70) Cedar Point amusement park management is preparing the park’s annual promotional plan for the coming
season. Several advertising alternatives exist: newspaper, television, radio, and displays at recreational shows.
The information below shows the characteristics associated with each of the advertising alternatives, as well as
the maximum number of placements available in each medium. Given an advertising budget of $250,000, how
many placements should be made in each medium to maximize total audience exposure? Formulate this as a
linear programming problem.
Type
Cost
Maximum
number
Exposure
(1000s)
Newspaper
1500
100
80
Television
2200
50
120
Radio
750
50
45
Shows
150
3
10
71) A computer start–up named Pear is considering entering the U.S. market with what they believe to be a
smaller and faster computer than some of the existing products on the market. They want to get a feel for whether
or not customers would be willing to switch from some of the existing bigger brands to consider their product.
They want to collect a reasonable sample from across the U.S. representative of all age brackets. They have split
the U.S. into 2 regions: East and West. They want to at least 65% of their sample to cover the East and no fewer
than 25% of the West. They also have divided the age groups into 3 categories: 18–35, 36–69, and 70 and up. They
want at least 50% of their sample to be between 18–35 and at least 40% to be between 36–69. The costs per person
surveyed is given in the table below:
Region
18–35
36–69
70 and up
East
$2.50
$2.00
$1.50
West
$3.50
$3.00
$2.00
Assume that exactly 1,000 people are to be surveyed. The problem is for Pear Company to decide how many
people to survey from each age bracket within each region in order to minimize costs while meeting
requirements. Formulate this problem as a linear program.
72) A computer start–up named Pear is considering entering the U.S. market with what they believe to be a
smaller and faster computer than some of the existing products on the market. They want to get a feel for whether
or not customers would be willing to switch from some of the existing bigger brands to consider their product.
They want to collect a reasonable sample from across the U.S. representative of all age brackets. They have split
the U.S. into 2 regions: East and West. They want to at least 65% of their sample to cover the East and no fewer
than 25% of the West. They also have divided the age groups into 3 categories: 18–35, 36–69, and 70 and up. They
want at least 50% of their sample to be between 18–35 and at least 40% to be between 36–69. The costs per person
surveyed is given in the table below:
Region
18–35
36–69
70 and up
East
$2.50
$2.00
$1.50
West
$3.50
$3.00
$2.00
Assume that exactly 1,000 people are to be surveyed. How many people should Pear Company survey from each
age bracket within each region in order to minimize costs while meeting all requirements?
73) A small furniture manufacturer produces tables and chairs. Each product must go through three stages of the
manufacturing process: assembly, finishing, and inspection. Each table requires 3 hours of assembly, 2 hours of
finishing, and 1 hour of inspection. Each chair requires 2 hours of assembly, 2 hours of finishing, and 1 hour of
inspection. The profit per table is $120, while the profit per chair is $80. Currently, each week there are 200 hours
of assembly time available, 180 hours of finishing time, and 40 hours of inspection time. To keep a balance, the
number of chairs produced should be at least twice the number of tables. Also, the number of chairs cannot
exceed six times the number of tables. Formulate this as a linear programming problem. Carefully define all
decision variables.
74) A small furniture manufacturer produces tables and chairs. Each product must go through three stages of the
manufacturing process: assembly, finishing, and inspection. Each table requires 3 hours of assembly, 2 hours of
finishing, and 1 hour of inspection. Each chair requires 2 hours of assembly, 2 hours of finishing, and 1 hour of
inspection. The profit per table is $120, while the profit per chair is $80. Currently, each week there are 200 hours
of assembly time available, 180 hours of finishing time, and 40 hours of inspection time. To keep a balance, the
number of chairs produced should be at least twice the number of tables. Also, the number of chairs cannot
exceed six times the number of tables. How many tables and chairs should the furniture manufacturer produce
to maximize profit?
