Three types of gasoline are manufactured by a company – Regular, Super, and Extr

Question

Three types of gasoline are manufactured by a company – Regular, Super, and Extra. Regular should have at least 11 percent additive 1 and 17 percent additive 2. Super should have at least 13 percent additive 1 and 22 percent additive 2. Extra should have at least 17 percent additive 1 and 19 percent additive 2. These are made by using two crudes – A and B. Crude A cost $28 per barrel and is 14 percent additive 1 and 18 percent additive 2. Crude B costs $30 per barrel and is 20 percent additive 1 and 24 percent additive 2. The demand for Regular is projected to be 1,000 barrels, while each of the others has a demand of 2,000 barrels. Formulate this as a linear programming problem to minimize cost while meeting all constraints. Define all decision variables.

NOTE:

I know there are some available answers, but there are all inconsistant answers. Would you please show the step by step solution with an explanation?

My prior question/answer done by Chegg was wrong?

Explanation / Answer

[Note: all percentages are expressed as decimal - 10% = 0.1, 11% = 0.11 etc.]

1000 barrels of Regular will require at least (1000 x 0.11) = 110 of Additive 1

2000 barrels of Super will require at least (2000 x 0.13) = 260 of Additive 1

2000 barrels of Extra will require at least (2000 x 0.17) = 340 of Additive 1

Thus, total requirement of Additive 1 to meet a demand of 1000 barrels of Regular, 2000 barrels of Super and 2000 barrels of Extra (110 + 260 + 340) = 710.

So, quantum of Additive 1   710 .............. (1)

Similarly, total requirement of Additive 2 to meet a demand of 1000 barrels of Regular, 2000 barrels of Super and 2000 barrels of Extra (1000 x 0.17) + (2000 x 0.22) + (2000 x 0.19) = 170 + 440 + 380 = 990.

So, quantum of Additive 2 990 .............. (2)

Now, the above reuirement have to be met from Crude A and Crude B. So, let quantity(all in barrels) of Crude A = x and that of Crude B = y. Then, quantum of Additive 1 derived will be 0.14x + 0.20y ....... (3)

Connecting (1) and (3), 0.14x + 0.20y 710 ....... (4)

Similarly, for Additive 2, 0.18x + 0,24y 990 ........(5)

The total cost of x of Crude A and y of Crude B = 28x + 30y ........ (6)

Combining (4), (5) and (6), the mathematical formulation is:

Min Z = 28x + 30y

subject to

0.14x + 0.20y 710

0.18x + 0,24y 990

x, y 0

And that completes the solution. [This can be confortably solved by graphical method.]

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