Chapter 4 – Linear Programming Applications in Marketing, Finance, and Operations Management
Production scheduling
45. 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.
R = the number of Wildlife Raft Trips to schedule
S = the number of Group Sailing Lessons to schedule
R + S ≤ 12
R + 2S ≤ 18
R ≤ 10
R , S ≥ 0
46. 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
Chapter 4 – Linear Programming Applications in Marketing, Finance, and Operations Management
four kinds of rooms to plan for, the Evans family will consider the following information.
After adjusting for expected occupancy, the average nightly revenue for a beachfront non-
smoking room is $175. The average nightly revenue for a lagoon view non-smoking room is
$130. Smokers will be charged an extra $15.
Construction costs vary. The cost estimate for a lagoon view room is $12,000 and for a
beachfront room is $15,000. Air purifying systems and additional smoke detectors and
sprinklers ad $3000 to the cost of any smoking room. Evans Enterprises has raised $6.3
million in construction guarantees for this portion of the building.
There will be at least 100 but no more than 180 beachfront rooms.
Design considerations require that the number of lagoon view rooms be at least 1.5 times the
number of beachfront rooms, and no more than 2.5 times that number.
Industry trends recommend that the number of smoking rooms be no more than 50% of the
number of non-smoking rooms.
Develop the linear programming model to maximize revenue.
175BN + 190BS + 130LN + 145LS
15000BN + 18000BS + 12000LN + 15000LS ≤ 6,300,000
Product mix
47. Super City Discount Department Store is open 24 hours a day. The number of cashiers need in each four hour period
of a day is listed below.
Period
Cashiers Needed
10 p.m. to 2 a.m.
8
2 a.m. to 6 a.m.
4
6 a.m. to 10 a.m.
7
10 a.m. to 2 p.m.
12
2 p.m. to 6 p.m.
10
6 p.m. to 10 p.m.
15
If cashiers work for eight consecutive hours, how many should be scheduled to begin working in each period in order to
minimize the number of cashiers needed?
TNP = the number of cashiers who begin working at 10 p.m.
SXA = the number of cashiers who begin working at 6 a.m.
TNA = the number of cashiers who begin working at 10 a.m.
SXP = the number of cashiers who begin working at 6 p.m.
Chapter 4 – Linear Programming Applications in Marketing, Finance, and Operations Management
48. Winslow Savings has $20 million available for investment. It wishes to invest over the next four months in such a way
that it will maximize the total interest earned over the four month period as well as have at least $10 million available at
the start of the fifth month for a high rise building venture in which it will be participating.
For the time being, Winslow wishes to invest only in 2-month government bonds (earning 2% over the 2-month period)
and 3-month construction loans (earning 6% over the 3-month period). Each of these is available each month for
investment. Funds not invested in these two investments are liquid and earn 3/4 of 1% per month when invested locally.
Formulate and solve a linear program that will help Winslow Savings determine how to invest over the next four months
if at no time does it wish to have more than $8 million in either government bonds or construction loans.
Chapter 4 – Linear Programming Applications in Marketing, Finance, and Operations Management
49. National Wing Company (NWC) is gearing up for the new B-48 contract. Currently NWC has 100 equally qualified
workers. Over the next three months NWC has made the following commitments for wing production:
Month
Wing Production
May
20
June
24
July
30
Each worker can either be placed in production or can train new recruits. A new recruit can be trained to be an apprentice
in one month. The next month, he, himself, becomes a qualified worker (after two months from the start of training). Each
trainer can train two recruits. The production rate and salary per employee is estimated below.
Employee
Production Rate (Wings/Month)
Salary Per Month
Production
.6
$3,000
Trainer
.3
3,300
Apprentice
.4
2,600
Recruit
.05
2,200
At the end of July, NWC wishes to have no recruits or apprentices but have at least 140 full-time workers. Formulate and
solve a linear program for NWC to accomplish this at minimum total cost.
Chapter 4 – Linear Programming Applications in Marketing, Finance, and Operations Management
50. The SMM Company, which is manufacturing a new instant salad machine, has $280,000 to spend on advertising. The
product is only to be test marketed initially in the Dallas area. The money is to be spent on an advertising blitz during one
weekend (Friday, Saturday, and Sunday) in January, and SMM is limited to television advertising.
The company has three options available: daytime advertising, evening news advertising and the Super Bowl. Even
though the Super Bowl is a national telecast, the Dallas Cowboys will be playing in it, and hence, the viewing audience
will be especially large in the Dallas area. A mixture of one-minute TV spots is desired. The table below gives pertinent
data:
Estimated New Audience
Cost Per Ad
Reached With Each Ad
Daytime
$ 5,000
3,000
Evening News
$ 7,000
4,000
Super Bowl
$100,000
75,000
SMM has decided to take out at least one ad in each option. Further, there are only two Super Bowl ad spots available.
There are 10 daytime spots and 6 evening news spots available daily. SMM wants to have at least 5 ads per day, but spend
no more than $50,000 on Friday and no more than $75,000 on Saturday. Formulate and solve a linear program to help
SMM decide how the company should advertise over the weekend.
Total cost = $1,098,000.
1
Production scheduling
Chapter 4 – Linear Programming Applications in Marketing, Finance, and Operations Management
51. John Sweeney is an investment advisor who is attempting to construct an “optimal portfolio” for a client who has
$400,000 cash to invest. There are ten different investments, falling into four broad categories that John and his client
have identified as potential candidates for this portfolio.
The following table lists the investments and their important characteristics. Note that Unidyde Equities (stocks) and
Unidyde Debt (bonds) are two separate investments, whereas First General REIT is a single investment that is considered
both an equities and a real estate investment.
