Linear Programming Application in Marketing: Media Selection for SMM Company

Chapter 3

Linear Programming Applications

n

The process of problem formulation

n

Marketing and media applications

n

Financial Applications

n

Transportation Problem

1.

Provide a detailed verbal description of the problem

2.

Determine the overall objective that appears to be

relevant.

3.

Determine the factors (constraints) that appear to

restrict the attainment of the objective function.

4.

Define the decision variables and state their units of

measurement.

5.

Using these decision variables, formulate an

objective function.

6.

Formulate a mathematical equations for each of the

identified constraints.

7.

Check the entire formulation to ensure linearity.

The process of problem formulation

n

One application of linear programming in marketing

is 

media selection

.

n

LP can be used to help marketing managers allocate a

fixed budget to various advertising media.

n

The objective is to maximize reach, frequency, and

quality of exposure.

n

Restrictions on the allowable allocation usually arise

during consideration of company policy, contract

requirements, and media availability.

Marketing Applications

Marketing Applications

Media Selection

Media Selection

     SMM Company recently developed a new instant

     SMM Company recently developed a new instant

salad machine, has $282,000 to spend on advertising.

salad machine, has $282,000 to spend on advertising.

The product is to be initially test marketed in the Dallas

The product is to be initially test marketed in the Dallas

area.  The money is to be spent on

area.  The money is to be spent on

a TV advertising blitz during one

a TV advertising blitz during one

weekend (Friday, Saturday, and

weekend (Friday, Saturday, and

Sunday) in November.

Sunday) in November.

     The three options available

     The three options available

are:  daytime advertising,

are:  daytime advertising,

evening news advertising, and

evening news advertising, and

Sunday game-time advertising.  A mixture of one-

Sunday game-time advertising.  A mixture of one-

minute TV spots is desired.

minute TV spots is desired.

Media Selection

Media Selection

Estimated Audience

Estimated Audience

Ad Type

Ad Type

Reached With Each Ad

Reached With Each Ad

Cost Per Ad

Cost Per Ad

Daytime           

Daytime           

3,000      

3,000      

              $5,000 

              $5,000 

Evening News       

Evening News       

            4,000     

            4,000     

              $7,000

              $7,000

Sunday Game 

Sunday Game 

          75,000   

          75,000   

          $100,000

          $100,000

     SMM wants to take out at least one ad of each type

     SMM wants to take out at least one ad of each type

(daytime, evening-news, and game-time).  Further, there

(daytime, evening-news, and game-time).  Further, there

are only two game-time ad spots available.  There are

are only two game-time ad spots available.  There are

ten daytime spots and six evening news spots available

ten daytime spots and six evening news spots available

daily.  SMM wants to have at least 5 ads per day, but

daily.  SMM wants to have at least 5 ads per day, but

spend no more than $50,000 on Friday and no more than

spend no more than $50,000 on Friday and no more than

$75,000 on Saturday.

$75,000 on Saturday.

Media Selection

Media Selection

DFR

DFR

 = number of daytime ads on Friday

 = number of daytime ads on Friday

DSA

DSA

 = number of daytime ads on Saturday

 = number of daytime ads on Saturday

DSU

DSU

 =

 =

 number of daytime ads on Sunday 

 number of daytime ads on Sunday 

EFR

EFR

 =

 =

 number of evening ads on Friday 

 number of evening ads on Friday 

ESA

ESA

 =

 =

 number of evening ads on Saturday

 number of evening ads on Saturday

ESU

ESU

 =

 =

 number of evening ads on Sunday

 number of evening ads on Sunday

GSU

GSU

 =

 =

 number of game-time ads on Sunday

 number of game-time ads on Sunday

n

Define the Decision Variables

Define the Decision Variables

Media Selection

Media Selection

n

Define the Objective Function

Define the Objective Function

Maximize the total audience reached:

Maximize the total audience reached:

Max  (audience reached per ad of each type)

Max  (audience reached per ad of each type)

           x  (number of ads used of each type)

           x  (number of ads used of each type)

Max  3000

Max  3000

DFR

DFR

 +3000

 +3000

DSA

DSA

 +3000

 +3000

DSU

DSU

 +4000

 +4000

EFR

EFR

 +4000

 +4000

ESA

ESA

 +4000

 +4000

ESU

ESU

 +75000

 +75000

GSU

GSU

Media Selection

Media Selection

n

Define the Constraints

Define the Constraints

Take out at least one ad of each type:

