Chapter 4
Linear Programming Applications
in Marketing, Finance, and Operations
Marketing Applications: Media selection
Financial Applications: Portfolio selection
Operations Management Applications
Slide 1
Marketing Applications
Media Selection
• One application of linear programming in
marketing is media selection.
• LP can be used to help marketing managers allocate
a fixed budget to various advertising media.
• The objective is to maximize reach, frequency, and
quality of exposure.
• Restrictions on the allowable allocation usually
arise during consideration of company policy,
contract requirements, and media availability.
Slide 2
Marketing Applications: Media Selection
The Camden County Chamber of Commerce (CCCC) would like to
promote the region to area businesses. Advertising alternatives include
television, radio and internet. Estimates of audience reached and costs per
advertisement and maximum media usage limitations are shown below:
Television
Radio
Internet
Audience/adv.
100,000
18,000
40,000
Cost/adv.
$2,000
$300
$600
Maximum usage
10
20
10
To ensure a balance use of advertisement media, television advertisements
must not exceed 50% of the total number authorized. In addition, internet
ads should account for at least 10% of the total advertisements used.
CCCC has a total budget of $18,000. Determine the number of each type of
media to use such that the number of audience reached is maximized.
If CCCC had an additional $100, what is the net increase of audience
reached. Would the allocation of the media stay the same?
Slide 3
Media Selection
Define the Decision Variables
TV = number of television advertisements to use
RA = number of radio advertisements to use
IN = number of internet advertisements to use
Define the Objective Function
Slide 4
Media Selection
Define the Constraints
Slide 5
Financial Applications
LP can be used in financial decision-making that
involves capital budgeting, make-or-buy, asset
allocation, portfolio selection, financial planning, and
more.
Portfolio selection problems involve choosing specific
investments – for example, stocks and bonds – from a
variety of investment alternatives.
This type of problem is faced by managers of banks,
mutual funds, and insurance companies.
The objective function usually is maximization of
expected return or minimization of risk.
Slide 6
Portfolio Selection
Willie Maykit, who recently inherited $1,000,000 from his late aunt,
would like to invest all his inheritance in a mix of 4 investment
instruments, each with characteristics as listed below:
Investment Price Annual Return
Maximum
Risk
Type
/unit
/unit
Investment
Factor
A
$60
$3
$500,000
0.7
B
$25
$5
$200,000
0.8
C
$20
$5
$300,000
1.2
D
$40
$8
$250,000
1.4
Due to its low risk, Willie would like to invest at least 30% of the total
dollar investments in Type A. Furthermore, he specified that the dollar
average weighted risk factor of his investments should not exceed 1.1
Determine the number of units of each investment type to buy so that
Willie will maximize his investment returns. How many dollars should
Willie invest in each of the 4 types of investments?
Slide 7
Portfolio Selection
Define the Decision Variables
A=
B=
C=
D=
Define the Objective Function
Slide 8
Portfolio Selection
Define the Constraints
Total investment amount limited to principal
30% of the total dollar investments in Type A
Risk factor of his investments should not exceed 1.1
Slide 9
Portfolio Selection
Define the Constraints (continued)
Slide 10
Operations Management Applications
LP can be used in operations management to aid in
decision-making about product mix, production
scheduling, staffing, blending, inventory control,
capacity planning, and other issues.
An important application of LP is multi-period
planning such as production scheduling.
Usually the objective is to establish an efficient, lowcost production schedule for one or more products
over several time periods.
Typical constraints include limitations on production
capacity, labor capacity, storage space, and more.
Slide 11
PRODUCT MIX
A MANUFACTURER HAS TWO PRODUCTS, BOTH OF
WHICH ARE MADE IN TWO STEPS BY MACHINES 1
AND 2. THE PROCESS TIMES FOR THE TWO
PRODUCTS ON THE TWO MACHINES ARE AS
FOLLOWS:
PRODUCT
MACHINE A (hrs). MACHINE B (hrs)
1
4
5
2
5
2
FOR THE COMING PERIOD, MACHINE A HAS 100
HOURS AVAILABLE AND MACHINE B HAS 80
HOURS AVAILABLE. THE PROFIT FOR PRODUCT 1 IS
$10 PER UNIT AND PRODUCT 2 IS $5 PER UNIT.