75) Swearingen and McDonald, a small furniture manufacturer, produces fine hardwood tables and chairs. Each
product must go through three stages of the manufacturing process: assembly, finishing, and inspection. Each
table requires 12 hours of assembly, 20 hours of finishing, and 2 hours of inspection. Each chair requires 4 hours
of assembly, 16 hours of finishing, and 3 hours of inspection. The profit per table is $150 while the profit per chair
is $100. Currently, each week there are 300 hours of assembly time available, 220 hours of finishing time, and 30
hours of inspection time. To keep a balance, the number of chairs produced should be at least twice the number
of tables. Also, the number of chairs cannot exceed 6 times the number of tables. Formulate this as a linear
programming problem. Carefully define all decision variables. Find the solution.
76) A manufacturer of microcomputers produces four models: Portable, Student, Office, and Network. The profit
per unit on each of these four models is $500, $350, $700, and $1000, respectively. The models require the labor
and materials per unit shown below.
Portable
Student
Office
Network
Total
Labor (hrs/week)
5
5
6
8
4000
Chassis (unit/week)
1
1
1
1
400
Disk Drive (unit/week)
2
1
2
1
300
Hard Disk (unit/week)
0
0
0
1
20
Memory Chip (unit/week)
16
8
32
64
22,000
Circuit Bds. (unit/week)
1
1
2
4
10,000
Formulate this product mix problem using linear programming.
77) Ivana Miracle wishes to invest her full inheritance of $300,000, and her goal is to minimize her risk subject to
an expected annual return of at least $30,000. She has decided to invest her money in any of three possible ways–
CDs, which pay a guaranteed 6 percent; stocks, which have an expected return of 15 percent; and a money market
mutual fund, which is expected to return 8 percent. Risk factors are 1.0 for the CDs, 3.6 for the stocks, and 1.8 for
the money market fund. Formulate this as a linear program.
78) A fast food restaurant uses full–time and part–time help to meet fluctuating demand during the day. The
following table presents projected need for workers at different times of the day:
Time
Workers needed
9:00–10:00
4
10:00–11:00
5
11:00–12:00
9
12:00–1:00
10
1:00–2:00
8
2:00–3:00
4
3:00–4:00
3
4:00–5:00
6
There is a maximum of four full–time workers and the other workers are part–time workers. Each full–time
worker is there from 9:00 until 5:00, while the part–time workers will work for 4 consecutive hours at a cost of
$4.00 per hour. The cost of the full–time worker is $50 per day. The company wishes to minimize total cost while
meeting the demands. Formulate this as a linear programming problem. Carefully define all decision variables.
79) First Securities, Inc., an investment firm, has $380,000 on account. The chief investment officer would like to
reinvest the $380,000 in a portfolio that would maximize return on investment while at the same time maintaining
a relatively conservative mix of stocks and bonds. The following table shows the investment opportunities and
rates of return.
Investment Opportunity
Rate of Return
Municipal Bonds
0.095
High Tech Stock
0.146
Blue Chip Stock
0.075
Federal Bonds
0.070
The Board of Directors has mandated that at least 60 percent of the investment consist of a combination of
municipal and federal bonds, 25 percent Blue Chip Stock, and no more than 15 percent High Tech Stock.
Formulate this portfolio selection problem using linear programming.
80) Dr. Malcomb Heizer wishes to invest his retirement fund of $2,000,000 so that his return on investment is
maximized, but he also wishes to keep the risk level relatively low. He has decided to invest his money in any of
three possible ways: CDs that pay a guaranteed 4 percent; stocks that have an expected return of 14 percent; and a
money market mutual fund that is expected to return 18 percent. He has decided that the total $2,000,000 will be
invested, but any part (or all) of it may be put in any of the three alternatives. Thus, he may have some money
invested in all three alternatives. He has also decided to invest, at most, 30 percent of this in stocks and at least 20
percent of this in money market funds. Formulate this as a linear programming problem and carefully define all
the decision variables.
