# Marketing and Finance Applications by 5R42zo5

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```									Readings

Chapter 4
Linear Programming Applications in Marketing, Finance,
and Operations Management

BA 452 Lesson I.8 Marketing and Finance Applications   1
Overview

Overview

BA 452 Lesson I.8 Marketing and Finance Applications   2
Overview

Media Selection Problems help marketing managers allocate a fixed advertising
budget to various advertising media. The objective is to maximize reach,
frequency, and quality of exposure.

Marketing Research Problems help marketing managers learn about consumer
characteristics and preferences, by surveying sufficiently many people at
minimum cost.

Portfolio Selection Problems with Dynamic Constraints help select specific
investments (stocks, bonds, …) over time to either maximize expected return or
minimize risk.

Financial Planning Problems help financial managers meet future budget
obligations by selecting specific investments (stocks, bonds, …) over time to
produce future income at minimum cost.

BA 452 Lesson I.8 Marketing and Finance Applications               3
Overview

Tool Summary
   One possible way to solve a program where some variables are
restricted to be integers (like the number of ads in Example 1): Do
not restrict the variables to be integers, and maybe the variables at
an optimum will be integers.
• Warning: This does not always work: without integer restrictions,
some variables at an optimum may not be integers --- even if all
coefficients in the linear program are integers (like in an Example
in Lesson I-3).
   Use compound variables:
• First Example: EFR = number of evening ads on Friday
• Second Example: Gi = amount of new investment in
government bonds in month i
• Third Example: SO1 = number of standard wheels made with
overtime labor in month 1

BA 452 Lesson I.8 Marketing and Finance Applications              4
Tool Summary

Tool Summary
   Use dynamic or recursive constraints:
• Second Example: Constrain that Month 2's total investment be
limited to principle and interest invested locally in Month 1:
G2 + C2 + L2 = 1.0075L1
   Use inventory variables:
• Second Example: SI = number of standard wheels held in
inventory from month 1 to month 2

BA 452 Lesson I.8 Marketing and Finance Applications        5
Media Selection

Media Selection

BA 452 Lesson I.8 Marketing and Finance Applications   6
Media Selection

Overview

Media Selection Problems help marketing managers
media. Potential media include newspapers, magazines,
radio, television, and direct mail. The objective is to
maximize reach, frequency, and quality of exposure.
Constraints may restrict permissible allocations by
company policy, contract requirements, and media
availability.

BA 452 Lesson I.8 Marketing and Finance Applications   7
Media Selection

Question: SMM Company recently developed a new instant
The product is to be initially test marketed in the Dallas
area. The money is to be spent on a TV advertising blitz
during one weekend (Friday, Saturday, and Sunday) in
November.

The three options available are: daytime advertising,
evening news advertising, and Sunday game-time
advertising. A mixture of one-minute TV spots is desired.

BA 452 Lesson I.8 Marketing and Finance Applications   8
Media Selection

Estimated Audience
Daytime                         3,000                       \$5,000
Evening News                     4,000                      \$7,000
Sunday Game                     75,000                    \$100,000

SMM wants to take out at least one ad of each type
(daytime, evening-news, and game-time). Further, there
are only two game-time ad spots available. There are
ten daytime spots and six 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.

BA 452 Lesson I.8 Marketing and Finance Applications         9
Media Selection

What should be SMM’s objective?
Are you making any implicit assumptions?

BA 452 Lesson I.8 Marketing and Finance Applications   10
Media Selection

BA 452 Lesson I.8 Marketing and Finance Applications   11
Media Selection
  Define the decision variables
DFR = number of daytime ads on Friday
DSA = number of daytime ads on Saturday
DSU = number of daytime ads on Sunday
EFR = number of evening ads on Friday
ESA = number of evening ads on Saturday
ESU = number of evening ads on Sunday
GSU = number of game-time ads on Sunday
 Define the objective function
Maximize the total audience reached
= Max (audience per ad of each type)x(number of ads used of each type)
Max 3000DFR +3000DSA +3000DSU +4000EFR
+4000ESA +4000ESU +75000GSU

