Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
MATH 105: Finite Mathematics
3-3: Linear Programming Story Problems
Prof. Jonathan Duncan
Walla Walla College
Winter Quarter, 2006
Conclusion
Investment Allocation
Widget Production
Outline
1
Investment Allocation
2
Widget Production
3
Hotel Occupancy
4
Cruise Ships
5
Conclusion
Hotel Occupancy
Cruise Ships
Conclusion
Investment Allocation
Widget Production
Outline
1
Investment Allocation
2
Widget Production
3
Hotel Occupancy
4
Cruise Ships
5
Conclusion
Hotel Occupancy
Cruise Ships
Conclusion
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Investments: Problem Set-up
Example
An investment broker wants to invest up to $20,000. She can purchase
type A bonds yielding a 10% return and she can purchase a more risky
type B bounds yielding a 15% return. She wants to invest at least as
much in the type A bond as in the type B bond to reduce her risk. The
minimum investment for the type A bond is $5,000, and there are only
$8,000 worth of type B bonds available. How much should the broker
invest in each type of bond to maximize her returns?
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Investments: Problem Set-up
Example
An investment broker wants to invest up to $20,000. She can purchase
type A bonds yielding a 10% return and she can purchase a more risky
type B bounds yielding a 15% return. She wants to invest at least as
much in the type A bond as in the type B bond to reduce her risk. The
minimum investment for the type A bond is $5,000, and there are only
$8,000 worth of type B bonds available. How much should the broker
invest in each type of bond to maximize her returns?
Let:
x be amount invested in A
y the amount invested in B
Objective:
Maximize 0.10x + 0.15y
Constraints:

x + y




x

x


y



y
≤
≥
≥
≤
≥
20, 000
y
5, 000
8, 000
0
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Investments: Locating Corner Points
Example Continued. . .
The region obtained from the constraints is shown below.
Conclusion
x+y=20000
y=8000
Investment Allocation
Widget Production
x=5000
y=x
Hotel Occupancy
Cruise Ships
Investments: Locating Corner Points
Example Continued. . .
The region obtained from the constraints is shown below.
Conclusion
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Investments: Locating Corner Points
Example Continued. . .
The region obtained from the constraints is shown below.
x=5000
Intersecting Lines
y = 8000 and y = x
(8000, 8000)
y = 8000 and x + y = 20000
(12000, 8000)
y = 0 and x + y = 20000
(20000, 0)
(5000, 5000)
(5000, 0)
x = 5000 and y = x
x = 5000 and y = 0
x+y=20000
y=8000
y=x
Point
Conclusion
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Investments: Locating Corner Points
Example Continued. . .
The region obtained from the constraints is shown below.
x=5000
Intersecting Lines
y = 8000 and y = x
(8000, 8000)
y = 8000 and x + y = 20000
(12000, 8000)
y = 0 and x + y = 20000
(20000, 0)
(5000, 5000)
(5000, 0)
x = 5000 and y = x
x = 5000 and y = 0
x+y=20000
y=8000
y=x
Point
Conclusion
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Investments: Locating Corner Points
Example Continued. . .
The region obtained from the constraints is shown below.
x=5000
Intersecting Lines
y = 8000 and y = x
(8000, 8000)
y = 8000 and x + y = 20000
(12000, 8000)
y = 0 and x + y = 20000
(20000, 0)
(5000, 5000)
(5000, 0)
x = 5000 and y = x
x = 5000 and y = 0
x+y=20000
y=8000
y=x
Point
Conclusion
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Investments: Locating Corner Points
Example Continued. . .
The region obtained from the constraints is shown below.
x=5000
Intersecting Lines
y = 8000 and y = x
(8000, 8000)
y = 8000 and x + y = 20000
(12000, 8000)
y = 0 and x + y = 20000
(20000, 0)
(5000, 5000)
(5000, 0)
x = 5000 and y = x
x = 5000 and y = 0
x+y=20000
y=8000
y=x
Point
Conclusion
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Investments: Locating Corner Points
Example Continued. . .