Take out at least one ad of each type:

       (1)  

       (1)  

DFR

DFR

 + 

 + 

DSA

DSA

 + 

 + 

DSU

DSU

>

>

 1

 1

       (2)  

       (2)  

EFR

EFR

 + 

 + 

ESA

ESA

 + 

 + 

ESU

ESU

>

>

 1

 1

       (3)  

       (3)  

GSU

GSU

>

>

 1

 1

Ten daytime spots available:

Ten daytime spots available:

       (4)  

       (4)  

DFR

DFR

<

<

 10

 10

       (5)  

       (5)  

DSA

DSA

<

<

 10

 10

       (6)  

       (6)  

DSU

DSU

<

<

 10

 10

Six evening news spots available:

Six evening news spots available:

       (7)  

       (7)  

EFR

EFR

<

<

 6  

 6  

       (8)  

       (8)  

ESA

ESA

<

<

 6

 6

       (9)  

       (9)  

ESU

ESU

<

<

 6

 6

Media Selection

Media Selection

n

Define the Constraints (continued)

Define the Constraints (continued)

Only two Sunday game-time ad spots available:

Only two Sunday game-time ad spots available:

       (10)  

       (10)  

GSU

GSU

<

<

 2

 2

At least 5 ads per day:

At least 5 ads per day:

       (11)  

       (11)  

DFR

DFR

 + 

 + 

EFR

EFR

>

>

 5

 5

       (12)  

       (12)  

DSA

DSA

 + 

 + 

ESA

ESA

>

>

 5

 5

       (13)  

       (13)  

DSU

DSU

 + 

 + 

ESU

ESU

 + 

 + 

GSU

GSU

>

>

 5

 5

Spend no more than $50,000 on Friday:

Spend no more than $50,000 on Friday:

       (14)  5000

       (14)  5000

DFR

DFR

 + 7000

 + 7000

EFR

EFR

<

<

 50000

 50000

Media Selection

Media Selection

n

Define the Constraints (continued)

Define the Constraints (continued)

Spend no more than $75,000 on Saturday:

Spend no more than $75,000 on Saturday:

     (15)  5000

     (15)  5000

DSA

DSA

 + 7000

 + 7000

ESA

ESA

<

<

 75000

 75000

Spend no more than $282,000 in total:

Spend no more than $282,000 in total:

     (16)  5000

     (16)  5000

DFR

DFR

 + 5000

 + 5000

DSA

DSA

 + 5000

 + 5000

DSU

DSU

 + 7000

 + 7000

EFR

EFR

             + 7000

             + 7000

ESA

ESA

 + 7000

 + 7000

ESU

ESU

 + 100000

 + 100000

GSU

GSU

7 

7 

< 

< 

 282000

 282000

Non-negativity:      

Non-negativity:      

DFR

DFR

, 

, 

DSA

DSA

, 

, 

DSU

DSU

, 

, 

EFR

EFR

, 

, 

ESA

ESA

, 

, 

ESU

ESU

, 

, 

GSU 

GSU 

>

>

 0

 0

Media Selection

Media Selection

n

The Management Scientist

The Management Scientist

  Solution

  Solution

Objective Function Value =      199000.000

Objective Function Value =      199000.000

Variable

Variable

Value

Value

Reduced Costs

Reduced Costs

DFR

DFR

8.000              

8.000              

     0.000

     0.000

DSA

DSA

5.000              

5.000              

     0.000

     0.000

DSU

DSU

2.000              

2.000              

     0.000

     0.000

EFR

EFR

0.000              

0.000              

     0.000

     0.000

ESA

ESA

0.000              

0.000              

     0.000

     0.000

ESU

ESU

1.000              

1.000              

     0.000

     0.000

GSU

GSU

2.000     

2.000     

     0.000

     0.000

Media Selection

Media Selection

n

Solution Summary

Solution Summary

Total new audience reached  =  199,000

Total new audience reached  =  199,000

  Number of daytime ads on Friday  

  Number of daytime ads on Friday  

=  8

=  8

  Number of daytime ads on Saturday  

  Number of daytime ads on Saturday  

=  5

=  5

  Number of daytime ads on Sunday  

  Number of daytime ads on Sunday  

=  2

=  2

  Number of evening ads on Friday  

  Number of evening ads on Friday  

=  0

=  0

  Number of evening ads on Saturday 

  Number of evening ads on Saturday 

=  0

=  0

  Number of evening ads on Sunday       

  Number of evening ads on Sunday       

=  1

=  1

  Number of game-time ads on Sunday

  Number of game-time ads on Sunday

=  2

=  2

Financial Applications

Financial Applications

n

LP can be used in financial decision-making that

LP can be used in financial decision-making that

involves capital budgeting, make-or-buy, asset

involves capital budgeting, make-or-buy, asset

allocation, portfolio selection, financial planning, and

allocation, portfolio selection, financial planning, and

more.

