Chapter 9 – LP Applications
True / False
1. Media selection problems can maximize exposure quality and use number of customers reached as a constraint, or
maximize the number of customers reached and use exposure quality as a constraint.
a. True
b. False
2. Revenue management methodology was originally developed for the banking industry.
a. True
b. False
3. Portfolio selection problems should acknowledge both risk and return.
a. True
b. False
4. If an LP problem is not correctly formulated, the computer software will indicate it is infeasible when trying to solve it.
a. True
b. False
5. It is improper to combine manufacturing costs and overtime costs in the same objective function.
a. True
b. False
6. Production constraints frequently take the form:
beginning inventory + sales − production = ending inventory
a. True
b. False
7. If a real-world problem is correctly formulated, it is not possible to have alternative optimal solutions.
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a. True
b. False
8. To properly interpret dual prices, one must know how costs were allocated in the objective function.
a. True
b. False
9. A company makes two products from steel; one requires 2 tons of steel and the other requires 3 tons. There are 100 tons
of steel available daily. A constraint on daily production could be written as: 2x1 + 3x2 ≤ 100.
a. True
b. False
10. Double-subscript notation for decision variables should be avoided unless the number of decision variables exceeds
nine.
a. True
b. False
11. Using minutes as the unit of measurement on the left-hand side of a constraint and using hours on the right-hand side
is acceptable since both are a measure of time.
a. True
b. False
12. Compared to the problems in the textbook, real-world problems generally require more variables and constraints.
a. True
b. False
13. For the multiperiod production scheduling problem in the textbook, period n − 1’s ending inventory variable was also
used as period n‘s beginning inventory variable.
a. True
b. False
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14. A company makes two products, A and B. A sells for $100 and B sells for $90. The variable production costs are $30
per unit for A and $25 for B. The company’s objective could be written as: MAX 190x1 − 55x2.
a. True
b. False
15. The primary limitation of linear programming’s applicability is the requirement that all decision variables be
nonnegative.
a. True
b. False
16. A decision maker would be wise to not deviate from the optimal solution found by an LP model because it is the best
solution.
a. True
b. False
17. Data collection for large-scale LP models can be more time-consuming than either the formulation of the model or the
development of the computer solution.
a. True
b. False
18. The media selection model presented in the textbook involves maximizing the number of potential customers reached
subject to a minimum total exposure quality rating.
a. True
b. False
19. The marketing research model presented in the textbook involves minimizing total interview cost subject to interview
quota guidelines.
a. True
b. False
Chapter 9 – LP Applications
20. A marketing research firm must determine how many daytime interviews (D) and evening interviews (E) to conduct.
At least 40% of the interviews must be in the evening. A correct modeling of this constraint is: -0.4D + 0.6E > 0.
a. True
b. False
Multiple Choice
21. Media selection problems usually determine
a. how many times to use each media source.
b. the coverage provided by each media source.
c. the cost of each advertising exposure.
d. the relative value of each medium.
22. To study consumer characteristics, attitudes, and preferences, a company would engage in
a. client satisfaction processing.
b. marketing research.
c. capital budgeting.
d. production planning.
23. A marketing research application uses the variable HD to represent the number of homeowners interviewed during the
day. The objective function minimizes the cost of interviewing this and other categories and there is a constraint that HD
≥ 100. The solution indicates that interviewing another homeowner during the day will increase costs by 10.00. What do
you know?
a. the objective function coefficient of HD is 10.
b. the dual price for the HD constraint is 10.
c. the objective function coefficient of HD is −10.
d. the dual price for the HD constraint is −10.
24. The dual price for a constraint that compares funds used with funds available is .058. This means that
a. the cost of additional funds is 5.8%.
b. if more funds can be obtained at a rate of 5.5%, some should be.
c. no more funds are needed.
d. the objective was to minimize.