The region obtained from the constraints is shown below.
x=5000
Intersecting Lines
y = 8000 and y = x
(8000, 8000)
y = 8000 and x + y = 20000
(12000, 8000)
y = 0 and x + y = 20000
(20000, 0)
(5000, 5000)
(5000, 0)
x = 5000 and y = x
x = 5000 and y = 0
x+y=20000
y=8000
y=x
Point
Conclusion
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Investments: Locating Corner Points
Example Continued. . .
The region obtained from the constraints is shown below.
x=5000
Intersecting Lines
y = 8000 and y = x
(8000, 8000)
y = 8000 and x + y = 20000
(12000, 8000)
y = 0 and x + y = 20000
(20000, 0)
(5000, 5000)
(5000, 0)
x = 5000 and y = x
x = 5000 and y = 0
x+y=20000
y=8000
y=x
Point
Conclusion
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Investments: The Solution
Example Continued. . .
Finally, plug each corner point into the objective function.
Corner Point
(5000, 0)
(5000, 5000)
Value of 0.10x + 0.15y
.1(5000) + .15(0) =
.10(5000) + .15(5000) =
$500
$1350
(8000, 8000)
.10(8000) + .15(8000) =
$2000
(12000, 8000)
.10(12000) + .15(8000) =
$2400
.10(20000) + .15(0) =
$2000
(20000, 0)
Maximum profit of $2,400 when she invests $12,000 in type A and
$8,000 in type B bonds.
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Investments: The Solution
Example Continued. . .
Finally, plug each corner point into the objective function.
Corner Point
(5000, 0)
(5000, 5000)
Value of 0.10x + 0.15y
.1(5000) + .15(0) =
.10(5000) + .15(5000) =
$500
$1350
(8000, 8000)
.10(8000) + .15(8000) =
$2000
(12000, 8000)
.10(12000) + .15(8000) =
$2400
.10(20000) + .15(0) =
$2000
(20000, 0)
Maximum profit of $2,400 when she invests $12,000 in type A and
$8,000 in type B bonds.
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Investments: The Solution
Example Continued. . .
Finally, plug each corner point into the objective function.
Corner Point
(5000, 0)
(5000, 5000)
Value of 0.10x + 0.15y
.1(5000) + .15(0) =
.10(5000) + .15(5000) =
$500
$1350
(8000, 8000)
.10(8000) + .15(8000) =
$2000
(12000, 8000)
.10(12000) + .15(8000) =
$2400
.10(20000) + .15(0) =
$2000
(20000, 0)
Maximum profit of $2,400 when she invests $12,000 in type A and
$8,000 in type B bonds.
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Investments: The Solution
Example Continued. . .
Finally, plug each corner point into the objective function.
Corner Point
(5000, 0)
(5000, 5000)
Value of 0.10x + 0.15y
.1(5000) + .15(0) =
$500
.10(5000) + .15(5000) =
$1350
(8000, 8000)
.10(8000) + .15(8000) =
$2000
(12000, 8000)
.10(12000) + .15(8000) =
$2400
.10(20000) + .15(0) =
$2000
(20000, 0)
Maximum profit of $2,400 when she invests $12,000 in type A and
$8,000 in type B bonds.
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Investments: The Solution
Example Continued. . .
Finally, plug each corner point into the objective function.
Corner Point
(5000, 0)
(5000, 5000)
Value of 0.10x + 0.15y
.1(5000) + .15(0) =
$500
.10(5000) + .15(5000) =
$1350
(8000, 8000)
.10(8000) + .15(8000) =
$2000
(12000, 8000)
.10(12000) + .15(8000) =
$2400
.10(20000) + .15(0) =
$2000
(20000, 0)
Maximum profit of $2,400 when she invests $12,000 in type A and
$8,000 in type B bonds.
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Investments: The Solution
Example Continued. . .
Finally, plug each corner point into the objective function.
Corner Point
(5000, 0)
(5000, 5000)
Value of 0.10x + 0.15y
.1(5000) + .15(0) =
$500
.10(5000) + .15(5000) =
$1350
(8000, 8000)
.10(8000) + .15(8000) =
$2000
(12000, 8000)
.10(12000) + .15(8000) =
$2400
.10(20000) + .15(0) =
$2000
(20000, 0)
Maximum profit of $2,400 when she invests $12,000 in type A and
$8,000 in type B bonds.
