What is a marketing problem in applications of linear progra

What is a marketing problem in applications of linear programming?

Briefly discuss the objective function and constraint requirements in a marketing problem.

Give a real world example of a marketing problem.

Solution

Marketing Applications

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.

Media Selection

SMM Company recently developed a new instant salad machine, has $282,000 to spend on advertising. 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.

Estimated Audience

Ad Type Reached With Each Ad Cost Per Ad

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.

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 reached per ad of each type) x (number of ads used of each type)

Max 3000DFR +3000DSA +3000DSU +4000EFR +4000ESA +4000ESU +75000GSU

Define the Constraints

Take out at least one ad of each type:

(1) DFR + DSA + DSU > 1

(2) EFR + ESA + ESU > 1

(3) GSU > 1

Ten daytime spots available:

(4) DFR < 10

(5) DSA < 10

(6) DSU < 10

Six evening news spots available:

(7) EFR < 6

(8) ESA < 6

(9) ESU < 6

Only two Sunday game-time ad spots available:

(10) GSU < 2

At least 5 ads per day:

(11) DFR + EFR > 5

(12) DSA + ESA > 5

(13) DSU + ESU + GSU > 5

Spend no more than $50,000 on Friday:

(14) 5000DFR + 7000EFR < 50000

Spend no more than $75,000 on Saturday:

(15) 5000DSA + 7000ESA < 75000

Spend no more than $282,000 in total:

(16) 5000DFR + 5000DSA + 5000DSU + 7000EFR + 7000ESA + 7000ESU + 100000GSU7 < 282000

Non-negativity:

DFR, DSA, DSU, EFR, ESA, ESU, GSU > 0

Objective Function Value = 199000.000

Variable Value Reduced Costs

DFR = 8.000 0.000

DSA = 5.000 0.000

DSU = 2.000 0.000

EFR = 0.000 0.000

ESA = 0.000 0.000

ESU = 1.000 0.000

GSU = 2.000 0.000