BA 452 Lesson I.8 Marketing and Finance Applications         12
Media Selection

   Define the constraints to take out at least one ad of each type:
(1) DFR + DSA + DSU > 1
(2) EFR + ESA + ESU > 1
(3) GSU > 1
   Define the constraints of ten daytime spots available:
(4) DFR < 10; (5) DSA < 10;           (6) DSU < 10
   Define the constraints of six evening news spots available:
(7) EFR < 6;      (8) ESA < 6;         (9) ESU < 6

BA 452 Lesson I.8 Marketing and Finance Applications     13
Media Selection
   Define the constraint of only two Sunday game-time ad spots available:
(10) GSU < 2
   Define the constraint of at least 5 ads per day:
(11) DFR + EFR > 5; (12) DSA + ESA > 5;
(13) DSU + ESU + GSU > 5
   Define the constraint of spending no more than \$50,000 on Friday:
(14) 5000DFR + 7000EFR < 50000
   Define the constraint of spending no more than \$75,000 on Saturday:
(15) 5000DSA + 7000ESA < 75000
   Define the constraint of spending no more than \$282,000 in total:
(16) 5000DFR + 5000DSA + 5000DSU + 7000EFR
+ 7000ESA + 7000ESU + 100000GSU7 < 282000

BA 452 Lesson I.8 Marketing and Finance Applications            14
Media Selection

Interpretation: Total new audience reached = 199,000
Number of daytime ads on Friday      = 8
Number of daytime ads on Saturday = 5
Number of daytime ads on Sunday = 2
Number of evening ads on Friday      = 0
Number of evening ads on Saturday = 0
Number of evening ads on Sunday             = 1
Number of game-time ads on Sunday = 2

BA 452 Lesson I.8 Marketing and Finance Applications       15
Marketing Research

Marketing Research

BA 452 Lesson I.8 Marketing and Finance Applications   16
Marketing Research

Overview

Marketing Research Problems help marketing managers
learn about consumer characteristics and preferences.
Marketing research firms design and solve such problems
to survey sufficiently many people at minimum cost.

BA 452 Lesson I.8 Marketing and Finance Applications   17
Marketing Research

Question: C R Market Surveys, Inc. specializes in
evaluating consumer reaction to new products and
advertising campaigns. A client hired Market Survey Inc. to
measure consumer reaction to a recently marketed
product. Market Survey Inc. agreed to conduct interviews
from households, both with or without children, during
either the day or evening.

BA 452 Lesson I.8 Marketing and Finance Applications   18
Marketing Research

The client’s contract calls for 1000 interviews under the following
quotas:
1. Interview at least 400 households with children.
2. Interview at least 400 households without children.
3. Interview at least as many households at night, as during the day.
4. Of the interviews of households with children, at least 40 percent
must be during the evening.
5. Of the interviews of households without children, at least 60
percent must be during the evening.
Based on previous studies, here are the estimated interview costs:
Household             Day             Evening
Children             \$20                \$25
No Children             \$18               \$20
Formulate the Marketing Research Problem for finding the minimum-
cost household, time-of-day interview plan satisfying all quotas.

BA 452 Lesson I.8 Marketing and Finance Applications          19
Marketing Research

DC = the number of daytime interviews of households with
children
DN = the number of daytime interviews of households with
no children
EC = the number of evening interviews of households with
children
EN = the number of evening interviews of households with
no children

BA 452 Lesson I.8 Marketing and Finance Applications   20
Marketing Research

Household              Day             Evening
Children              \$20                \$25
No Children            \$18                \$20

Given cost per unit, minimize total cost:
Min 20DC + 18DN + 25EC + 20EN

BA 452 Lesson I.8 Marketing and Finance Applications   21
Marketing Research

Formulate Constraints:
1. Interview at least 1000 households:
DC + DN + EC + EN > 1000
2. Interview at least 400 households with children:
DC + EC > 400
3. Interview at least 400 households without children:
DN + EN > 400
4. Interview at least as many households at night, as during the day:
EC + EN > DC + DN
5. Of the interviews of households with children, at least 40 percent
must be during the evening:
EC > 0.40(DC + EC)
6. Of the interviews of households without children, at least 60
percent must be during the evening:
EN > 0.60(DN + EN)

BA 452 Lesson I.8 Marketing and Finance Applications          22
Portfolio Selection with Dynamic Constraints

Portfolio Selection with Dynamic
Constraints

BA 452 Lesson I.8 Marketing and Finance Applications   23
Portfolio Selection with Dynamic Constraints

Overview

Portfolio Selection Problems with Dynamic Constraints help
financial managers select specific investments (stocks,
bonds, …) over time to either maximize expected return or
minimize risk. Dynamic Constraints may restrict new
investment in one period to equal the return from
investments in previous periods. For example, there may
be no new investments one month if all investments in
previous months have not yet matured.