more.

n

Portfolio selection

Portfolio selection

 problems involve choosing specific

 problems involve choosing specific

investments – for example, stocks and bonds – from a

investments – for example, stocks and bonds – from a

variety of investment alternatives.

variety of investment alternatives.

n

This type of problem is faced by managers of banks,

This type of problem is faced by managers of banks,

mutual funds, and insurance companies.

mutual funds, and insurance companies.

n

The objective function usually is maximization of

The objective function usually is maximization of

expected return or minimization of risk.

expected return or minimization of risk.

Portfolio Selection

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.

Portfolio Selection

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.

Portfolio Selection

Formulate 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.

Portfolio Selection

n

Define the Decision Variables

Define the Decision Variables

G

G

i

i

  =  amount of new investment in government

  =  amount of new investment in government

               bonds in month

               bonds in month

 i 

 i 

(for 

(for 

i

i

 = 1, 2, 3, 4)

 = 1, 2, 3, 4)

C

C

i

i

  =  amount of new investment in construction

  =  amount of new investment in construction

               loans in month 

               loans in month 

i

i

 (for 

 (for 

i

i

 = 1, 2, 3, 4)

 = 1, 2, 3, 4)

L

L

i

i

  =  amount invested locally in month

  =  amount invested locally in month

 i

 i

,

,

   (for

   (for

 i

 i

 = 1, 2, 3, 4)

 = 1, 2, 3, 4)

Portfolio Selection

n

Define the Objective Function

Define the Objective Function

     Maximize total interest earned in the 4-month period:

     Maximize total interest earned in the 4-month period:

     Max  (interest rate on investment) X (amount invested)

     Max  (interest rate on investment) X (amount invested)

     Max    .02G

     Max    .02G

1

1

 + .02

 + .02

G

G

2

2

 + .02

 + .02

G

G

3

3

 + .02

 + .02

G

G

4

4

  + .06

  + .06

C

C

1

1

 + .06

 + .06

C

C

2

2

 + .06

 + .06

C

C

3

3

 + .06

 + .06

C

C

4

4

  + .0075

  + .0075

L

L

1

1

 + .0075

 + .0075

L

L

2

2

 + .0075

 + .0075

L

L

3

3

 + .0075

 + .0075

L

L

4

4

Portfolio Selection

n

Define the Constraints

Define the Constraints

    Month 1's total investment limited to $20 million:

    Month 1's total investment limited to $20 million:

(1)   

(1)   

G

G

1

1

 + 

 + 

C

C

1

1

 + 

 + 

L

L

1

1

 = 20,000,000

 = 20,000,000

    Month 2's total investment limited to principle and

    Month 2's total investment limited to principle and

interest invested locally in Month 1:

interest invested locally in Month 1:

(2)   

(2)   

G

G

2

2

 + 

 + 

C

C

2

2

 + 

 + 

L

L

2

2

 = 1.0075

 = 1.0075

L

L

1

1

        or    

        or    

G

G

2

2

 + 

 + 

C

C

2

2

 - 1.0075

 - 1.0075

L

L

1

1

 + 

 + 

L

L

2

2

 = 0

 = 0

Portfolio Selection

n

Define the Constraints (continued)

Define the Constraints (continued)

Month 3's total investment amount limited to

Month 3's total investment amount limited to

principle and interest invested in government bonds

principle and interest invested in government bonds

in Month 1 and locally invested in Month 2:

in Month 1 and locally invested in Month 2:

(3)   

(3)   

G

G

3

3

 + 

 + 

C

C

3

3

 + 

 + 

L

L

3

3

 = 1.02

 = 1.02

G

G

1

1

 + 1.0075

 + 1.0075

L

L

2

2

        or    - 1.02

        or    - 1.02

G

G

1

1

 + 

 + 

G

G

3

3

 + 

 + 

C

C

3

3

 - 1.0075

 - 1.0075

L

L

2

2

 + 

 + 

L

L

3

3

 = 0

 = 0

Portfolio Selection

n

Define the Constraints (continued)

