by

f (x1,x2,x3) = 38 ⋅ (4x1 + x2 + 3x3) + 33 ⋅ (3x1 + x2 + 4x3) − 51x1 − 11x2 − 40

under the condition that the amount of crude oil A is limited by 8 million barrels

3x1 + x2 + 5x3 ≤ 8

and the amount of crude oil B is limited by 5 million barrels

5x1 + x2 + 3x3 ≤ 5.

These conditions result in the following linear optimization problem:

maximize 200x1 + 60x2 + 206x3 subject to3x1 + x2 + 5x3 ≤ 8 5x1 + x2 + 3x3 ≤ 5 x1,x2,x3 ≥ 0.
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2022-02-12 13:56
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