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25. Let M be the number of units to make and B be the number of units to buy. If it costs $2 to make a unit and $3 to buy
a unit and 4000 units are needed, the objective function is
a. Max 2M + 3B
b. Min 4000 (M + B)
c. Max 8000M + 12000B
d. Min 2M + 3B
26. If Pij = the production of product i in period j, then to indicate that the limit on production of the company’s three
products in period 2 is 400,
a. P21 + P22 + P23 ≤ 400
b. P12 + P22 + P32 ≤ 400
c. P32 ≤ 400
d. P23 ≤ 400
27. Let Pij = the production of product i in period j. To specify that production of product 1 in period 3 and in period 4
differs by no more than 100 units,
a. P13 − P14 ≤ 100; P14 − P13 ≤ 100
b. P13 − P14 ≤ 100; P13 − P14 ≥ 100
c. P13 − P14 ≤ 100; P14 − P13 ≥ 100
d. P13 − P14 ≥ 100; P14 − P13 ≥ 100
28. Let A, B, and C be the amounts invested in companies A, B, and C. If no more than 50% of the total investment can
be in company B, then
a. B ≤ 5
b. A − .5B + C ≤ 0
c. .5A − B − .5C ≤ 0
d. −.5A + .5B − .5C ≤ 0
29. Department 3 has 2500 hours. Transfers are allowed to departments 2 and 4, and from departments 1 and 2. If Ai
measures the labor hours allocated to department i and Tij the hours transferred from department i to department j, then
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a. T13 + T23 − T32 − T34 − A3 = 2500
b. T31 + T32 − T23 − T43 + A3 = 2500
c. A3 + T13 + T23 − T32 − T34 = 2500
d. A3 − T13 − T23 + T32 + T34 = 2500
30. Modern revenue management systems maximize revenue potential for an organization by helping to manage
a. pricing strategies.
b. reservation policies.
c. short-term supply decisions.
d. All of the alternatives are correct.
31. Blending problems arise whenever a manager must decide how to
a. mix several different asset types in one investment strategy.
b. mix two or more resources to produce one or more products.
c. combine the results of two or more research studies into one.
d. allocate workers with different skill levels to various work shifts.
32. In a production scheduling LP, the demand requirement constraint for a time period takes the form
a. beginning inventory + production + ending inventory > demand.
b. beginning inventory − production + ending inventory = demand.
c. beginning inventory + production − ending inventory = demand.
d. beginning inventory − production − ending inventory > demand.
33. A mutual fund manager must decide how much money to invest in Atlantic Oil (A) and how much to invest in Pacific
Oil (P). At least 60% of the money invested in the two oil companies must be in Pacific Oil. A correct modeling of this
constraint is
a. 0.4A + 0.6P > 0.
b. -0.4A + 0.6P > 0.
c. 0.6A + 0.4P > 0.
d. -0.6A + 0.4P > 0.
34. The production scheduling problem modeled in the textbook involves capacity constraints on all of the following
types of resources except
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a. material.
b. labor.
c. machine.
d. storage.
35. Which of the operations management applications modeled in the text has an objective function that minimizes the
sum of manufacturing costs, purchasing costs, and overtime costs?
a. make-or-buy decision
b. blending
c. production scheduling
d. workforce assignment
36. A mutual fund manager is deciding how much money to invest in each of three companies: Atlantic Oil (A), Pacific
Oil (P), and Gulf Oil (G). The money invested in Gulf Oil must equal at least 30% of the money invested in the other two
companies. A correct modeling of this constraint is
a. 0.7A + 0.7P + .3G > 0
b. -0.3A – 0.3P + G > 0
c. 0.7A + 0.7P – G > 0
Subjective Short Answer
37. A&C Distributors is a company that represents many outdoor products companies and schedules deliveries to discount
stores, garden centers, and hardware stores. Currently, scheduling needs to be done for two lawn sprinklers, the Water
Wave and Spring Shower models. Requirements for shipment to a warehouse for a national chain of garden centers are
shown below.
Month Shipping
Capacity Product Minimum
Requirement Unit Cost
to Ship Per Unit
Inventory Cost
March 8000 Water Wave 3000 .30 .06
Spring Shower 1800 .25 .05
April 7000 Water Wave 4000 .40 .09
Spring Shower 4000 .30 .06
May 6000 Water Wave 5000 .50 .12
Spring Shower 2000 .35 .07
Let Sij be the number of units of sprinkler i shipped in month j, where i = 1 or 2, and j = 1, 2, or 3. Let Wij be the number
of sprinklers that are at the warehouse at the end of a month, in excess of the minimum requirement.
a Write the portion of the objective function that minimizes shipping costs.
b. An inventory cost is assessed against this ending inventory. Give the portion of the objective function that represents
inventory cost.
