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Please see the attached: MAT540 Quantitative Method_WK10Homework.docx

Christy Tallant
MAT540 Quantitative Methods
Week 7
August 11, 2014
Chapter 3 Homework
Problem 8 (solve using the computer), 10, 11 (solve using the computer), 12, 14, 15, 16, 18,
24
8.) Solve the model formulated in Problem 7 for Southern Sporting Goods Company
graphically.
a. Identify the amount of unused resources (i.e., slack) at each of the graphical extreme points.
A: 3(0) + 2(160) + s1 = 500
s1 = 180
4(0) + 5(160) + s2 = 800
s2 = 0
B: 3(128.5) + 2(57.2) + s1 = 500
s1 = 0
4(128.5) + 2(57.2) + s2 = 800
s2 = 0
C: 2(167) + 2(0) + s1 = 500
s1 = 0
4(167) + 5(0) + s2 = 800
s2 = 132
b. What would be the effect on the optimal solution if the profit for a basketball changed from
$12 to $13? What would be the effect if the profit for a football changed from $16 to $15?
Z = 12x1 + 16x2
and,
x2 = Z/16 − 12x1/16
The slope of the objective function, −12/16, would have to become steeper (i.e., greater)
than the slope of the constraint line 4x1 + 5x2 = 800, for the solution to change. The profit,
c1, for a basketball that would change the solution point is,
−4/5 = −c1/16
5c1 = 64
c1 = 12.8
Since $13 > 12.8 the solution point would change to B where x1 = 128.5, x2 = 57.2.
The new Z value is $2,585.70.
For a football,
−4/5 = −12/c2
4c2 = 60
c2 = 15
Thus, if the profit for a football decreased to $15 or less, point B will also be optimal (i.e.,
multiple optimal solutions). The solution at B is x1 = 128.5, x2 = 57.2, and Z = $2,400.
c. What would be the effect on the optimal solution if 500 additional pounds of rubber could be
obtained? What would be the effect if 500 additional square feet of leather could be obtained?
If the constraint line for rubber changes to 3x1 + 2x2 = 1,000, it moves outward, eliminating points
B and C. However, since A is the optimal point, it will not change and the optimal solution
remains the same, x1 = 0, x2 = 160, and Z = 2,560. There will be an increase in slack, s1,
to 680 lbs.
If the constraint line for leather changes to 4x1 + 5x2 = 1,300, point A will move to a new
location, x1 = 0, x2 = 250, Z = $4,000.
10.) A company produces two products, A and B, which have profits of $9 and $7, respectively.
Each unit of product must be processed on two assembly lines, where the required production
times are as follows:
Product
A
B
Total Hours
Hours/Units
Line 1
12
4
60
Hours/Units
Line 2
4
8
40
a. Formulate a linear programming model to determine the optimal product mix that will
maximize profit.
Maximize...Z = 9 x1 + 7 x 2
subject..to..
12 x1 + 4 x 2 60
4 x1 + 8 x 2 40
x1 , x 2 0
b. Transform this model into standard form
Maximize...Z = 9 x1 + 7 x 2 + 0s1 + 0s 2
subject..to..
12 x1 + 4 x 2 + s1 = 60
4 x1 + 8 x 2 + s 2 = 40
x1 , x 2 , s1 , s 2 0
11.) Solve Problem 10 graphically
a. Identify the amount of unused resources (i.e., slack) at each of the graphical extreme points
b. What would be the effect on the optimal solution if the production time on line 1 was reduced
to 40 hours?
c. What would be the effect on the optimal solution if the profit for product B was increased
from $7 to $15? To $20.
12.) For the linear programming model formulated in Problem 10 and solved graphically in
Problem 11:
a. Determine the sensitivity ranges for the objective function coefficients, using graphical
analysis.
b. Verify the sensitivity ranges determined in (a) by using the computer.
c. Using the computer, determine the shadow prices for additional hours of production time on
line 1 and line 2 and indicate whether the company would prefer additional line 1 or line 2
hours.
14.) Solve the model formulated in Problem 13 for Irwin Textile Mills graphically.
a. How much extra cotton and processing time are left over at the optimal solution? Is the
demand for corduroy met?
