Question

Bay Oil produces two types of fuels (regular and super) by mixing three ingredients. The major distinguishing feature of the two products is the octane level required. Regular fuel must have a minimum octane level of 80 while super must have a level of at least 95. The cost per barrel, octane levels, and available amounts (in barrels) for the upcoming two-week period are shown in the following table. Likewise, the maximum demand for each end product and the revenue generated per barrel are shown.

Input	Cost/Barrel	Octane	Available (barrels)
1	$17.5	90	120000
2	$17	85	320000
3	$18	110	310000

	Revenue/Barrel	Max Demand (barrels)
Regular	$18.5	360000
Super	$22	510000

Develop and solve a linear programming model to maximize contribution to profit.

Let	Ri = the number of barrels of input i to use to produce Regular, i=1,2,3
	Si = the number of barrels of input i to use to produce Super, i=1,2,3

If required, round your answers to two decimal places. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300)

Max

R1

+

R2

+

R3

+

S1

+

S2

+

S3

s.t.

R1

+

S1

≤

R2

+

+

S2

≤

R3

+

S3

≤

R1

+

R2

+

R3

≤

S1

+

S2

+

S3

≤

R1

+

R2

+

R3

≥

R1

+

R2

+

R3

S1

+

S2

+

S3

≥

S1

+

S2

+

S3

R1, R2, R3, S1, S2, S3 ≥ 0

What is the optimal contribution to profit?

Maximum Profit = $   by making  barrels of Regular and  barrels of Super.

Answer 1

Solution :

Revenue from Regular fuel = 18.5 (R1+R2+R3)
Revenue from Super fuel = 22 (S1+S2+S3)

Objective is to maximize profit = Max Revenue - cost =

Max 18.5(R1+R2+R3) + 22(S1+S2+S3) - 17.5(R1+S1) - 17(R2+S2) - 18(R3+S3)

=> Max 1*R1+1.5R2+0.5R3+4.5S1+5S2+4S3

s.t

1R1+1S1 <= 120000 (Available barrels of input 1)

1R2+1S2 <= 320000 (Available barrels of input 2)

1R3+1S3 <= 310000 (Available barrels of input 3)

1R1+1R2+1R3 <= 360000 (Max demand of Regular fuel)

1S1+1S2+1S3 <= 510000 (Max demand of Super fuel)

90R1+85R2+110R3 >= 80R1 +80R2 +80R3 (Minimum octane level of Regular fuel)

90S1+85S2+110S3 = 95S1+95S2+95S3 (Minimum octane level of Super fuel)  

Solving in excel solver we get,

Total barrels of regular = R1+R2+R3 = 0+110000+130000 = 240000

Total barrels of Super= S1+S2+S3 =120000+210000+180000 = 510000

Maximum profit = $2540000 by making 24000 barrels of regular and 510000 barrels of Super

Solver screenshot :

Sensitivity report

Let me know in the comment section if anything is not clear. I will reply ASAP!
If you liked the answer, please give a thumbs-up. This will be quite encouraging for me.Thank-you!