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Investments: The Solution
Example Continued. . .
Finally, plug each corner point into the objective function.
Corner Point
(5000, 0)
(5000, 5000)
Value of 0.10x + 0.15y
.1(5000) + .15(0) =
$500
.10(5000) + .15(5000) =
$1350
(8000, 8000)
.10(8000) + .15(8000) =
$2000
(12000, 8000)
.10(12000) + .15(8000) =
$2400
.10(20000) + .15(0) =
$2000
(20000, 0)
Maximum profit of $2,400 when she invests $12,000 in type A and
$8,000 in type B bonds.
Investment Allocation
Widget Production
Outline
1
Investment Allocation
2
Widget Production
3
Hotel Occupancy
4
Cruise Ships
5
Conclusion
Hotel Occupancy
Cruise Ships
Conclusion
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Widgets: Problem Set-up
Example
Ms. Li produces widgets. To make 100 left-handed widgets she uses 1
pound of metal and 5 pounds of fiberglass. To make 100 right-handed
widgets she uses 2 pounds of metal and 3 pounds of fiberglass. Each
week Ms. Li has 65 poinds of metal and 150 pounds of fiberglass
delivered. She makes a profit of $2.50 per right-handed widget and $2.00
per left-handed widget. How many widgets of each type should Ms. Li
produce to maximize profit?
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Widgets: Problem Set-up
Example
Ms. Li produces widgets. To make 100 left-handed widgets she uses 1
pound of metal and 5 pounds of fiberglass. To make 100 right-handed
widgets she uses 2 pounds of metal and 3 pounds of fiberglass. Each
week Ms. Li has 65 poinds of metal and 150 pounds of fiberglass
delivered. She makes a profit of $2.50 per right-handed widget and $2.00
per left-handed widget. How many widgets of each type should Ms. Li
produce to maximize profit?
Let:
x be # 100s of right-handed
y be # 100s of left-handed
Objective:
Maximize 250x + 200y
Constraints:

x + 2y



5x + 3y
x



y
≤
≤
≥
≥
65
150
0
0
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Widgets: Locating Corner Points
Example Continued. . .
The region obtained from the constraints is shown below.
Conclusion
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Widgets: Locating Corner Points
Example Continued. . .
The region obtained from the constraints is shown below.
5x+3y=150
x+2y=65
Conclusion
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Widgets: Locating Corner Points
Example Continued. . .
The region obtained from the constraints is shown below.
Intersecting Lines
Point
y = 0 and 5x + 3y = 150
(30, 0)
x = 0 and y = 0
(0, 0)
x = 0 and x + 2y = 65
(0, 32.5)
5x + 3y = 150 and x + 2y = 65
(15, 25)
5x+3y=150
x+2y=65
Conclusion
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Widgets: Locating Corner Points
Example Continued. . .
The region obtained from the constraints is shown below.
Intersecting Lines
Point
y = 0 and 5x + 3y = 150
(30, 0)
x = 0 and y = 0
(0, 0)
x = 0 and x + 2y = 65
(0, 32.5)
5x + 3y = 150 and x + 2y = 65
(15, 25)
5x+3y=150
x+2y=65
Conclusion
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Widgets: Locating Corner Points
Example Continued. . .
The region obtained from the constraints is shown below.
Intersecting Lines
Point
y = 0 and 5x + 3y = 150
(30, 0)
x = 0 and y = 0
(0, 0)
x = 0 and x + 2y = 65
(0, 32.5)
5x + 3y = 150 and x + 2y = 65
(15, 25)
5x+3y=150
x+2y=65
Conclusion
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Widgets: Locating Corner Points
Example Continued. . .
The region obtained from the constraints is shown below.
Intersecting Lines
Point
y = 0 and 5x + 3y = 150
(30, 0)
x = 0 and y = 0
(0, 0)
x = 0 and x + 2y = 65
(0, 32.5)
5x + 3y = 150 and x + 2y = 65
(15, 25)
5x+3y=150
x+2y=65
Conclusion
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Widgets: Locating Corner Points
Example Continued. . .
The region obtained from the constraints is shown below.