Define the Constraints (continued)

Month 4's total investment limited to principle and

Month 4's total investment limited to principle and

interest invested in construction loans in Month 1,

interest invested in construction loans in Month 1,

goverment bonds in Month 2, and locally invested in

goverment bonds in Month 2, and locally invested in

Month 3:

Month 3:

     (4)   

     (4)   

G

G

4

4

 + 

 + 

C

C

4

4

 + 

 + 

L

L

4

4

 = 1.06

 = 1.06

C

C

1

1

 + 1.02

 + 1.02

G

G

2

2

 + 1.0075

 + 1.0075

L

L

3

3

 or    - 1.02

 or    - 1.02

G

G

2

2

 + 

 + 

G

G

4

4

 - 1.06

 - 1.06

C

C

1

1

 + 

 + 

C

C

4

4

 - 1.0075

 - 1.0075

L

L

3

3

 + 

 + 

L

L

4

4

 = 0

 = 0

$10 million must be available at start of Month 5:

$10 million must be available at start of Month 5:

     (5)   1.06

     (5)   1.06

C

C

2

2

 + 1.02

 + 1.02

G

G

3

3

 + 1.0075

 + 1.0075

L

L

4

4

>

>

 10,000,000

 10,000,000

Portfolio Selection

n

Define the Constraints (continued)

Define the Constraints (continued)

No more than $8 million in government bonds at any

No more than $8 million in government bonds at any

time:

time:

(6)    

(6)    

G

G

1

1

<

<

  8,000,000

  8,000,000

(7)    

(7)    

G

G

1

1

 + 

 + 

G

G

2

2

<

<

  8,000,000

  8,000,000

(8)    

(8)    

G

G

2

2

 + 

 + 

G

G

3

3

<

<

  8,000,000

  8,000,000

(9)    

(9)    

G

G

3

3

 + 

 + 

G

G

4

4

<

<

  8,000,000

  8,000,000

Portfolio Selection

n

Define the Constraints (continued)

Define the Constraints (continued)

No more than $8 million in construction loans at

No more than $8 million in construction loans at

any time:

any time:

(10)   

(10)   

C

C

1

1

<

<

  8,000,000

  8,000,000

(11)   

(11)   

C

C

1

1

 + 

 + 

C

C

2

2

<

<

  8,000,000

  8,000,000

(12)   

(12)   

C

C

1

1

 + 

 + 

C

C

2

2

 + 

 + 

C

C

3

3

<

<

  8,000,000

  8,000,000

(13)           

(13)           

C

C

2

2

 + 

 + 

C

C

3

3

 + 

 + 

C

C

4

4

<

<

  8,000,000

  8,000,000

Non-negativity:

Non-negativity:

G

G

i

i

, 

, 

C

C

i

i

, 

, 

L

L

i

i

>

>

 0  for 

 0  for 

i

i

 = 1, 2, 3, 4

 = 1, 2, 3, 4

Portfolio Selection

Portfolio Selection

n

The Management Scientist

The Management Scientist

  Solution

  Solution

     Objective Function Value =    1429213.7987

     Objective Function Value =    1429213.7987

Variable

Variable

Value

Value

Reduced Costs

Reduced Costs

G

G

1        

1        

8000000.0000                   0.0000

8000000.0000                   0.0000

G

G

2       

2       

            0.0000                   0.0000

            0.0000                   0.0000

G

G

3         

3         

5108613.9228                   0.0000

5108613.9228                   0.0000

G

G

4         

4         

2891386.0772                   0.0000

2891386.0772                   0.0000

C

C

1        

1        

8000000.0000                   0.0000

8000000.0000                   0.0000

C

C

2         

2         

            0.0000                   0.0453

            0.0000                   0.0453

C

C

3          

3          

            0.0000                   0.0076

            0.0000                   0.0076

C

C

4        

4        

8000000.0000                   0.0000

8000000.0000                   0.0000

L

L

1       

1       

4000000.0000                   0.0000

4000000.0000                   0.0000

L

L

2       

2       

4030000.0000                   0.0000

4030000.0000                   0.0000

L

L

3      

3      

7111611.0772                   0.0000

7111611.0772                   0.0000

L

L

4     

4     

4753562.0831                   0.0000

4753562.0831                   0.0000

Transportation Problem

n

The 

transportation problem

 seeks to minimize the

total shipping costs of transporting goods from 

m

origins (each with a supply 

s

i

) to 

n

 destinations

(each with a demand 

d

j

), when the unit shipping

cost from an origin, 

i

, to a destination, 

j

, is 

c

ij

.

n

The 

network representation

 for a transportation

problem with two sources and three destinations is

given on the next slide.