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c. There will be three constraints that guarantee, for each month, that the total number of sprinklers shipped will not
exceed the shipping capacity. Write these three constraints.
d. There are six constraints that work with inventory and the number of units shipped, making sure that enough
sprinklers are shipped to meet the minimum requirements. Write these six constraints.
38. An ad campaign for a new snack chip will be conducted in a limited geographical area and can use TV time, radio
time, and newspaper ads. Information about each medium is shown below.
Medium Cost Per Ad # Reached Exposure Quality
TV 500 10000 30
Radio 200 3000 40
Newspaper 400 5000 25
If the number of TV ads cannot exceed the number of radio ads by more than 4, and if the advertising budget is $10000,
develop the model that will maximize the number reached and achieve an exposure quality if at least 1000.
39. Information on a prospective investment for Wells Financial Services is given below.
Period
1 2 3 4
Loan Funds Available 3000 7000 4000 5000
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Investment Income
(% of previous period’s investment) 110% 112% 113%
Maximum Investment 4500 8000 6000 7500
Payroll Payment 100 120 150 100
In each period, funds available for investment come from two sources: loan funds and income from the previous period’s
investment. Expenses, or cash outflows, in each period must include repayment of the previous period’s loan plus 8.5%
interest, and the current payroll payment. In addition, to end the planning horizon, investment income from period 4 (at
110% of the investment) must be sufficient to cover the loan plus interest from period 4. The difference in these two
quantities represents net income, and is to be maximized. How much should be borrowed and how much should be
40. Tots Toys makes a plastic tricycle that is composed of three major components: a handlebar-front wheel-pedal
assembly, a seat and frame unit, and rear wheels. The company has orders for 12,000 of these tricycles. Current schedules
yield the following information.
Requirements Cost to Cost to
Component Plastic Time Space Manufacture Purchase
Front 3 10 2 8 12
Seat/Frame 4 6 2 6 9
Rear wheel (each) .5 2 .1 1 3
Available 50000 160000 30000
The company obviously does not have the resources available to manufacture everything needed for the completion of
12000 tricycles so has gathered purchase information for each component. Develop a linear programming model to tell
the company how many of each component should be manufactured and how many should be purchased in order to
provide 12000 fully completed tricycles at the minimum cost.
Chapter 9 – LP Applications
41. The Tots Toys Company is trying to schedule production of two very popular toys for the next three months: a rocking
horse and a scooter. Information about both toys is given below.
Begin. Invty. Required Required Production Production
Toy June 1 Plastic Time Cost Cost
Rocking Horse 25 5 2 12 1
Scooter 55 4 3 14 1.2
Plastic Time Monthly Demand Monthly Demand
Summer Schedule Available Available Horse Scooter
June 3500 2100 220 450
July 5000 3000 350 700
August 4800 2500 600 520
Develop a model that would tell the company how many of each toy to produce during each month. You are to minimize
total cost. Inventory cost will be levied on any items in inventory on June 30, July 31, or August 31 after demand for the
month has been satisfied. Your model should make use of the relationship
Beginning Inventory + Production − Demand = Ending Inventory
for each month. The company wants to end the summer with 150 rocking horses and 60 scooters as beginning inventory
for Sept. 1. Don’t forget to define your decision variables.
Chapter 9 – LP Applications
42. Larkin Industries manufactures several lines of decorative and functional metal items. The most recent order has been
for 1200 door lock units for an apartment complex developer. The sales and production departments must work together
to determine delivery schedules. Each lock unit consists of three components: the knob and face plate, the actual lock
itself, and a set of two keys. Although the processes used in the manufacture of the three components vary, there are three
areas where the production manager is concerned about the availability of resources. These three areas, their usage by the
three components, and their availability are detailed in the table.
Resource Knob and Plate Lock Key (each) Available
Brass Alloy 12 5 1 15000 units
Machining 18 20 10 36000 minutes
Finishing 15 5 1 12000 minutes
A quick look at the amounts available confirms that Larkin does not have the resources to fill this contract. A
subcontractor, who can make an unlimited number of each of the three components, quotes the prices below.
