Expert answer:Quantitative Methods Homework
Answer & Explanation:Instructions for
Homework
1)
Homework problems are listed in the pdf file “mat540 hw wk8.pdf”.
There are 6 problems. You
are required to do problems 1,2,3,4 and 5.
To reduce your workload, ignore Q3 part b.
2)
You are required to use the Excel Solver to solve the LP
problems. Answers without embedded
formulas for solver will not be given credit.
3)
Download the homework template Excel file “Hw8_2015_Fall_Template”. You are required to use the template for the
homework.4)
Do the homework and enter your answers in the
cells highlighted in yellow. I will
grade only answers in the cells highlighted in yellow. You must show how you obtain the answers
using embedded Excel formula.
5)
Click on the link “Week 8 Homework
Submission”. Submit the completed
homework as attachment.
Notes
These homework problems are quite
challenging. To help you, I have left
some of the “usage” and “objective” formulas in the template. You are to complete the rest and run the
solver to solve it. The cells highlighted in gray are variables. Cells highlighted in yellow are places where
formulas need to be inserted. Solver
setups for Q1,2,3 are quite similar to Week 6 and week 7 homework. It just has more variables and
constraints. Solver layout for Q4 and Q5
are different from Q1,2,3. Study the
comments I have left with cells. You
should be able to figure out the rest. I
will go over these problems during Supplemental Instruction. I have also uploaded a file “Chapter 4
Textbook Examples”. It is the Excel
solver solutions for the examples given in Chapter 4 of your textbook. Study them with the equations given in the
textbook. You will get a better idea how
to use solver. To see the embedded
formula, hold the Ctrl key and click ~ key or double-click the highlighted
cells, it will show you which cells are being multiplied and added. Study the pattern and how it relates to the
LP equations. To go back to normal
display mode, hold the Ctrl key and click ~ key. It will toggle between the two modes.
mat540_hw_wk8_2.pdfmat540_hw_wk8_2.pdfhw8_fall_2015_template_v1.xlsx
mat540_hw_wk8_2.pdf
mat540_hw_wk8_2.pdf
hw8_fall_2015_template_v1.xlsx
Unformatted Attachment Preview
MAT540 Homework
Week 8
Page 1 of 4
MAT540
Week 8 Homework
Chapter 4
1.
Betty Malloy, owner of the Eagle Tavern in Pittsburgh, is preparing for Super Bowl Sunday, and
she must determine how much beer to stock. Betty stocks three brands of beer- Yodel, Shotz, and
Rainwater. The cost per gallon (to the tavern owner) of each brand is as follows:
Brand
Cost/Gallon
Yodel
$1.50
Shotz
0.90
Rainwater
0.50
The tavern has a budget of $2,000 for beer for Super Bowl Sunday. Betty sells Yodel at a rate of
$3.00 per gallon, Shotz at $2.50 per gallon, and Rainwater at $1.75 per gallon. Based on past
football games, Betty has determined the maximum customer demand to be 400 gallons of Yodel,
500 gallons of shotz, and 300 gallons of Rainwater. The tavern has the capacity to stock 1,000
gallons of beer; Betty wants to stock up completely. Betty wants to determine the number of gallons
of each brand of beer to order so as to maximize profit.
a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer.
2. As result of a recently passed bill, a congressman’s district has been allocated $3 million for
programs and projects. It is up to the congressman to decide how to distribute the money. The
congressman has decide to allocate the money to four ongoing programs because of their
importance to his district- a job training program, a parks project, a sanitation project, and a
mobile library. However, the congressman wants to distribute the money in a manner that will
please the most voters, or, in other words, gain him the most votes in the upcoming election. His
staff’s estimates of the number of votes gained per dollar spent for the various programs are as
follows.
MAT540 Homework
Week 8
Page 2 of 4
Program
Votes/Dollar
Job training
0.03
Parks
0.08
Sanitation
0.05
Mobile library
0.03
In order also to satisfy several local influential citizens who financed his election, he is obligated to
observe the following guidelines:
None of the programs can receive more than 30% of the total allocation
The amount allocated to parks cannot exceed the total allocated to both the sanitation
project and the mobile library.
The amount allocated to job training must at least equal the amount spent on the sanitation
project.
Any money not spent in the district will be returned to the government; therefore, the congressman
wants to spend it all. Thee congressman wants to know the amount to allocate to each program to
maximize his votes.
a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer.