5.0(456) + 7.5(510) + s1 = 6,500
s1 = 6,500 − 6,105
s1 = 395 lbs.
3.0(456) + 3.2(510) + s2 = 3,000
s2 = 0 hrs.
510 + s3 = 510
s3 = 0
Therefore demand for corduroy is met.
b. What is the effect on the optimal solution if the profit per yard of denim is increased
from $2.25 to $3.00? What is the effect if the profit per yard of corduroy is increased
from $3.10 to $4.00?
In order for the optimal solution point to change from B to C the slope of the objective
function must be at least as great as the slope of the constraint line, 3.0x1 + 3.2x2 = 3,000,
which is −3/3.2. Thus, the profit for denim would have to be,
c. −c1/3.0 = −3/3.2
d. c1 = 2.91
e.
f. If the profit for denim is increased from $2.25 to $3.00 the optimal solution would
change to point C, where x1 = 1,000, x2 = 0, Z = 3,000. Profit for corduroy has no
upper limit that would change the optimal solution point.
c. What would be the effect on the optimal solution if Irwin Mills could obtain only 6,000
pounds per cotton per month?
The constraint line for cotton would move inward as shown in the following graph where
point C is optimal.
15.) Solve the linear programming model formulated in Problem 13 for Irwin Mills by using the
computer.
Z = 2,607.000
a. If Irwin Mills can obtain additional cotton or processing time, but not both, which should
it select? How much? Explain your answer.
The company should select 237 additional hours of processing time, with a shadow price
of $0.75 per hour. Cotton has a shadow price of $0 because there is already extra (slack)
cotton available and not being used, so any more would have no marginal value.
b. Identify the sensitivity ranges for the objective function coefficients and for the constraint
quantity values. Then explain the sensitivity range for the demand for corduroy.
0 ≤ c1 ≤ 2.906
6,105 ≤ q1 ≤ ∞
2.4 ≤ c2 ≤ ∞
1,632 ≤ q2 ≤ 3,237
0 ≤ q3 ≤ 692.308
The demand for corduroy can decrease to zero or increase to 692.308 yds. without changing
the current solution mix of denim and corduroy. If the demand increases beyond 692.308
yds., then denim would no longer be produced and only corduroy would be produced.
16.) United Aluminum Company of Cincinnati produces three grades (high, medium, and low) of
aluminum at two mills. Each mill has a different production capacity (in tons per day) for each
grade as follows:
Aluminum
Grade
High
Medium
Low
Mill
1
6
2
4
Mill
2
2
2
10
The company has contracted with a manufacturing firm to supply at least 12 tons of high-grade
aluminum, 8 tons of medium-grade aluminum, and 5 tons of low grade aluminum. It costs United
$6,000 per day to operate mill 1 and $7,000 per day to operate mill 2. The company wants to
know the number of days to operate each mill in order to meet the contract at the minimum cost.
Formulate a linear programming model for this problem.
Let X1= No of days Mill 1 is operated
X2= no of days mill 2 is operated
Minimize, Z= $6000X1+$7000X2
Subject to:
6X1+2X2>=12(Demand for High grade of aluminium)
2X1+2X2>=8(Demand for High grade of aluminium)
4X1+10X2>=5(Demand for High grade of aluminium)
X1, X2>=0
18.) Solve the linear programming model formulated in Problem 16 for United Aluminum
Company by using the computer.
a. Identify and explain the shadow prices for each of the aluminum grade contract requirements.
Cost/Day
No of days to run
Aluminium grade
High
Medium
Low
X1
X2
$6,000
4
$7,000
0
6
2
4
2
2
10
$
24,000
24
8
16
>=
>=
>=
12
8
5
The United Aluminium company should run Mill 1 for 4 days at a minimum total cost of $24,000.
b. Identify the sensitivity ranges for the objective function coefficients and the constraint
quantity values.
24.) Solve the linear programming model developed in Problem 22 for the Burger Doodle
restaurant by using the computer.
