A firm produces and sells four grades of industrial solvents – A, B, C, and D. T
ID: 442907 • Letter: A
Question
A firm produces and sells four grades of industrial solvents – A, B, C, and D. The selling price per gallon of each grade of solvent is $6.40, $5.00, $4.20, and $3.50 respectively. Because of demand limitations, the firm can sell at most 100,000 gallons of solvent A; 300,000 gallons of solvent B; 360,000 gallons of solvent C; and 220,000 gallons of solvent D.
The solvents are produced by blending two types of liquid ingredients: Ingredient1 and Ingredient2. The cost price per gallon for the ingredients are $3.20 for Ingredient1 and $2.40 for Ingredient2. At most 400,000 gallons of Ingredient1 and 600,000 gallons of Ingredient2 are available.
Regulations require a minimum percentage by volume of Ingredient1 in each grade of solvent: 60% for A, 50% for B, 40% for C, and 10% for D.
For your convenience, the information presented above is summarized in the tables below:
Solvent grade
A
B
C
D
Selling price per gallon
$ 6.40
$ 5.00
$ 4.20
$ 3.50
Maximum quantity allowed (gallons)
100,000
300,000
360,000
220,000
Minimum % of Ingredient1 required
60%
50%
40%
10%
Availability (gallons)
Price per gallon
Ingredient1
400,000
$ 3.20
Ingredient2
600,000
$ 2.40
The firm must determine an optimal production plan so as to maximize their profits subject to the applicable constraints.
Formulate the problem as a linear program
Define the decision variables:
Specify the objective function:
Specify the constraints:
Solve the linear program and report your optimal solutions
What is the maximum profit attainable under an optimal plan?
Maximum Profit =
$
How many gallons of each ingredient should be used to produce each grade of solvent under this optimal plan?
Quantity (in gallons)
A
B
C
D
Ingredient1
Ingredient2
How many gallons of each ingredient is used up under this optimal plan?
Quantity (in gallons)
Used
Available
Ingredient1
400,000
Ingredient2
600,000
At most how much should the firm be willing to pay per gallon for additional quantities of the ingredients? Justify your answer.
The maximum amount that the firm should be willing to pay for each additional gallon:
Ingredient1 :
$
per gallon.
Ingredient2 :
$
per gallon.
Reasoning:
Solvent grade
A
B
C
D
Selling price per gallon
$ 6.40
$ 5.00
$ 4.20
$ 3.50
Maximum quantity allowed (gallons)
100,000
300,000
360,000
220,000
Minimum % of Ingredient1 required
60%
50%
40%
10%
Explanation / Answer
A firm produces and sells four grades of industrial solvents – A, B, C, and D. T
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