Component Subcontractor Cost Larkin Cost
Knob and Plate 10.00 6.00
Lock 9.00 4.00
Keys (set of 2) 1.00 .50
Develop a linear programming model that would tell Larkin how to fill the order for 1200 lock sets at the minimum cost.
Chapter 9 – LP Applications
43. G and P Manufacturing would like to minimize the labor cost of producing dishwasher motors for a major appliance
manufacturer. Although two models of motors exist, the finished models are indistinguishable from one another; their cost
difference is due to a different production sequence. The time in hours required for each model in each production area is
tabled here, along with the labor cost.
Model 1 Model 2
Area A 15 3
Area B 4 10
Area C 4 8
Cost 80 65
Currently labor assignments provide for 10,000 hours in each of Areas A and B and 18000 hours in Area C. If 2000 hours
are available to be transferred from area B to Area A, 3000 hours are available to be transferred from area C to either
Areas A or B, develop the linear programming model whose solution would tell G&P how many of each model to
produce and how to allocate the workforce.
Chapter 9 – LP Applications
44. FarmFresh Foods manufactures a snack mix called TrailTime by blending three ingredients: a dried fruit mixture, a
nut mixture, and a cereal mixture. Information about the three ingredients (per ounce) is shown below.
Ingredient Cost Volume Fat Grams Calories
Dried Fruit .35 1/4 cup 0 150
Nut Mix .50 3/8 cup 10 400
Cereal Mix .20 1 cup 1 50
The company needs to develop a linear programming model whose solution would tell them how many ounces of each
mix to put into the TrailTime blend. TrailTime is packaged in boxes that will hold between three and four cups. The blend
should contain no more than 1000 calories and no more than 25 grams of fat. Dried fruit must be at least 20% of the
volume of the mixture, and nuts must be no more than 15% of the weight of the mixture. Develop a model that meets
these restrictions and minimizes the cost of the blend.
45. The Meredith Ribbon Company produces paper and fabric decorative ribbon which it sells to paper products
companies and craft stores. The demand for ribbon is seasonal. Information about projected demand and production for a
particular type of ribbon is given.
Demand (yards) Production Cost Per Yard Production Capacity (yards)
Quarter 1 10,000 .03 30,000
Quarter 2 18,000 .04 20,000
Quarter 3 16,000 .06 20,000
Quarter 4 30,000 .08 15,000
An inventory holding cost of $.005 is levied on every yard of ribbon carried over from one quarter to the next.
a. Define the decision variables needed to model this problem.
b. The objective is to minimize total cost, the sum of production and inventory holding cost. Give the objection
function.
c. Write the production capacity constraints.
d. Write the constraints that balance inventory, production, and demand for each quarter. Assume there is no beginning
inventory in quarter 1.
e. To attempt to balance the production and avoid large changes in the workforce, production in period 1 must be within
5000 yards of production in period 2. Write this constraint.
Chapter 9 – LP Applications
46. Island Water Sports is a business that provides rental equipment and instruction for a variety of water sports in a resort
town. On one particular morning, a decision must be made of how many Wildlife Raft Trips and how many Group Sailing
Lessons should be scheduled. Each Wildlife Raft Trip requires one captain and one crew person, and can accommodate
six passengers. The revenue per raft trip is $120. Ten rafts are available, and at least 30 people are on the list for
reservations this morning. Each Group Sailing Lesson requires one captain and two crew people for instruction. Two
boats are needed for each group. Four students form each group. There are 12 sailboats available, and at least 20 people
are on the list for sailing instruction this morning. The revenue per group sailing lesson is $160. The company has 12
captains and 18 crew available this morning. The company would like to maximize the number of customers served while
generating at least $1800 in revenue and honoring all reservations.
47. Evans Enterprises has bought a prime parcel of beachfront property and plans to build a luxury hotel. After meeting
with the architectural team, the Evans family has drawn up some information to make preliminary plans for construction.
Excluding the suites, which are not part of this decision, the hotel will have four kinds of rooms: beachfront non-smoking,
beachfront smoking, lagoon view non-smoking, and lagoon view smoking. In order to decide how many of each of the
four kinds of rooms to plan for, the Evans family will consider the following information.