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:
|
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 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:
Specify the objective function:
Specify the constraints:
(b) Solve the linear program and report your optimal solutions.
i. What is the maximum profit attainable under an optimal plan?
ii. 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
|
|
|
|
|
iii. How many gallons of each ingredientis used up under this optimal plan?
|
Quantity (in gallons)
|
Used
|
Available
|
|
Ingredient1
|
|
400,000
|
|
Ingredient2
|
|
600,000
|
(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:
|
Ingredient1:
|
$
|
per gallon.
|
|
Ingredient2:
|
$
|
per gallon.
|
Reasoning:
Question 2: Decision Analysis and Bayes Rule
Two trained classifiers - A and B - are available to classify tissue samples as benign or malignant. Each classifier is prone to two types of errors. The table below summarizes the probability of these errors:
|
Classifiers
|
False Positive Error Probability
|
False Negative Error Probability
|
|
A
|
0.06
|
0.01
|
|
B
|
0.04
|
0.02
|
- False Positive error probability is defined as the conditional probability of classifying ahealthy tissue sample as malignant.
- False Negative error probability is defined as the conditional probability of classifying an infected tissue sample as benign.
Historical data suggests that 10 percent of the tissue samples are infected.
a. Based on the information specified above, what is the conditional probability that:
(i) A tissue sample classified as benignbyclassifierB is actually infected?
(ii) A tissue sample classified as malignantbyclassifierA is actually healthy?
b. If the cost of classifying an infected tissue sample as benign is 100 times the cost of classifying a healthy tissue as malignant, which classifier should a risk neutral rational decision maker use? Why?
c. We assumed that 10% of the tissue samples are infected. At least how low should the percentage of infected tissues be for a risk neutral rational decision maker to prefer classifierB? Assume that all other parameters remain as specified in (a) and (b).
d. We assumed that the ratio of the cost of classifying an infected tissue sample as benign to the cost of classifying a healthy tissue as malignant is 100. At least how low must this ratio be for a risk neutral rational decision maker to prefer classifierB? Assume that all other parameters remain as specified in (a) and (b).