Question 1. Constrained Optimization
A company produces and sells four grades of industrial solvents - A, B, C, and D. The selling price per gallon of each grade of solventis $6.40, $5.00, $4.20, and $3.50 respectively. Because of demand limitations, the company can sell at most 100,000 gallons of solvent A; 300,000 gallons of solvent B; 360,000 gallons of solvent C; and 220,000gallons 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 $2.40 for Ingredient2. At most 400,000 gallons of Ingredient1 and 600,000 gallons of Ingredient2 are available.
Regulations require a minimum percentage by volume of Ingredient1 in each grade of solvent: 60% for A, 50% for B, 40% for C, and 10% for D.
For your convenience, the information presented above is summarized in the tables below:
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Solvent grade
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A
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B
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C
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D
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Selling price per gallon
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$ 6.40
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$ 5.00
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$ 4.20
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$ 3.50
|
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Maximum quantity allowed (gallons)
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100,000
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300,000
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360,000
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220,000
|
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Minimum % of Ingredient1 required
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60%
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50%
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40%
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10%
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|
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Availability (gallons)
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Price per gallon
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|
Ingredient1
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400,000
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$ 3.20
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|
Ingredient2
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600,000
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$ 2.40
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The company must determine an optimal production plan so as to maximize their profits subject to the applicable constraints.
(a) Formulate the problem as a linear program
Define the decision variables:
Y11 = # of gallons of Ingredient1 blend to produce solvent A
Y21 = # of gallons of Ingredient2 blend to produce solvent A
Y12 = # of gallons of Ingredient1 blend to produce solvent B
Y22 = # of gallons of Ingredient2 blend to produce solvent B
Y13 = # of gallons of Ingredient1 blend to produce solvent C
Y23 = # of gallons of Ingredient2 blend to produce solvent C
Y14 = # of gallons of Ingredient1 blend to produce solvent D
Y24 = # of gallons of Ingredient2 blend to produce solvent D
Specify the objective function:
Max Z = 6.40*(Y11+Y21)+5.00*(Y12+Y22)+4.20*(Y13+Y23)+3.50*(Y14+Y24) - 3.20*(Y11+Y12+Y13+Y14) -2.40*(Y21+Y22+Y23+Y24)
Specify the constraints:
(Y11+Y12+Y13+Y14) <= 400,000
(Y21+Y22+Y23+Y24) <= 600,000
(Y11+Y21) <= 100,000
(Y12+Y22) <= 300,000
(Y13+Y23)<=360,000
(Y14+Y24)<= 220,000
Y11 => 1.5*Y21
Y12 =>Y22
Y13 => 0.67*Y23
Y14 => 0.11*Y24
Yij => 0 for all i , j
(b) Solve the linear program and report your optimal solutions
i. What is the maximum profit attainable under an optimal plan?
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Maximum Profit =
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$ 1,766,000
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ii. How many gallons of each ingredient should be used to produce each grade of solvent under this optimal plan?
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Quantity (in gallons)
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A
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B
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C
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D
|
|
Ingredient1
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60,000
|
150,000
|
144,431
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25,569
|
|
Ingredient2
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40,000
|
150,000
|
215,569
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194,431
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iii. How many gallons of each ingredientis used up under this optimal plan?
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Quantity (in gallons)
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Used
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Available
|
|
Ingredient1
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380,000
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400,000
|
|
Ingredient2
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600,000
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600,000
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(c) At most how much should the company be willing to pay per gallon for additional quantities of the ingredients? Justify your answer.
The maximum amount that the company should be willing to pay for each additional gallon:
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Ingredient1:
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$ 0
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per gallon.
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|
Ingredient2:
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$ 0.8
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per gallon.
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Reasoning:
Refer to the Shadow price in column E of Sensitivity report in the excel file.Each constraint's shadow price tells how much objective function's total value change if you were to increase that constraint's limit by one.
The objective function is profit. The objective function's total value is total profit. The shadow price for each constraint is therefore how much profit would go up if you had one more unit of Ingredient available of that constraint's type of Ingredient.
The use of the shadow price is following:
It tells what it would be worth to get one more gallon of any one type of Ingredient. For example, the F10 "Ingredient2 Total constraint" has a shadow price of $0.8. For one more gallon of Ingredient2, the production plan can be changed in a way which will make $0.8 more in profit. The most someone would be willing to pay for another gallon of Ingredient2 would therefore be $0.8.
Again, the F9 Ingredient1 Total constraint has a shadow price of $ 0.For any additional gallon of Ingredient1, the company will not pay anything extra.