Rutgers University Statistics Questions

Homework 2 – Due on Oct 30 23:59•
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Total score of this homework is 30 pt. – each of the sub-problems is for 1 pt.
Late homework will be penalized 20% per day for a maximum of one days.
All homework needs to be submitted through Canvas using a PDF file.
Show your work to get full marks. (Attach screenshot of excel part if needed)
1. A linear programming model is given as follows:
Minimize
Subject to
𝑍 = 5.2𝑥1 + 2.5𝑥2 + 6.0𝑥3
3𝑥1 + 2𝑥2 + 2𝑥3 ≥ 200
𝑥1
≥ 0.4
𝑥1 + 𝑥2 + 𝑥3
𝑥2 + 𝑥3
≤ 0.2
2𝑥1
𝑥1 ≥ 𝑥2 + 𝑥3
𝑥1 , 𝑥2 , 𝑥3 ≥ 0
(a) Solve the problem by using the computer. What are the minimum Z and the optimal point?
(b) Obtain the values of the slack/surplus variables at the optimal solution in (a)
(c) Identify the sensitivity range of the objective function coefficient of 𝑥2.
(d) Identify the sensitivity range of the value of the 1st resource constraint (right-hand side).
(e) Identify the sensitivity range of the value of the 3rd resource constraint (right-hand side).
(f) which of the following makes the model infeasible? (Choose one)
i.
Increase of the coefficient of 𝑥1 on the objective function to 2000
ii.
Decrease of the coefficient of 𝑥1 on the 1st constraint to -5
iii.
Addition of a new constraint, 𝑥1 + 2𝑥2 + 3𝑥3 ≤ 100
iv.
Removal of the non-negativity constraints for 𝑥1, 𝑥2 , 𝑥3
v.
None
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2. Lakeside Boatworks is planning to manufacture three types of molded fiberglass recreational
boats – a fishing (bass) boat, a ski boat, and a small speedboat. The estimated selling price and
variable cost for each type of boat are summarized in the following table:
Boat
Variable Cost ($)
Selling Price ($)
Bass
12,000
24,000
Ski
9,000
15,000
Speed
15,000
30,000
The company has incurred fixed costs of $3,000,000 to set up its manufacturing operation and
begin production. Lakeside has also entered into agreements with several boat dealers in the
region to provide a minimum of 50 bass boats, 60 ski boats, and 70 speedboats. Alternatively,
the company is unsure of what actual demand will be, so it has decided to limit production to no
more than 100 of any one boat. The company wants to determine the number of boats that it must
sell to break even, while minimizing its total variable cost.
(a) Formulate this problem in a linear programming model.
(b) Solve the problem by using the computer. What are the minimum variable cost and the
optimal point?
(c) To produce $500,000 profit, how many boats the company needs to produce?
(d) If the agreements were changed to a minimum of 40 bass boats, 70 ski boats, and 60
speedboats, how many boats the company needs to produce?
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3. The Mill Mountain Coffee Shop blends coffee on the premises for its customers. It sells three
basic blends in 1-pound bags, Special, Mountain Dark, and Mill Regular. It uses four different
types of coffee to produce the blends – Brazilian, mocha, Colombian, and mild. The shop has the
following blend recipe requirements:
Blend
Mix requirements
Selling price ($)/pound
Special
At least 30% Columbian, at least 40% mocha
6.20
Dark
At least 50% Brazilian, no more than 10% mild
5.50
Regular
No more than 70% mild, at least 30% Brazilian
4.00
The cost of Brazilian coffee is $2.00 per pound, the cost of mocha is $2.75 per pound, the cost
of Colombian is $2.90 per pound, and the cost of mild is $1.70 per pound. The shop has 110
pounds of Brazilian coffee, 70 pounds of mocha, 80 pounds of Colombian, and 150 pounds of
mild coffee available per week. The shop wants to know the amount of each blend it should
prepare each week to maximize profit.
(a) Formulate this problem in a linear programming model.
(b) Solve the problem by using the computer. What are the maximum Z and the optimal point?
(c) Based on the optimal solution (b), how much cost can be caused by the use of Colombian
coffee?
(d) Based on the optimal solution (b), how much sales can be generated by selling Mountain
Dark blend?