BA 452 Lesson I.8 Marketing and Finance Applications   24
Portfolio Selection with Dynamic Constraints

Question: Pepperdine University 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 (after
all investments mature) 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,
Pepperdine 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.

How should Pepperdine 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?

BA 452 Lesson I.8 Marketing and Finance Applications          25
Portfolio Selection with Dynamic Constraints

Gi = amount of new investment in government
bonds in month i (for i = 1, 2, 3, 4)
Ci = amount of new investment in construction
loans in month i (for i = 1, 2, 3, 4)
Li = amount invested locally in month i (for i = 1, 2, 3, 4)

BA 452 Lesson I.8 Marketing and Finance Applications   26
Portfolio Selection with Dynamic Constraints

   Define the objective function
Maximize total interest earned after all investments mature:
Max (interest rate on investment) X (amount invested)
Max .02G1 + .02G2 + .02G3 + .02G4
+ .06C1 + .06C2 + .06C3 + .06C4
+ .0075L1 + .0075L2 + .0075L3 + .0075L4

BA 452 Lesson I.8 Marketing and Finance Applications   27
Portfolio Selection with Dynamic Constraints
 Constrain that Month 1's total investment be \$20 million:
(1) G1 + C1 + L1 = 20,000,000
 Constrain that Month 2's total investment be limited to principle and interest
invested locally in Month 1:
(2) G2 + C2 + L2 = 1.0075L1 , or G2 + C2 - 1.0075L1 + L2 = 0
 Constrain that Month 3's total investment be limited to principle and interest
invested in government bonds in Month 1 and locally invested in Month 2:
(3) G3 + C3 + L3 = 1.02G1 + 1.0075L2 , or - 1.02G1 + G3 + C3 -
1.0075L2 + L3 = 0

BA 452 Lesson I.8 Marketing and Finance Applications           28
Portfolio Selection with Dynamic Constraints

   Constrain that Month 4's total investment be limited to principle and interest
invested in construction loans in Month 1, government bonds in Month 2,
and locally invested in Month 3:
(4) G4 + C4 + L4 = 1.06C1 + 1.02G2 + 1.0075L3 , or
- 1.02G2 + G4 - 1.06C1 + C4 - 1.0075L3 + L4 = 0
   Constrain that \$10 million must be available at start of Month 5:
(5) 1.02G3 + 1.06C2 + 1.0075L4 > 10,000,000

BA 452 Lesson I.8 Marketing and Finance Applications             29
Portfolio Selection with Dynamic Constraints
   Constrain that no more than \$8 million be in government bonds at any time:
(6) G1       < 8,000,000;       (7) G1 + G2 < 8,000,000
(8) G2 + G3 < 8,000,000;        (9) G3 + G4 < 8,000,000
   Constrain that no more than \$8 million be in construction loans at any time:
(10) C1             < 8,000,000; (11) C1 + C2           < 8,000,000
(12) C1 + C2 + C3 < 8,000,000; (13) C2 + C3 + C4 < 8,000,000

BA 452 Lesson I.8 Marketing and Finance Applications              30
Portfolio Selection with Dynamic Constraints

Interpretation: Total interest earned in the 4-month
period = \$1,429,213.80.
Gi = invested in government bonds
Ci = invested in construction loans
Li = invested locally, in month i (for i = 1, 2, 3, 4)
BA 452 Lesson I.8 Marketing and Finance Applications   31
Financial Planning

Financial Planning

BA 452 Lesson I.8 Marketing and Finance Applications   32
Financial Planning

Overview

Financial Planning Problems help financial managers meet
future budget obligations by selecting specific investments
(stocks, bonds, …) over time to produce future income.
The goal is to meet those obligations at minimum cost.