Exp. Annual
Liquidity
Risk
Category
Investment
After Tax Return
Factor
Factor
Equities
Unidyne Corp.
15.0%
100
60
Col. Mustard Restaurant
17.0%
100
70
First General REIT
17.5%
100
75
Debt
Metropolitan Electric
11.8%
95
20
Unidyne Corp.
12.2%
92
30
Lemonville Transit
12.0%
79
22
Real Estate
Fairview Apartment Partnership
22.0%
0
50
First General REIT
(see above)
(see above)
(see above)
Money
T-Bill Account
9.6%
80
0
Chapter 4 – Linear Programming Applications in Marketing, Finance, and Operations Management
Money Market Fund
10.5%
100
10
All Saver’s Certificate
12.6%
0
0
Formulate and solve a linear program to accomplish John’s objective as an investment advisor which is to construct a
portfolio that maximizes his client’s total expected after-tax return over the next year, subject to a number of constraints
placed upon him by the client for the portfolio:
Its (weighted) average liquidity factor must be at least 65.
The (weighted) average risk factor must be no greater than 55.
At most, $60,000 is to be invested in Unidyde stocks or bonds.
No more than 40% of the investment can be in any one category except the money category.
No more than 20% of the investment can be in any one investment except the money market
fund.
At least $1,000 must be invested in the money market fund.
The maximum investment in All Saver’s Certificates is $15,000.
The minimum investment desired for debt is $90,000.
At least $10,000 must be placed in a T-Bill account.
Chapter 4 – Linear Programming Applications in Marketing, Finance, and Operations Management
52. BP Cola must decide how much money to allocate for new soda and traditional soda advertising over the coming year.
The advertising budget is $10,000,000. Because BP wants to push its new sodas, at least one-half of the advertising
budget is to be devoted to new soda advertising. However, at least $2,000,000 is to be spent on its traditional sodas. BP
estimates that each dollar spent on traditional sodas will translate into 100 cans sold, whereas, because of the harder sell
needed for new products, each dollar spent on new sodas will translate into 50 cans sold.
To attract new customers BP has lowered its profit margin on new sodas to 2 cents per can as compared to 4 cents per can
for traditional sodas. How should BP allocate its advertising budget if it wants to maximize its profits while selling at least
750 million cans?
53. Wes Wheeler is the production manager of Wheeler Wheels, Inc. Wes has just received orders for 1,000 standard
wheels and 1,250 deluxe wheels next month and for 800 standard and 1,500 deluxe wheels the following month. All
orders are to be filled.
The cost of producing standard wheels is $10 and deluxe wheels is $16. Overtime rates are 50% higher. There are 1,000
hours of regular time and 500 hours of overtime available each month. It takes .5 hour to make a standard wheel and .6
hour to make a deluxe wheel. The cost of storing one wheel from one month to the next is $2.
Wes wants to develop a two-month production schedule for standard and deluxe wheels. Formulate this production
planning problem as a linear program.
Chapter 4 – Linear Programming Applications in Marketing, Finance, and Operations Management
54. Comfort Plus Inc. (CPI) manufactures a standard dining chair used in restaurants. The demand forecasts for quarter 1
(January-March) and quarter 2 (April-June) are 3700 chairs and 4200 chairs, respectively. CPI has a policy of satisfying
all demand in the quarter in which it occurs.
The chair contains an upholstered seat that can be produced by CPI or purchased from DAP, a subcontractor. DAP
currently charges $12.50 per seat, but has announced a new price of $13.75 effective April 1. CPI can produce the seat at
a cost of $10.25. CPI can produce up to 3800 seats per quarter.
Seats that are produced or purchased in quarter 1 and used to satisfy demand in quarter 2 cost CPI $1.50 each to hold in
inventory, but maximum inventory cannot exceed 300 seats.
Formulate this problem as a linear programming problem.
Chapter 4 – Linear Programming Applications in Marketing, Finance, and Operations Management
55. Target Shirt Company makes three varieties of shirts: Collegiate, Traditional and European. These shirts are made
from different combinations of cotton and polyester.
The cost per yard of unblended cotton is $5 and for unblended polyester is $4. Target can receive up to 4,000 yards of
raw cotton and 3,000 yards of raw polyester fabric weekly. The table below pertinent data concerning the manufacture of
the shirts.
Shirt
Total
Yards
Fabric
Requirement
Weekly
Contracts
Weekly
Demand
Selling
Price
Collegiate
1.00
At least
50% cotton
500
600
$14.00
Traditional
1.20
No more than
20% polyester
650
850
$15.00
European
.90
As much as
80% polyester
280
675
$18.00
Formulate and solve this blending problem as a linear program.
Chapter 4 – Linear Programming Applications in Marketing, Finance, and Operations Management
Essay
56. Discuss the need for the use of judgment or other subjective methods in mathematical modeling.
57. What benefits exist in using linear programming for production scheduling problems?
Chapter 4 – Linear Programming Applications in Marketing, Finance, and Operations Management
58. Describe some common feature of multiperiod financial planning models.
59. Why should decision makers who are primarily concerned with marketing or finance or production know about linear
programming?
60. Discuss several resource allocation problems that can be modeled by varying the workforce assignment model.
61. Discuss at least three additional considerations that might be included in media selection LP models more complex
than the one covered in the text.
62. Describe three situations where variations of the workforce assignment model could be applied.
63. Interpret the dual price for any two of the following constraints: 1) the available funds in a portfolio selection model,
2) the cash requirement due at the beginning of a year in a financial planning model, and 3) the manufacturing
capacity in a make-or–buy decision model.