3. Anna Broderick is the dietician for the State University football team, and she is attempting to
determine a nutritious lunch menu for the team. She has set the following nutritional guidelines
for each lunch serving:
Between 1,300 and 2,100 calories
At least 4 mg of iron
At least 15 but no more than 55g of fat
At least 30g of protein
At least 60g of carbohydrates
No more than 35 mg of cholesterol
She selects the menu from seven basic food items, as follows, with the nutritional contributions
per pound and the cost as given:
MAT540 Homework
Week 8
Page 3 of 4
Calories Iron
(per lb.)
Protein
(mg/lb.) (g/lb.)
Carbo-
Fat
Cholesterol Cost
hydrates
(g/lb.)
(mg/lb)
($/lb.)
(g/lb.)
Chicken
500
4.2
17
0
30
180
0.85
Fish
480
3.1
85
0
5
90
3.35
Ground beef
840
0.25
82
0
75
350
2.45
Dried beans
590
3.2
10
30
3
0
0.85
Lettuce
40
0.4
6
0
0
0
0.70
Potatoes
450
2.25
10
70
0
0
0.45
Milk (2%)
220
0.2
16
22
10
20
0.82
The dietician wants to select a menu to meet the nutritional guidelines while minimizing the total
cost per serving.
a. Formulate a linear programming model for this problem and solve.
b. If a serving of each of the food items (other than milk) was limited to no more than a half
pound, what effect would this have on the solution?
4. Dr. Maureen Becker, the head administrator at Jefferson County Regional Hospital, must
determine a schedule for nurses to make sure there are enough of them on duty throughout the
day. During the day, the demand for nurses varies. Maureen has broken the day in to twelve 2hour periods. The slowest time of the day encompasses the three periods from 12:00 A.M. to 6:00
A.M., which beginning at midnight; require a minimum of 30, 20, and 40 nurses, respectively.
The demand for nurses steadily increases during the next four daytime periods. Beginning with
the 6:00 A.M.- 8:00 A.M. period, a minimum of 50, 60, 80, and 80 nurses are required for these
four periods, respectively. After 2:00 P.M. the demand for nurses decreases during the afternoon
and evening hours. For the five 2-hour periods beginning at 2:00 P.M. and ending midnight, 70,
70, 60, 50, and 50 nurses are required, respectively. A nurse reports for duty at the beginning of
one of the 2-hour periods and works 8 consecutive hours (which is required in the nurses’
contract).
Dr. Becker wants to determine a nursing schedule that will meet the hospital’s
minimum requirement throughout the day while using the minimum number of nurses.
a. Formulate a linear programming model for this problem.
MAT540 Homework
Week 8
Page 4 of 4
b. Solve the model by using the computer.
5. The production manager of Videotechnics Company is attempting to determine the
upcoming 5-month production schedule for video recorders. Past production records
indicate that 2,000 recorders can be produced per month. An additional 600 recorders can
be produced monthly on an overtime basis. Unit cost is $10 for recorders produced
during regular working hours and $15 for those produced on an overtime basis.
Contracted sales per month are as follows:
Month
Contracted Sales (units)
1
1200
2
2100
3
2400
4
3000
5
4000
Inventory carrying costs are $2 per recorder per month. The manager does not want any
inventory carried over past the fifth month. The manager wants to know the monthly
production that will minimize total production and inventory costs.
a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer.
MAT540 Homework
Week 8
Page 1 of 4
MAT540
Week 8 Homework
Chapter 4
1.
Betty Malloy, owner of the Eagle Tavern in Pittsburgh, is preparing for Super Bowl Sunday, and
she must determine how much beer to stock. Betty stocks three brands of beer- Yodel, Shotz, and
Rainwater. The cost per gallon (to the tavern owner) of each brand is as follows:
Brand
Cost/Gallon
Yodel
$1.50
Shotz
0.90
Rainwater
0.50
The tavern has a budget of $2,000 for beer for Super Bowl Sunday. Betty sells Yodel at a rate of
$3.00 per gallon, Shotz at $2.50 per gallon, and Rainwater at $1.75 per gallon. Based on past
football games, Betty has determined the maximum customer demand to be 400 gallons of Yodel,
500 gallons of shotz, and 300 gallons of Rainwater. The tavern has the capacity to stock 1,000
gallons of beer; Betty wants to stock up completely. Betty wants to determine the number of gallons
of each brand of beer to order so as to maximize profit.
a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer.