Intersecting Lines
Point
y = 0 and 5x + 3y = 150
(30, 0)
x = 0 and y = 0
(0, 0)
x = 0 and x + 2y = 65
(0, 32.5)
5x + 3y = 150 and x + 2y = 65
(15, 25)
5x+3y=150
x+2y=65
Conclusion
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Widgets: The Solution
Example Continued. . .
Finally, plug each corner point into the objective function.
Corner Point
(30, 0)
(0, 0)
Value of 250x + 200y
250(30) + 200(0) =
250(0) + 200(0) =
$7500
$0
(0, 32.5)
250(0) + 200(32.5) =
$6500
(15, 25)
250(15) + 200(25) =
$8750
Maximum profit of $8750 when Ms. Li makes 150 right- and 250
left-handed widgets.
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Widgets: The Solution
Example Continued. . .
Finally, plug each corner point into the objective function.
Corner Point
(30, 0)
(0, 0)
Value of 250x + 200y
250(30) + 200(0) =
250(0) + 200(0) =
$7500
$0
(0, 32.5)
250(0) + 200(32.5) =
$6500
(15, 25)
250(15) + 200(25) =
$8750
Maximum profit of $8750 when Ms. Li makes 150 right- and 250
left-handed widgets.
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Widgets: The Solution
Example Continued. . .
Finally, plug each corner point into the objective function.
Corner Point
(30, 0)
(0, 0)
Value of 250x + 200y
250(30) + 200(0) =
250(0) + 200(0) =
$7500
$0
(0, 32.5)
250(0) + 200(32.5) =
$6500
(15, 25)
250(15) + 200(25) =
$8750
Maximum profit of $8750 when Ms. Li makes 150 right- and 250
left-handed widgets.
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Widgets: The Solution
Example Continued. . .
Finally, plug each corner point into the objective function.
Corner Point
(30, 0)
(0, 0)
Value of 250x + 200y
250(30) + 200(0) =
250(0) + 200(0) =
$7500
$0
(0, 32.5)
250(0) + 200(32.5) =
$6500
(15, 25)
250(15) + 200(25) =
$8750
Maximum profit of $8750 when Ms. Li makes 150 right- and 250
left-handed widgets.
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Widgets: The Solution
Example Continued. . .
Finally, plug each corner point into the objective function.
Corner Point
(30, 0)
(0, 0)
Value of 250x + 200y
250(30) + 200(0) =
250(0) + 200(0) =
$7500
$0
(0, 32.5)
250(0) + 200(32.5) =
$6500
(15, 25)
250(15) + 200(25) =
$8750
Maximum profit of $8750 when Ms. Li makes 150 right- and 250
left-handed widgets.
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Widgets: The Solution
Example Continued. . .
Finally, plug each corner point into the objective function.
Corner Point
(30, 0)
(0, 0)
Value of 250x + 200y
250(30) + 200(0) =
250(0) + 200(0) =
$7500
$0
(0, 32.5)
250(0) + 200(32.5) =
$6500
(15, 25)
250(15) + 200(25) =
$8750
Maximum profit of $8750 when Ms. Li makes 150 right- and 250
left-handed widgets.
Investment Allocation
Widget Production
Outline
1
Investment Allocation
2
Widget Production
3
Hotel Occupancy
4
Cruise Ships
5
Conclusion
Hotel Occupancy
Cruise Ships
Conclusion
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Hotels: Problem Set-up
Example
A company owns two hotels: the Luxar and the Grand. The Luxar has 10
single rooms and 12 double rooms on each of its 10 floors. The Grand
has 12 single rooms and 8 double rooms on each of its 15 floors. The
floors can be staffed individually. It costs $10,000 per floor to staff the
Luxar and $8,50 per floor to staff the Grand. If there is a demand for 100
single rooms and 80 double rooms, how many floors should be staffed in
each hotel to meet the demand while minimizing costs?
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Hotels: Problem Set-up
Example
A company owns two hotels: the Luxar and the Grand. The Luxar has 10
single rooms and 12 double rooms on each of its 10 floors. The Grand
has 12 single rooms and 8 double rooms on each of its 15 floors. The
floors can be staffed individually. It costs $10,000 per floor to staff the
Luxar and $8,50 per floor to staff the Grand. If there is a demand for 100
single rooms and 80 double rooms, how many floors should be staffed in
each hotel to meet the demand while minimizing costs?