DEFINE THE DECISION VARIABLES AND FORMULATE
THE LP PROBLEM.
Slide 12
PRODUCT MIX
Define the Decision Variables
Objective Function
Constraints
Slide 13
PRODUCT MIX
A MANUFACTURER HAS TWO PRODUCTS, BOTH OF
WHICH ARE CAN BE MADE BY EITHER BY MACHINES 1
AND 2. THE PROCESS TIMES FOR THE TWO PRODUCTS
ARE AS FOLLOWS:
PRODUCT
MACHINE A (hrs). MACHINE B (hrs)
1
4
5
2
5
2
FOR THE COMING PERIOD, MACHINE A HAS 100
HOURS AVAILABLE AND MACHINE B HAS 80 HOURS
AVAILABLE. THE PROFIT FOR PRODUCT 1 IS $10 PER
UNIT AND PRODUCT 2 IS $5 PER UNIT. DEFINE THE
DECISION VARIABLES AND FORMULATE THE LP
PROBLEM.
Slide 14
PRODUCT MIX
Define the Decision Variables
Objective Function
Constraints
Slide 15
Workforce Assignment
Due to recent budget cuts, Gotham City Police department
would like to better utilize its police force. The minimum
required number of police officers for each time period is as
follows:
Time Period
Minimum Patrol Officers
Midnight-4 a.m
6
4-8 a.m.
4
8-noon
14
12-4 p.m.
8
4-8 p.m.
12
8-midnight
16
Police officers report for duty at midnight or each of the
following 4 hour intervals and they work continuously fir 8
hours. What is the minimum number of officers needed per
day?
Slide 16
Workforce Assignment
Define the Decision Variables
Define the Objective Function
Slide 17
Workforce Assignment
Define the Constraints (continued)
Slide 18
Blending Problem
BURN-CLEAN OIL COMPANY PRODUCES REGULAR AND
PREMIUM GASOLINE. BOTH TYPES OF GASOLINE ARE
PRODUCED BY BLENDING TWO TYPES OF CRUDE OILS. THE
CRUDE OILS CONTAIN TWO INGREDIENTS, NAMELY A AND B.
THE PERCENTAGES OF INGREDIENTS A AND B, AND THE COST
PER GALLON OF EACH TYPE OF THE CRUDE OIL ARE AS
FOLLOWS:
CRUDE OIL COST INGREDIENT A
INGREDIENT B
1
$0.20
20%
60%
2
$0.30
50%
30%
EACH GALLON OF REGULAR MUST CONTAIN AT LEAST 40% OF
INGREDIENT A, AND EACH GALLON OF PREMIUM CAN
CONTAIN AT MOST 50% OF INGREDIENT B. THE DAILY
DEMAND FOR REGULAR AND PREMIUM GASOLINE ARE 800,000
GALLONS AND 500,000 GALLONS, RESPECTIVELY. FORMULATE
THE LP PROBLEM.
Slide 19
Blending Problem
Define the decision variables
Define the objective function
Slide 20
Blending Problem
Define the constraints
Slide 21
MULTIPERIOD PRODUCTION PLANNING MODEL
XYZ CO. USES A 3 MONTH PLANNING HORIZON TO
PLAN THE PRODUCTION OF WIDGETS. INVENTORY AT
THE END OF MARCH IS 500 UNITS. MANAGEMENT
WANTS AT LEAST 300 UNITS IN INVENTORY AT THE END
OF JUNE. THE INVENTORY HOLDING COST IS
$2.00/UNIT/MONTH. PRODUCTION RATE IS 2 UNITS /
HOUR. EACH WIDGET REQUIRES 5 SQ FT. AND THE
WAREHOUSE STORAGE CAPACITY IS 10,000 SQ FT.
USE THE FOLLOWING DATA TO FORMULATE THE LP
MODEL TO FIND THE OPTIMAL PRODUCTION SCHEDULE
THAT MEETS THE DEMAND.
Month
Demand
Prod cost/unit
Prod. capacity
April
600 units
$ 10
1000 hrs
May
1500
15
800
June
2500
15
600
Slide 22
MULTIPERIOD PRODUCTION PLANNING MODEL
Define the decision variables
Define the objective function
Slide 23
MULTIPERIOD PRODUCTION PLANNING MODEL
Define the constraints
Slide 24