2. As result of a recently passed bill, a congressman’s district has been allocated $3 million for
programs and projects. It is up to the congressman to decide how to distribute the money. The
congressman has decide to allocate the money to four ongoing programs because of their
importance to his district- a job training program, a parks project, a sanitation project, and a
mobile library. However, the congressman wants to distribute the money in a manner that will
please the most voters, or, in other words, gain him the most votes in the upcoming election. His
staff’s estimates of the number of votes gained per dollar spent for the various programs are as
follows.
MAT540 Homework
Week 8
Page 2 of 4
Program
Votes/Dollar
Job training
0.03
Parks
0.08
Sanitation
0.05
Mobile library
0.03
In order also to satisfy several local influential citizens who financed his election, he is obligated to
observe the following guidelines:
None of the programs can receive more than 30% of the total allocation
The amount allocated to parks cannot exceed the total allocated to both the sanitation
project and the mobile library.
The amount allocated to job training must at least equal the amount spent on the sanitation
project.
Any money not spent in the district will be returned to the government; therefore, the congressman
wants to spend it all. Thee congressman wants to know the amount to allocate to each program to
maximize his votes.
a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer.
3. Anna Broderick is the dietician for the State University football team, and she is attempting to
determine a nutritious lunch menu for the team. She has set the following nutritional guidelines
for each lunch serving:
Between 1,300 and 2,100 calories
At least 4 mg of iron
At least 15 but no more than 55g of fat
At least 30g of protein
At least 60g of carbohydrates
No more than 35 mg of cholesterol
She selects the menu from seven basic food items, as follows, with the nutritional contributions
per pound and the cost as given:
MAT540 Homework
Week 8
Page 3 of 4
Calories Iron
(per lb.)
Protein
(mg/lb.) (g/lb.)
Carbo-
Fat
Cholesterol Cost
hydrates
(g/lb.)
(mg/lb)
($/lb.)
(g/lb.)
Chicken
500
4.2
17
0
30
180
0.85
Fish
480
3.1
85
0
5
90
3.35
Ground beef
840
0.25
82
0
75
350
2.45
Dried beans
590
3.2
10
30
3
0
0.85
Lettuce
40
0.4
6
0
0
0
0.70
Potatoes
450
2.25
10
70
0
0
0.45
Milk (2%)
220
0.2
16
22
10
20
0.82
The dietician wants to select a menu to meet the nutritional guidelines while minimizing the total
cost per serving.
a. Formulate a linear programming model for this problem and solve.
b. If a serving of each of the food items (other than milk) was limited to no more than a half
pound, what effect would this have on the solution?
4. Dr. Maureen Becker, the head administrator at Jefferson County Regional Hospital, must
determine a schedule for nurses to make sure there are enough of them on duty throughout the
day. During the day, the demand for nurses varies. Maureen has broken the day in to twelve 2hour periods. The slowest time of the day encompasses the three periods from 12:00 A.M. to 6:00
A.M., which beginning at midnight; require a minimum of 30, 20, and 40 nurses, respectively.
The demand for nurses steadily increases during the next four daytime periods. Beginning with
the 6:00 A.M.- 8:00 A.M. period, a minimum of 50, 60, 80, and 80 nurses are required for these
four periods, respectively. After 2:00 P.M. the demand for nurses decreases during the afternoon
and evening hours. For the five 2-hour periods beginning at 2:00 P.M. and ending midnight, 70,
70, 60, 50, and 50 nurses are required, respectively. A nurse reports for duty at the beginning of
one of the 2-hour periods and works 8 consecutive hours (which is required in the nurses’
contract).
Dr. Becker wants to determine a nursing schedule that will meet the hospital’s
minimum requirement throughout the day while using the minimum number of nurses.
a. Formulate a linear programming model for this problem.
MAT540 Homework
Week 8
Page 4 of 4
b. Solve the model by using the computer.
5. The production manager of Videotechnics Company is attempting to determine the
upcoming 5-month production schedule for video recorders. Past production records
indicate that 2,000 recorders can be produced per month. An additional 600 recorders can
be produced monthly on an overtime basis. Unit cost is $10 for recorders produced
during regular working hours and $15 for those produced on an overtime basis.