Let:
x be # floors in Luxar
y be # floors in Grand
Objective:
Minimize 10000x + 8500y
Constraints:

10x +




12x +










12y
8y
x
y
x
y
≥
≥
≤
≤
≥
≥
100
80
10
15
0
0
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Hotels: Locating Corner Points
Example Continued. . .
The region obtained from the constraints is shown below.
Conclusion
12x+8y=80
Investment Allocation
Widget Production
y=15
x=10
10x+12y=100
Hotel Occupancy
Cruise Ships
Hotels: Locating Corner Points
Example Continued. . .
The region obtained from the constraints is shown below.
Conclusion
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Hotels: Locating Corner Points
Example Continued. . .
The region obtained from the constraints is shown below.
Intersecting Lines
Point
x = 0 and y = 15
(0, 15)
x = 10 and y = 15
(10, 15)
(10, 0)
x=10
12x + 8y = 80 and 10x + 12y = 100
“
10 , 25
4
4
12x + 8y = 80 and x = 0
x = 10 and y = 0
12x+8y=80
y=15
10x+12y=100
(0, 10)
”
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Hotels: Locating Corner Points
Example Continued. . .
The region obtained from the constraints is shown below.
Intersecting Lines
Point
x = 0 and y = 15
(0, 15)
x = 10 and y = 15
(10, 15)
(10, 0)
x=10
12x + 8y = 80 and 10x + 12y = 100
“
10 , 25
4
4
12x + 8y = 80 and x = 0
x = 10 and y = 0
12x+8y=80
y=15
10x+12y=100
(0, 10)
”
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Hotels: Locating Corner Points
Example Continued. . .
The region obtained from the constraints is shown below.
Intersecting Lines
Point
x = 0 and y = 15
(0, 15)
x = 10 and y = 15
(10, 15)
(10, 0)
x=10
12x + 8y = 80 and 10x + 12y = 100
“
10 , 25
4
4
12x + 8y = 80 and x = 0
x = 10 and y = 0
12x+8y=80
y=15
10x+12y=100
(0, 10)
”
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Hotels: Locating Corner Points
Example Continued. . .
The region obtained from the constraints is shown below.
Intersecting Lines
Point
x = 0 and y = 15
(0, 15)
x = 10 and y = 15
(10, 15)
(10, 0)
x=10
12x + 8y = 80 and 10x + 12y = 100
“
10 , 25
4
4
12x + 8y = 80 and x = 0
x = 10 and y = 0
12x+8y=80
y=15
10x+12y=100
(0, 10)
”
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Hotels: Locating Corner Points
Example Continued. . .
The region obtained from the constraints is shown below.
Intersecting Lines
Point
x = 0 and y = 15
(0, 15)
x = 10 and y = 15
(10, 15)
(10, 0)
x=10
12x + 8y = 80 and 10x + 12y = 100
“
10 , 25
4
4
12x + 8y = 80 and x = 0
x = 10 and y = 0
12x+8y=80
y=15
10x+12y=100
(0, 10)
”
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Hotels: Locating Corner Points
Example Continued. . .
The region obtained from the constraints is shown below.
Intersecting Lines
Point
x = 0 and y = 15
(0, 15)
x = 10 and y = 15
(10, 15)
(10, 0)
x=10
12x + 8y = 80 and 10x + 12y = 100
“
10 , 25
4
4
12x + 8y = 80 and x = 0
x = 10 and y = 0
12x+8y=80
y=15
10x+12y=100
(0, 10)
”
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Hotels: The Solution
Example Continued. . .
Finally, plug each corner point into the objective function.
Corner Point
Value of 10000x + 8500y
(0, 15)
10000(0) + 8500(15) =
$127,500
(10, 15)
10000(10) + 8500(15) =
$227,500
(10, 0)
10000(10) + 8500(0) =
$100,000
“
10 , 25
4
4
”
10000
“
10
4
”
+ 8500
“
25
4
”
=
$78,125
(0, 10)
10000(0) + 8500(10) =
$85,000
(3, 6)
10000(3) + 8500(6) =
$81,000
(2, 7)
10000(2) + 85000(7) =
$79,000
Minimum cost of $79,000 when 2 floors in the Luxar and 7 in the
Grand are opened.