Transportation Problem

n

Network Representation

Network Representation

2

c

c

11

11

c

c

12

12

c

c

13

13

c

c

21

21

c

c

22

22

c

c

23

23

d

d

1

1

d

d

2

2

d

d

3

3

s

s

1

1

s

2

Sources

Sources

Destinations

Destinations

3

2

1

1

Transportation Problem

n

LP Formulation

LP Formulation

The LP formulation in terms of the amounts

The LP formulation in terms of the amounts

shipped from the origins to the destinations, 

shipped from the origins to the destinations, 

x

x

ij 

ij 

, can

, can

be written as:

be written as:

          Min   

          Min   





c

c

ij

ij

x

x

ij

ij

i j

i j

                       s.t.    

                       s.t.    





x

x

ij

ij

<

<

s

s

i

i

for each origin 

for each origin 

i

i

j

j





x

x

ij

ij

 ≥ 

 ≥ 

d

d

j

j

for each destination 

for each destination 

j

j

i

i

x

x

ij

ij

>

>

 0 

 0 

for all 

for all 

i

i

 and 

 and 

j

j

n

Powerco has three electric power plants that supply

the electric needs of four cities.

n

The associated supply of each plant and demand of

each city is given in the table 1.

n

The cost of sending 1 million kwh of electricity from

a plant to a city depends on the distance the

electricity must travel.

n

A transportation problem is specified by the supply,

the demand, and the shipping costs. So the relevant

data can be summarized in a transportation tableau.

The transportation tableau implicitly expresses the

supply and demand constraints and the shipping cost

between each demand and supply point.

Transportation Problem

Transportation tableau

Transportation

Transportation

Tableau

Tableau

1.

Decision Variable:

Since we have to determine how much

electricity is sent from each plant to each

city;

X

ij 

= Amount of electricity produced at plant

i and sent to city j

X

14

 = Amount of electricity produced at

plant 1 and sent to city 4

Transportation Problem

Transportation Problem

2. Objective function

Since we want to minimize the total cost of

shipping from plants to cities;

Minimize Z = 8X

11

+6X

12

+10X

13

+9X

14

+9X

21

+12X

22

+13X

23

+7X

24

+14X

31

+9X

32

+16X

33

+5X

34

Transportation Problem

Transportation Problem

3. Supply Constraints

Since each supply point has a limited production

capacity;

X

11

+X

12

+X

13

+X

14 

<= 35

X

21

+X

22

+X

23

+X

24 

<= 50

X

31

+X

32

+X

33

+X

34 

<= 40

Transportation Problem

Transportation Problem

4. Demand Constraints

Since each supply point has a limited production

capacity;

X

11

+X

21

+X

31 

>= 45

X

12

+X

22

+X

32 

>= 20

X

13

+X

23

+X

33 

>= 30

X

14

+X

24

+X

34 

>= 30

Transportation Problem

Transportation Problem

5. Sign Constraints

Since a negative amount of electricity can not be

shipped all Xij’s must be non negative;

Xij >= 0 (i= 1,2,3; j= 1,2,3,4)

Transportation Problem

Transportation Problem

LP Formulation of Powerco’s Problem

Min Z = 8X

11

+6X

12

+10X

13

+9X

14

+9X

21

+12X

22

+13X

23

+7X

24

+14X

31

+9X

32

+16X

33

+5X

34

S.T.:

X

11

+X

12

+X

13

+X

14 

<= 35 

(Supply Constraints)

X

21

+X

22

+X

23

+X

24 

<= 50

X

31

+X

32

+X

33

+X

34 

<= 40

X

11

+X

21

+X

31 

>= 45 

(Demand Constraints)

X

12

+X

22

+X

32 

>= 20

X

13

+X

23

+X

33 

>= 30

X

14

+X

24

+X

34 

>= 30

Xij >= 0 (i= 1,2,3; j= 1,2,3,4)