Project 3 The Astro Refining Company produces two types of unleaded gasoline, re
ID: 1110043 • Letter: P
Question
Project 3 The Astro Refining Company produces two types of unleaded gasoline, regular and premium, which it sells to its chain of service stations for $12 and $14 per barrel, respectively Both types are blended from Astro's inventory of refined domestic oil and refined foreign oil, and must meet the following specifications Maximum Vapor Pressure Minimum Octane Ratin Maximum Demand, bbl/wk Minimum Deliveries, bbl/wk Regular Premiumm 23 23 100 000 20 000 50 000 5 000 93 The characteristics of the refined oils in inventory are as follows Vapor Pressure 25 15 Octane Inventory, Cost, bbl 40 000 60000 Ratin S/bbl Domestic Foreign 87 98 15 What quantities of the two oils should Astro blend into the two gasolines in order to maximize weekly profit? Hint: Set barrels of domestic blended into regular x2 barrels of foreign blended into regular Xy barrels of domestic blended into premium barrels of foreign blended into premiumExplanation / Answer
Let x1 and x2 denote the numbers of barrels of domestic and foreign oils,respectively, to be blended into regular gasoline per week, and
Let x3 and x4 denote the numbers of barrels of domestic and foreign oils, respectively, to be blended into premium gasoline per week. All of these variables are nonnegative.
Astro’s weekly profit is given by
(36 16)(x1) + (36 30)(x2) + (42 16)(x3) + (42 30)(x4) = 20(x1) + 6(x2)+ 26(x3) + 12(x4).
This is the objective function; the objective is to maximize this value. Under the assumption that vapor pressure combines linearly in a blend, the vapor pressure of the resulting regular gasoline will be (25DR + 15FR)/(DR + FR), so to meet the maximum vapor pressure constraint for regular gasoline we need
25 x1 + 15 x2
x1 + x2 23.
This constraint is not linear. But we can multiply both sides by x1 + x2 to get,
25x1 + 15x2 23x1 + 23x2, or 2x1 8x2 0.
Maximum vapor pressure constraint for premium gasoline can be expressed as 2x3 8x4 0.
Similarly, the minimum octane rating constraints for regular and premium gasoline respectively, are
87x1 + 98x2
x1 + x2 88 and
87x3 + 98x4
x3 + x4 93.
This becomes x1 + 10x2 0 and 6x3 + 5x4 0.
After including constraints for maximum demand, minimum deliveries, and inventory,.
maximize 20x1 + 6x2 + 26x3 + 12x4 [weekly profit]
subject to 2x1 8x2 0 [vapor pressure, regular]
2x3 8x4 0 [vapor pressure, premium]
x1 + 10x2 0 [octane rating, regular]
6x3 + 5x4 0 [octane rating, premium]
x1 + x2 100,000 [max. demand, regular]
x1 + x2 50,000 [min. deliveries, regular]
x3 + x4 20,000 [max. demand, premium]
x1 + x4 5,000 [min. deliveries, premium]
x1 + x3 40,000 [inventory, domestic]
x2 + x4 60,000 [inventory, foreign]
x1 0, x2 0, x3 0, x4 0.
The following Maple worksheet can be used to solve this linear program.
> restart;
> with(Optimization);
[ImportMPS,Interactive, LPSolve, LSSolve, Maximize, Minimize, NLPSolve,
QPSolve]
> f:=(x1,x2,x3,x4) --> 20*x1 +6*x2 +26*x3 +12*x4;
f := (x1, x2, x3, x4) 20x1 + 6x2 + 26x3 + 12x4
Constraints are given above
LP Solve(f(x1,x2,x3,x4), constraints,’maximize’,assume=nonnegative);
1.280000 106
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