The decision variables are
x1 = Number of Sausage biscuits and
x2 = Number of Ham biscuits
The objective function is
Maximize Z = $0.60 x1 + $0.50 x2
The model constraints are
0.010 x1 + 0.024 x2 < 6
(Labor)
0.10 x1 < 30
(Sausage)
0.15 x2 < 30
(Ham)
0.04 x1 + 0.04 x2 < 16
(Flour)
x1, x2 > 0
Thus, the linear programming model is
Maximize Z = $0.60 x1 + $0.50 x2
Subject to
0.010 x1 + 0.024 x2 < 6
0.10 x1 < 30
0.15 x2 < 30
0.04 x1 + 0.04 x2 < 16
x1, x2 > 0
The output of the above linear programming problem using Excel Solver is given below:
Target Cell (Max)
Cell
Name
Original Value Final Value
$B$14 Z =
0
230
Adjustable Cells
Cell
Name
Original Value Final Value
$B$12 x1 =
0.00
300.00
$B$13 x2 =
0.00
100.00
Constraints
Cell
Name
$E$6
Labor Used
5.4 $E$6<=$G$6 Not Binding
$E$7
Sausage Used
30 $E$7<=$G$7 Binding
$E$8
Ham Used
15 $E$8<=$G$8 Not Binding
$E$9
Flour Used
16 $E$9<=$G$9 Binding
Thus, the optimum solution is
Cell Value
Formula
Status
Slack
0.6
0
15
0
x1 = 300, x2 = 100
and the maximum value of Z is $230.
The sensitivity report of the LPP is given below:
Adjustable Cells
Cell
Name
Final Reduced Objective Allowable
Allowable
Value
Decrease
Cost
Coefficient Increase
$B$12 x1 =
300.00
0.00
0.6
1E+30
0.1
$B$13 x2 =
100.00
0.00
0.5
0.1
0.5
Constraints
Final
Shadow Constraint Allowable
Value
Price
R.H. Side
Increase
Allowable
Cell
Name
Decrease
$E$6
Labor Used
5.4
0
6
$E$7
Sausage Used
30
1
30
$E$8
Ham Used
15
0
30
1E+30
15
$E$9
Flour Used
16
12.5
16
1
4
1E+30
0.6
10 4.285714286
a. Indentify and explain the shadow prices for each of the resource constraints.
From the sensitivity report, the shadow price for both Labor and Ham are 0 since there are extra
labor hours and ham available. This means that additional amounts of those resources would add
nothing to profit.
The shadow price for Sausage is $1. This means that for every additional pound of sausage that
can be obtained, profit will increase by $1.
The shadow price for Flour is $12.5. This means that for every additional pound of flour that can
be obtained, profit will increase by $12.5.
b. Which of the resources constraints profit the most?
The shadow price for Flour is the highest. Thus, the resources constraint for Flour profits
the most.
c.
Identify the sensitivity ranges for the profit of a sausage biscuit and the amount of
sausage available. Explain these sensitivity ranges.
For a sausage biscuit (x1), the objective function coefficient is 0.6, allowable increase is
and allowable decrease is 0.1. Thus, the sensitivity range for the profit of a sausage biscuit
is
0.50 < c1 <
This sensitivity range for the profit of a sausage biscuit indicates that the optimal mix of
sausage and ham biscuits will remain optimal as long as profit of a sausage biscuit does
not fall below $0.50.
MAT540 Quantitative Methods
Week 10
August 25, 2014
Chapter 6 Homework
Complete the following problems from Chapter 6:
Problems 4, 6, 36, 48
4. Consider the following transportation problem:
To (cost)
From
A
B
C
Demand
1
$6
12
4
80
2
$9
3
8
110
3
$100
5
11
60
Supply
130
70
100
Formulate this problem as a linear programming model and solve it by using the computer.
6. Consider the following transportation problem:
To (cost)
From
A
B
C
Demand
1
$6
12
4
80
2
$9
3
8
110
3
$7
5
11
60
Supply
130
70
100
Solve it by using the computer.