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Hotels: The Solution
Example Continued. . .
Finally, plug each corner point into the objective function.
Corner Point
Value of 10000x + 8500y
(0, 15)
10000(0) + 8500(15) =
$127,500
(10, 15)
10000(10) + 8500(15) =
$227,500
(10, 0)
10000(10) + 8500(0) =
$100,000
“
10 , 25
4
4
”
10000
“
10
4
”
+ 8500
“
25
4
”
=
$78,125
(0, 10)
10000(0) + 8500(10) =
$85,000
(3, 6)
10000(3) + 8500(6) =
$81,000
(2, 7)
10000(2) + 85000(7) =
$79,000
Minimum cost of $79,000 when 2 floors in the Luxar and 7 in the
Grand are opened.
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Hotels: The Solution
Example Continued. . .
Finally, plug each corner point into the objective function.
Corner Point
Value of 10000x + 8500y
(0, 15)
10000(0) + 8500(15) =
$127,500
(10, 15)
10000(10) + 8500(15) =
$227,500
(10, 0)
10000(10) + 8500(0) =
$100,000
“
10 , 25
4
4
”
10000
“
10
4
”
+ 8500
“
25
4
”
=
$78,125
(0, 10)
10000(0) + 8500(10) =
$85,000
(3, 6)
10000(3) + 8500(6) =
$81,000
(2, 7)
10000(2) + 85000(7) =
$79,000
Minimum cost of $79,000 when 2 floors in the Luxar and 7 in the
Grand are opened.
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Hotels: The Solution
Example Continued. . .
Finally, plug each corner point into the objective function.
Corner Point
Value of 10000x + 8500y
(0, 15)
10000(0) + 8500(15) =
$127,500
(10, 15)
10000(10) + 8500(15) =
$227,500
(10, 0)
10000(10) + 8500(0) =
$100,000
“
10 , 25
4
4
”
10000
“
10
4
”
+ 8500
“
25
4
”
=
$78,125
(0, 10)
10000(0) + 8500(10) =
$85,000
(3, 6)
10000(3) + 8500(6) =
$81,000
(2, 7)
10000(2) + 85000(7) =
$79,000
Minimum cost of $79,000 when 2 floors in the Luxar and 7 in the
Grand are opened.
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Hotels: The Solution
Example Continued. . .
Finally, plug each corner point into the objective function.
Corner Point
Value of 10000x + 8500y
(0, 15)
10000(0) + 8500(15) =
$127,500
(10, 15)
10000(10) + 8500(15) =
$227,500
(10, 0)
10000(10) + 8500(0) =
$100,000
“
10 , 25
4
4
”
10000
“
10
4
”
+ 8500
“
25
4
”
=
$78,125
(0, 10)
10000(0) + 8500(10) =
$85,000
(3, 6)
10000(3) + 8500(6) =
$81,000
(2, 7)
10000(2) + 85000(7) =
$79,000
Minimum cost of $79,000 when 2 floors in the Luxar and 7 in the
Grand are opened.
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Hotels: The Solution
Example Continued. . .
Finally, plug each corner point into the objective function.
Corner Point
Value of 10000x + 8500y
(0, 15)
10000(0) + 8500(15) =
$127,500
(10, 15)
10000(10) + 8500(15) =
$227,500
(10, 0)
10000(10) + 8500(0) =
$100,000
“
10 , 25
4
4
”
10000
“
10
4
”
+ 8500
“
25
4
”
=
$78,125
(0, 10)
10000(0) + 8500(10) =
$85,000
(3, 6)
10000(3) + 8500(6) =
$81,000
(2, 7)
10000(2) + 85000(7) =
$79,000
Minimum cost of $79,000 when 2 floors in the Luxar and 7 in the
Grand are opened.
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Hotels: The Solution
Example Continued. . .
Finally, plug each corner point into the objective function.