Contracted sales per month are as follows:
Month
Contracted Sales (units)
1
1200
2
2100
3
2400
4
3000
5
4000
Inventory carrying costs are $2 per recorder per month. The manager does not want any
inventory carried over past the fifth month. The manager wants to know the monthly
production that will minimize total production and inventory costs.
a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer.
variables
maximize
subject to
Y,S,R
1.5*Y + 1.6*S + 1.25*R
1.5*Y + 0.9*S + 0.5*R <= 2000
Y + S + R <= 1000
Y <= 400
S <= 500
R <= 300
Profit
Yodel
Shotz
Rainwater
profit
budget
Storage capacity
Yodel demand
Shotz demand
Rainwater demand
Budget
1.5
1.6
1.25
Storage
1.5
0.9
0.5
0
Usage
<=
Available
Profit
0
Yodel
1
1
1
0
<=
2000
Rainwater
<=
<=
1
0
<=
1000
Shotz
400
variables
maximize
subject to
J,P,S,M
0.03*J + 0.08*P + 0.05*S + 0.03*M
J + P + S + M = 3000000
J <= 3000000*30%
P <= 3000000*30%
S <= 3000000*30%
M <= 3000000*30%
P - S - M <= 0
J - S => 0
Votes/$ Budget
J limit
P limit
0.03
1
1
0.08
1
0
0.05
1
0
0.03
1
0
Usage
0
=
<=
Available
3000000
900000
Job Allocation
Parks Allocation
Sanitation Allocation
Mobile Allocation
Total Votes
votes
budget
Job training allocation limit
Parks allocation limit
Sanitation allocation limit
Mobile Library allocation limit
Constraint 1
Constraint 2
0
S limit
g allocation limit
allocation limit
rary allocation limit
M limit
Const. 1
Const. 2
0
1
-1
-1
Anna Broderick Lunch Menu
Decision
Cost
Variables
Calories
Calories
Iron
Protein
$/lb.
(per lb.)
(per lb.)
(mg/lb.)
(g/lb.)
Chicken
Fish
Ground
beef
Dried
beans
Lettuce
Potatoes
0.85
3.35
500
480
500
480
4.2
3.1
17
85
2.45
840
840
0.25
82
0.85
590
590
3.2
10
0.7
0.45
40
450
40
450
0.4
2.25
6
10
Milk (2%)
0.82
220
220
0.2
16
>=
1300
<=
2100
>=
4
>=
30
Usage
Available
0
Minimize Cost
($)
(a)
Objective
(b)
if a serving of each of the the food items (other than milk) was limited to no more than a half pound, what effec
Decision Variables
Objective
Chicken
Fish
Ground beef
Dried beans
Lettuce
Potatoes
Milk (2%)
<=
<=
<=
<=
<=
<=
Minimize Cost
($)
Answer:
No feasible solution found
0.5
0.5
0.5
0.5
0.5
0.5
Carbohydrates
Fat (g/lb.)
Fat (g/lb.)
(g/lb.)
Cholesterol
(mg/lb.)
0
0
30
5
30
5
180
90
0
75
75
350
30
3
3
0
0
70
0
0
0
0
0
0
22
10
10
20
>=
60
>=
15
<=
55
<=
35
a half pound, what effect would this have on the solution?
Jefferson County Regional HospitalScheduling
Let Xi = number of nurses that begin their 8 hour shift in period I (I = 1,2,3,4, …., 12)
period 1 12:00AM -- 2:00Am
period 2 2:00AM -- 4:00 AM etc
DV
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
Objective Function
Period
1
2
3
4
5
6
7
8
9
10
11
12
0
# of Nurses
working
0
0
0
0
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
Minimum # of
Nurses
30
20
40
50
60
80
80
70
70
60
50
50
Videotechnics Company (Multi Production and Inventory)
Let Ri = Regular production in month I, where I = 1,2,3,4,5
Oi = Overtime production in month I, where I = 1,2,3,4,5
I_i = Inventory at end of month I, where I = 1,2,3,4
Week
1
2
3
4
5
Total Reg
Prod
Cost/unit
Total Cost
Regular
Prod (Ri)
Regular
Overtime
Capacity
Prod (Oi)
2000
2000
2000
2000
2000
Total OT
0
Prod
Cost/unit
10
-2400.00
Overtime
Capacity
600
600
600
600
600
0
15
Cost for recorders produced during r
Cost for recorders produced during r
Video
Recorders
Available
Total
for
Prod.
shipping
0
0
-1200
ost for recorders produced during regular working hours is $10
ost for recorders produced during regular working hours is $15
Video
Recorders Inventory
Sales
I_i
1200
-1200
2100
2400
3000
4000
Total
Inventory
Cost/unit
-1200
2
…
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