Question

2.    (18 pts) (Media selection) The Samulson Co. has a promotional plan for its new product. Advertising alternatives include television, radio, magazine and newspaper. Audience estimates, costs, and maximum media usage are as shown.

Constraint	Television	Radio	Magazine	Newspaper
Audience reached per ad	1,600	1,300	1,000	2,200
Cost ($) per ad	1,400	400	700	1,200
Maximum media available	40	45	55	70

         

Samulson has the promotional budget of $175,000. To ensure a balanced use of advertising media, radio advertisements must not exceed 35% of the total number of advertisements. Television should account for at least 30% of the total number of advertisements. Television and radio together may not exceed twice of the total of magazine and newspaper advertisements. In addition, the company has decided to use at least 20 ads in each media.

              Set up a linear programming model to maximize audience reached.

Answer 1

Decision Variables:

Let,

Number of Television ads = x1

Number of Radio ads = x2

Number of Magazine ads = x3

Number of Newspaper ads = x4

Objective Function:

Our objective is to maximize the audience reached.

Audience reached per unit ad on Television = 1600

Audience reached per unit ad on Radio = 1300

Audience reached per unit ad on Magazine = 1000

Audience reached per unit ad on Newspaper = 2200

So, the objective function is:

MAXIMIZE 1600*x1 + 1300*x2 + 1000*x3 + 2200*x4

Constraints:

The budget is $175,000. The total cost should not exceed the budget.

Cost per ad on Television = $1400

Cost per ad on Radio = $400

Cost per ad on Magazine = $700

Cost per ad on Newspaper = $1200

So,

1400*x1 + 400*x2 + 700*x3 + 1200*x4 <= 175000

Radio ads must not exceed 35% of total ads

x2 <= 0.35*(x1 + x2 + x3 + x4)

0.65*x2 – 0.35*x1 – 0.35*x3 – 0.35*x4 <= 0

Television should account for at least 30% of total ads.

x1 >= 0.30*(x1 + x2 + x3 + x4)

0.7*x1 – 0.3*x2 – 0.3*x3 – 0.3*x4 >= 0

Television and radio together may not exceed twice the total of magazine and newspaper ads.

x1 + x2 <= 2*(x3 + x4)

x1 + x2 – 2*x3 – 2*x4 <= 0

At least 20 ads in each media should be used.

x1, x2, x3, x4 >= 20

There is a maximum cap on number of media usage.

x1 <= 40

x2 <= 45

x3 <= 55

x4 <= 70

Number of ads cant be fractional.

x1, x2, x3, x4 = integer

Putting in Excel:

Solver Parameters:

Optimal Solution:

Television	Radio	Magazine	Newspaper
x1	x2	x3	x4
40	20	20	53

Number of Television ads = 40

Number of Radio ads = 20

Number of Magazine ads = 20

Number of Newspaper ads = 53

Optimal number of viewers = 226600

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