Corner Point
Value of 10000x + 8500y
(0, 15)
10000(0) + 8500(15) =
$127,500
(10, 15)
10000(10) + 8500(15) =
$227,500
(10, 0)
10000(10) + 8500(0) =
$100,000
“
10 , 25
4
4
”
10000
“
10
4
”
+ 8500
“
25
4
”
=
$78,125
(0, 10)
10000(0) + 8500(10) =
$85,000
(3, 6)
10000(3) + 8500(6) =
$81,000
(2, 7)
10000(2) + 85000(7) =
$79,000
Minimum cost of $79,000 when 2 floors in the Luxar and 7 in the
Grand are opened.
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Hotels: The Solution
Example Continued. . .
Finally, plug each corner point into the objective function.
Corner Point
Value of 10000x + 8500y
(0, 15)
10000(0) + 8500(15) =
$127,500
(10, 15)
10000(10) + 8500(15) =
$227,500
(10, 0)
10000(10) + 8500(0) =
$100,000
“
10 , 25
4
4
”
10000
“
10
4
”
+ 8500
“
25
4
”
=
$78,125
(0, 10)
10000(0) + 8500(10) =
$85,000
(3, 6)
10000(3) + 8500(6) =
$81,000
(2, 7)
10000(2) + 85000(7) =
$79,000
Minimum cost of $79,000 when 2 floors in the Luxar and 7 in the
Grand are opened.
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Hotels: The Solution
Example Continued. . .
Finally, plug each corner point into the objective function.
Corner Point
Value of 10000x + 8500y
(0, 15)
10000(0) + 8500(15) =
$127,500
(10, 15)
10000(10) + 8500(15) =
$227,500
(10, 0)
10000(10) + 8500(0) =
$100,000
“
10 , 25
4
4
”
10000
“
10
4
”
+ 8500
“
25
4
”
=
$78,125
(0, 10)
10000(0) + 8500(10) =
$85,000
(3, 6)
10000(3) + 8500(6) =
$81,000
(2, 7)
10000(2) + 85000(7) =
$79,000
Minimum cost of $79,000 when 2 floors in the Luxar and 7 in the
Grand are opened.
Investment Allocation
Widget Production
Outline
1
Investment Allocation
2
Widget Production
3
Hotel Occupancy
4
Cruise Ships
5
Conclusion
Hotel Occupancy
Cruise Ships
Conclusion
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Cruise Ships: Problem Set-up
Example
Columbus Cruise Lines offers one-week cruises on three ships: the Nina,
Pinta, and Santa Maria. The Nina has 500 regular rooms and 200 deluxe
rooms. The Pinta has 400 regular and 400 deluxe rooms. The Santa
Maria has 800 regular and 500 deluxe rooms. The costs to run the ships
are $100,000 a week, $120,000 a week, and $180,000 a week respectively.
For how many weeks of the 12 week season should each ship be active to
meet the season’s demand for 12,000 regular rooms and 8,000 deluxe
rooms while minimizing costs?
Investment Allocation
Widget Production
Outline
1
Investment Allocation
2
Widget Production
3
Hotel Occupancy
4
Cruise Ships
5
Conclusion
Hotel Occupancy
Cruise Ships
Conclusion
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Important Concepts
Things to Remember from Section 3-3
1
Be sure to define variables.
2
Find the objective function and specify minimize or maximize.
3
Identify the constraints using a table, when possible.
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Important Concepts
Things to Remember from Section 3-3
1
Be sure to define variables.
2
Find the objective function and specify minimize or maximize.
3
Identify the constraints using a table, when possible.
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Important Concepts
Things to Remember from Section 3-3
1
Be sure to define variables.
2
Find the objective function and specify minimize or maximize.
3
Identify the constraints using a table, when possible.
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Conclusion
Important Concepts
Things to Remember from Section 3-3
1
Be sure to define variables.
2
Find the objective function and specify minimize or maximize.
3
Identify the constraints using a table, when possible.
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Next Time. . .
This is the last section of the course. Please see the review sheet
as you start to study for the final.
For next time
Review for the Final
Conclusion
Investment Allocation
Widget Production
Hotel Occupancy
Cruise Ships
Next Time. . .
This is the last section of the course. Please see the review sheet
as you start to study for the final.
For next time
Review for the Final
Conclusion