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Riverside Oil Company in eastern Kentucky produces regular and supreme gasoline. Each barrel of regular sells for $21 and must have an octane rating of at least 90. Each barrel of supreme sells for $25 and must have an octane rating of at least 97. Each of these types of gasoline are manufactured by mixing different quantities of the following three inputs:
Input Cost per Barrel Octane Rating Barrels Available in (1000s)
1 $17.25 100 150
2 $15.75 87 350
3 $17.75 110 300
Riverside has orders for 300,000 barrels of regular and 450,000 barrels of supreme. How should the company allocate the available inputs to the production of regular and supreme gasoline to maximize profits?
a. Formulate and LP model for this problem.
b. What is the optimal solution?

Respuesta :

Solution :

Here,

[tex]$X_{iR}$[/tex] = the number of the barrels mixed i to manufacture the regular gasoline

[tex]$X_{iS}$[/tex] = the number of the barrels mixed i to manufacture the supreme gasoline.

The [tex]$\text{selling price}$[/tex]  of each of the barrel of both gasoline is [tex]$\$ 21$[/tex] and [tex]$\$25$[/tex]. So the total [tex]$\text{selling price}$[/tex] of both types of gasoline is represented by :

[tex]$21 \times \sum X_{iR} +25 \times \sum X_{iS}$[/tex]

The cost prices of one barrel of the three types of input are 17.25, 1575 and 17.75.

So the total price is represented by :

[tex]$17.25 \times (X_{iR}+X_{iS})+15.75 \times (X_{2R}+X_{2S})+17.75 \times (X_{3R}+X_{3S})$[/tex]

The company wants to increase the profit. So maximize objective function will be used.

Max Z = [tex]$(21. \times \sum X_{iR} +24 \times \sum X_{iS})-[17.25 \times (X_{iR}+X_{iS})+17.75 \times (X_{2R}+X_{2S})+17.75 \times (X_{3R}+X_{3S})]$[/tex]The company has 150,000 barrels of input 1 available. So,

[tex]$X_{1R}+ X_{1S} \leq 150,000$[/tex]

[tex]$X_{2R}+ X_{2S} \leq 350,000$[/tex]

[tex]$X_{3R}+ X_{3S} \leq 300,000$[/tex]

The company got an order to sell 300,000 barrels of regular and 450,000 barrels of supreme gasoline. So,

[tex]$X_{1R}+X_{2R}+X_{3R} = 300,000$[/tex]

[tex]$X_{1S}+X_{2S}+X_{3S} = 450,000$[/tex]

The company wishes the regular gasoline to have octane number of at least 90. So,

[tex]$\frac{100 \times X_{1R}+87 \times X_{2R} +10 \times X_{3R}}{\sum X_{iR}}\geq 90$[/tex]

The company wishes the supreme gasoline to have octane number of at least 97. So,

[tex]$\frac{100 \times X_{1S}+87 \times X_{2S} +10 \times X_{3S}}{\sum X_{iR}}\geq 97$[/tex]

Formulating the LP model :

Max :

[tex]$[21 \times \sum X_{iR}+25 \times \sum X_{iS}]$[/tex] [tex]$-[17.25 \times (X_{1R}+X_{1S})+15.75 \times (X_{2R}+X_{2S})+17.75 \times (X_{3R}+X_{3S})]$[/tex]

Subject to :

[tex]$X_{1R}+ X_{1S} \leq 150,000$[/tex]

[tex]$X_{2R}+ X_{2S} \leq 350,000$[/tex]

[tex]$X_{3R}+ X_{3S} \leq 300,000$[/tex]

Also,

[tex]$X_{1R}+X_{2R}+X_{3R} = 300,000$[/tex]

[tex]$X_{1S}+X_{2S}+X_{3S} = 450,000$[/tex]

[tex]$\frac{100 \times X_{1R}+87 \times X_{2R} +10 \times X_{3R}}{\sum X_{iR}}\geq 90$[/tex]

[tex]$\frac{100 \times X_{1S}+87 \times X_{2S} +10 \times X_{3S}}{\sum X_{iR}}\geq 97$[/tex]

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