BA 452 Lesson I.8 Marketing and Finance Applications   33
Financial Planning

Question: Hewlitt Corporation established an early
retirement program as part of its corporate restructuring.
68 employees selected early retirement. The company
thus incurs the following obligations over the next eight
years:

Year     1         2         3         4         5         6         7         8
Cash   430,000   210,000   222,000   231,000   240,000   195,000   225,000   255,000
Req.

The requirements are due at the beginning of each year.

The corporate treasurer must determine how much money
must be set aside today to meet the eight yearly obligations
as they come due.

BA 452 Lesson I.8 Marketing and Finance Applications                            34
Financial Planning

The financial plan for the retirement program includes
savings, and government bonds. There are three bond
choices available at the beginning of year 1:
Market                      Years to
Bond                      Rate (%)
Price                       Maturity
1       \$1,150         8.875           5
2       \$1,000         5.500           6
3       \$1,350        11.750           7

The government bonds have a par value of \$1000, which
means that even with different prices, each bond pays
\$1000 at maturity. The rates shown are based on the par
value. (For example, each unit of Bond 1 pays \$88.75
each year for 5 years.) Each year, all money not invested
in bonds earns an annual rate of 4%.
BA 452 Lesson I.8 Marketing and Finance Applications   35
Financial Planning

Define decision variables
F = total dollars required to meet the retirement plan’s eight
year obligation
B1 = units of bond 1 purchased at the beginning of year 1
B2 = units of bond 2 purchased at the beginning of year 1
B3 = units of bond 3 purchased at the beginning of year 1
Si = dollars in savings at the beginning of year i, (i = 1,…,8)

BA 452 Lesson I.8 Marketing and Finance Applications   36
Financial Planning

Define objective
Minimize F

Define Constraints
For each of the eight years,
(Funds available at the beginning of the year)
- (Funds invested in bonds and placed in savings)
= (Cash obligation for the current year)

BA 452 Lesson I.8 Marketing and Finance Applications   37
Financial Planning

For year 1,
(Funds available at the beginning of the year) = F,
(Funds invested in bonds and placed in savings) =
1150B1 + 1000B2 + 1350B3 + S1,
(Cash obligation for the current year) = 430000,
F - 1150B1 - 1000B2 - 1350B3 - S1 = 430000

BA 452 Lesson I.8 Marketing and Finance Applications   38
Financial Planning

For year 2,
(Funds available at the beginning of the year) =
88.75B1 + 55B2 + 117.5B3 + 1.04S1,
(Funds placed in savings) = S1,
(Cash obligation for the current year) = 210000,
88.75B1 + 55B2 + 117.5B3 + 1.04S1 – S2 = 210000

BA 452 Lesson I.8 Marketing and Finance Applications   39
Financial Planning

For year 3,
88.75B1 + 55B2 + 117.5B3 + 1.04S2 – S3 = 222000
For year 4,
88.75B1 + 55B2 + 117.5B3 + 1.04S3 – S4 = 231000
For year 5,
88.75B1 + 55B2 + 117.5B3 + 1.04S4 – S5 = 240000
For year 6, since Bond 1 matures,
1088.75B1 + 55B2 + 117.5B3 + 1.04S5 – S6 = 195000
For year 7, since Bond 2 matures,
1055B2 + 117.5B3 + 1.04S6 – S7 = 225000
For year 8, since Bond 3 matures,
1117.5B3 + 1.04S6 – S7 = 225000

BA 452 Lesson I.8 Marketing and Finance Applications   40
Financial Planning

Management Scientist reveals the optimal solution to this
12-variable, 8-constraint problem,
B1 = 144.988, B2 = 187.856, B3 = 228.188
so
\$1150(144.988) = \$166,736 is invested in Bond 1,
\$1150(187.856) = \$187,856 is invested in Bond 2,
\$1150(228.188) = \$308,054 is invested in Bond 3.

F = 1728793.85, so \$1,728,793.85 is the minimum starting
investment for the company can make the specified
investments and generate future income to meet its budget
obligations.

BA 452 Lesson I.8 Marketing and Finance Applications   41
BA 452                                                Quantitative Analysis

End of Lesson I.8

BA 452 Lesson I.8 Marketing and Finance Applications            42

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