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(6’, 3 points for each sub-questions)
5. Consider two decisions A and B represented by two binary decision variables 𝑋𝐴 and 𝑋𝐵 , each
of which takes 1 if the decision is made to choose it, and 0 otherwise. Select an appropriate
expression for the constraint in each question.
① 𝑋𝐴 + 𝑋𝐵 = 0
⑤ 𝑋𝐴 + 𝑋𝐵 = 1
② 𝑋𝐴 − 𝑋𝐵 = 0
⑥ 𝑋𝐴 + 𝑋𝐵 ≤ 1
③ 𝑋𝐴 + 𝑋𝐵 ≥ 0
⑦ 𝑋𝐴 + 𝑋𝐵 ≥ 1
④ 𝑋𝐴 − 𝑋𝐵 ≥ 0
⑧ 𝑋𝐴 + 𝑋𝐵 = 2
(a) Both A and B should be always chosen.
(b) At least one of the decisions A and B should be chosen; and both can be chosen together.
(c) At least one of the decisions A and B should be chosen; and both should not be chosen
together.
(d) Each of the decisions A and B can be chosen individually; and both should not be chosen
together.
(e) B can be chosen when A is chosen; but A should be chosen when B is chosen.
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6. Northwoods Backpackers is a retail catalog store in Vermont that specializes in outdoor clothing
and camping equipment. Phone orders are taken each day by a large pool of computer operators,
which consists of permanent and temporary operators. A permanent operator can process an
average of 70 orders per day, whereas a temporary operator can process an average of 50 orders
per day. The company averages at least 850 orders per day. The store has 15 computer
workstations. A permanent operator processes averagely 1.5 orders with errors each day, whereas
a temporary operator processes averagely 6.0 orders with errors each day. The store wants to
limit total errors to 32 per day. A permanent operator is paid $100 per day, including benefits,
and a temporary operator is paid $50 per day. The company wants to know the number of
permanent and temporary operators to hire to minimize total cost.
(a) Formulating the problem as a linear programing model, find the minimal total cost and the
optimal solution point by using the computer.
(b) Choose which of the following constraint(s) will be violated by a new solution from rounding
down the coordinates of the optimal solution point in (a) to the nearest integer.
① The minimal number of orders
② Available computer workstations
③ Total error limitation
④ None
(c) Formulating the problem as a total integer programing model, find the minimal total cost
and the optimal solution point by using the computer.
(d) Illustrate the feasible solution area and optimal solutions in (a) and (c) graphically.
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Self-practice (No need to submit)
1. A linear programming model is given as follows:
Maximize
Subject to
𝑍 = 20𝑥1 + 47𝑥2
50𝑥1 + 24𝑥2 ≤ 1200
77𝑥1 − 50𝑥2 ≥ 300
3𝑥1
≥ 50
𝑥1 , 𝑥2 ≥ 0
(a) Solve the problem by using the computer. What are the maximum Z and the optimal point?
(b) Obtain the values of the slack/surplus variables at the optimal solution in (a)
(c) Identify the sensitivity range of the objective function coefficient of 𝑥1.
(d) Identify the sensitivity range of the value of the 1st resource constraint (right-hand side).
(e) Identify the sensitivity range of the value of the 2nd resource constraint (right-hand side).
(f) which of the following makes the model infeasible? (Choose one)
i.
Increase of the coefficient of 𝑥1 on the objective function to 3000
ii.
Decrease of the coefficient of 𝑥1 on the 2nd constraint to -5000
iii.
Addition of a new constraint, 𝑥2 ≥ 50
iv.
Removal of the non-negativity constraints for 𝑥1, 𝑥2
v.
None
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Solutions for Self-practice questions:
1. (a) 𝑍 ∗ = 1051.39 at (16.67, 15.28)
(b) 𝑠1 = 0, 𝑠2 = 219.44, 𝑠3 = 0
(c) 20 − ∞ ≤ 𝑐1 ≤ 20 + 77.92 where ∞ ≈ (1𝐸 + 30) ⇔ −∞ ≤ 𝑐1 ≤ 97.92
(d) 1200 − 366.67 ≤ 𝑏1 ≤ 1200 + 105.33 ⇔ 833.33 ≤ 𝑐1 ≤ 1305.33
(e) 300 − ∞ ≤ 𝑏2 ≤ 300 + 219.44 where ∞ ≈ (1𝐸 + 30) ⇔ −∞ ≤ 𝑏2 ≤ 519.44
(f) ii or iii
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