36. World foods, Inc., imports food products such as meats, cheesed, and pastries to the United
States from warehouses at ports in Hamburg, Marseilles, and Liverpool. Ships from these ports
deliver the products to Norfolk, New York, and Savannah, where they are stored in company
warehouses before being shipped to distribution centers in Dallas, St. Louis, and Chicago. The
products are then distributed to specialty food stores and sold through catalogs. The shipping
costs ($/1,000 lb.) from the European ports to the U.S. cities and the available supplies (1,000
lb.) at the European ports are provided in the following table:
U.S. Cities
European Port
1. Hamburg
2. Marseillies
3. Liverpool
4. Norfolk
$420
510
450
5. New York
$390
590
360
6. Savannah
$610
470
480
Supply
55
78
37
The transportation costs (#/1,000 lb.) from each U.S. city of the three distribution centers and the
demands (1,000 lb) at the distribution center are as follows:
Distribution Center
Warehouse
4. Norfolk
5. New York
6. Savannah
Demand
7. Dallas
$75
125
68
60
8. St. Louis
$63
110
82
45
9. Chicago
$81
95
95
50
Determine the optimal shipments between the European ports and the warehouses and the
distribution centers to minimize total transportation costs.
48.The Omega pharmaceutical firm has five salespersons, whom the firm wants to assign to five
sales regions. Given their various previous contracts, the salespersons are able to cover the
regions in different amounts of time. The amount of time (days) required by each salesperson to
cover each city is shown in the following table:
Region
Salesperson
1
2
3
4
5
A
17
12
11
14
13
B
10
9
16
10
12
C
15
16
14
10
9
D
16
9
15
18
15
E
20
14
12
17
11
Which salesperson should be assigned to each region to minimize total time? Identify the optimal
assignments and compute the total minimum time.
MAT540 Quantitative Methods
Week 10
August 25, 2014
Chapter 6 Homework
Complete the following problems from Chapter 6:
Problems 4, 6, 36, 48
4. Consider the following transportation problem:
To (cost)
From
A
B
C
Demand
1
$6
12
4
80
2
$9
3
8
110
3
$100
5
11
60
Supply
130
70
100
Formulate this problem as a linear programming model and solve it by using the computer.
6. Consider the following transportation problem:
To (cost)
From
A
B
C
Demand
1
$6
12
4
80
2
$9
3
8
110
3
$7
5
11
60
Supply
130
70
100
Solve it by using the computer.
36. World foods, Inc., imports food products such as meats, cheesed, and pastries to the United
States from warehouses at ports in Hamburg, Marseilles, and Liverpool. Ships from these ports
deliver the products to Norfolk, New York, and Savannah, where they are stored in company
warehouses before being shipped to distribution centers in Dallas, St. Louis, and Chicago. The
products are then distributed to specialty food stores and sold through catalogs. The shipping
costs ($/1,000 lb.) from the European ports to the U.S. cities and the available supplies (1,000
lb.) at the European ports are provided in the following table:
U.S. Cities
European Port
1. Hamburg
2. Marseillies
3. Liverpool
4. Norfolk
$420
510
450
5. New York
$390
590
360
6. Savannah
$610
470
480
Supply
55
78
37
The transportation costs (#/1,000 lb.) from each U.S. city of the three distribution centers and the
demands (1,000 lb) at the distribution center are as follows:
Distribution Center
Warehouse
4. Norfolk
5. New York
6. Savannah
Demand
7. Dallas
$75
125
68
60
8. St. Louis
$63
110
82
45
9. Chicago
$81
95
95
50
Determine the optimal shipments between the European ports and the warehouses and the
distribution centers to minimize total transportation costs.
48.The Omega pharmaceutical firm has five salespersons, whom the firm wants to assign to five
sales regions. Given their various previous contracts, the salespersons are able to cover the
regions in different amounts of time. The amount of time (days) required by each salesperson to
cover each city is shown in the following table:
Region
Salesperson
1
2
3
4
5
A
17
12
11
14
13
B
10
9
16
10
12
C
15
16
14
10
9
D
16
9
15
18
15
E
20
14
12
17
11
Which salesperson should be assigned to each region to minimize total time? Identify the optimal
assignments and compute the total minimum time.
...

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