81) Friendly Manufacturing has three factories (1, 2, and 3) and three warehouses (A, B, and C). The table below
shows the shipping costs between each factory (in dollars) and warehouse, the factory manufacturing capabilities
(in 1000s), and the warehouse capacities (in 1000s). Write the objective function and the constraint inequalities.
Let X1A = 1000s of units shipped from factory 1 to warehouse A, etc.
82) The following table provides shipping costs from each of two regional warehouses to each of three
destinations. The supplies available and the demands are also given in the table.
Formulate this as a linear programming problem. Carefully define all decision variables.
83) Green Grass, Inc. just ran out of stock and suddenly has two emergency orders for grass seed blends: one is
for 1500 pounds of normal, the other for 2300 pounds of special. At least each pound of normal should contain 60
percent annual seed, while each pound of special should contain at least 70 percent perennial seed. Green Grass
has two input mixtures, A and B. Mixture A contains 80 percent perennial and 15 percent annual seed. Mixture B
contains 70 percent annual and 25 percent perennial seed. Mixture A costs 90 cents per pound and mixture B
costs 50 cents per pound. Set up the constraints and the objective function to solve this blending problem.
84) Three types of gasoline are manufactured by a company — Regular, Super, and Extra. Regular should have at
least 11 percent additive 1 and 17 percent additive 2. Super should have at least 13 percent additive 1 and 22
percent additive 2. Extra should have at least 17 percent additive 1 and 19 percent additive 2. These are made by
using two crudes — A and B. Crude A cost $28 per barrel and is 14 percent additive 1 and 18 percent additive 2.
Crude B costs $30 per barrel and is 20 percent additive 1 and 24 percent additive 2. The demand for Regular is
projected to be 1,000 barrels, while each of the others has a demand of 2,000 barrels. Formulate this as a linear
programming problem to minimize cost while meeting all constraints. Carefully define all decision variables.
85) Three types of fertilizer are manufactured by a company: Regular, Supergro, and Jungle Feeder. Regular
should have at least 10 percent nitrogen and 16 percent phosphorous. Supergro should have at least 12 percent
nitrogen and 20 percent phosphorous, and Jungle Feeder should have at least 15 percent nitrogen and 18 percent
phosphorous. These are made by using two components: A and B. Component A costs $0.30 per pound and is 14
percent nitrogen and 18 percent phosphorous. Component B costs $0.50 per pound and is 20 percent nitrogen
and 24 percent phosphorous. The demand for Regular is projected to be 1,000 pounds, while each of the others
has a demand of 2,000 pounds. Formulate the appropriate linear program.
86) A cruise line is planning its menu for the next trip. Vacationers like eating steak, lobster, and chicken. The
cruise line has decided to plan for at least half of all booked passengers to have a steak dinner, for at least a
quarter of all passengers to have lobster, and the rest to have chicken. Steak dinners cost the company $8, lobsters
cost $15, and chicken costs the line $4. On the next cruise, there are 400 passengers booked. In addition, the
cruise line has decided to plan for an additional 25% more meals than bookings. Formulate the appropriate
linear program.
87) A mail order firm, AmazingCo, can use one of 3 shipping couriers. Ajax Shipping charges $6 per pound and
delivers in 2 days. Bilco Lanes charges $9 per pound but guarantees next day delivery. The final courier, Hobo
Ltd., charges only $3 per pound but takes 4 days to deliver. AmazingCo has a quarterly budget of $280,000 and a
reputation for timely delivery. Formulate the linear program so that the firm delivers as fast as possible within
budget for 70,000 pounds of shipments per quarter.
88) Media selection problems can be approached with LP from which two perspectives?
89) Describe the marketing research linear programming application.
90) Describe the production mix linear programming application.
91) Describe the production scheduling linear programming application.
92) Describe the portfolio selection linear programming application.
93) Describe the truck loading linear programming application.
94) Describe the shipping problem linear programming application.
95) Describe a diet problem linear programming application.
96) Describe the feed mix linear programming application.