Please provide a linear graph with details for the following problem/solution: h

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Please provide a linear graph with details for the following problem/solution:

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Case Problem: Advertising Strategy

Midtown Motors, Inc. has hired a marketing services firm to develop an advertising strategy for promoting Midtowns used-car sales. The marketing firm has recommended that Midtown use spot announcements on both television and radio as the advertising media for the proposed promotional campaign. Advertising strategy guidelines are expressed as follows:

1. Use at least 30 announcements for combined television and radio coverage.

2. Do not use more than 25 radio announcements.

3. The number of radio announcements cannot be less than the number of television announcements.

The television station has quoted a cost of $1200 per spot announcement, and the radio station has quoted a cost of $300 per spot announcement. Midtowns advertising budget has been set at $25,500. The marketing services firm has rated the various advertising media in terms of audience coverage and recall power of the advertisement. For Midtowns media alternatives, the television announcement is rated at 600, and the radio announcement is rated at 200. Midtowns president would like to know how many television and how many radio spot announcements should be used to maximize the overall rating of the advertising campaign.

?Midtowns president believes the television station will consider running the Midtown spot announcement on its highly rated evening news program (at the same cost) if Midtown will consider using additional television announcements.

Managerial Report

Perform an analysis of advertising strategy for Midtown Motors, and prepare a report to Midtowns president presenting your findings and recommendations. Include a consideration of the following:

1. The recommended number of television and radio spot announcements.

2. The relative merits, financial worthiness, of each advertising medium.

3. The rating that would be necessary for the news program before it would make sense to increase the number of television spots.

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Explanation / Answer

It is a linear programming problem. An advertisement campaign is being planned. Your objective is to maximize number of responses. You require announcement through Radio and Television.

Let,

Z= Total response

X1= Number of radio announcement

X2=Number of Television advertisement.

Then, linear programming formulation is:

Maximize:

Z= 200X1+600X2   {Total response from announcement. It is objective function}[

Subject to:

X1+X2 30            {Total announcement. It should be at least 30}

X1       25             { Radio announcement should not be below 25}

X1       X2                  { Radio announcement must be greater than Television announcement}

300X1+1200X2 25,500 {Total advertisement cost should be within $25,500}

X1 0; X2 0                   { Non negative restriction to make the solution feasible}

Let radio announcement X1 is measured in X axis and Television announcement in Y axis. It can be solved graphically.

Your diagram should be within first quadrant to satisfy non negative restrictions.

Now consider each restriction and identify extreme points on X and Y axis. For this substitute X1= 0 or X2 = 0. The results are:

If X1= 0 the X2= 30. It is point A.

If X2 = 0 then X1 = 30 . It is point B.

Join these two points to get line AB. Any area on or above this line line will satisfy this restriction.

It is shown by a vertical line drawn from X1=25. Any point on or to the left of this line will satisfy it.

It is shown by a line starting from origin and constituting an angle 45 degree with x axis. Any point on or its right will satisfy it.

If X1= 0 then X2=25,500/1,200 = 21.25 Let it as point E

If X2 = 0 then X1 = 25,500 / 300 = 85 Let it as point F

Join these two points to get line EF. Any point on or below it will satisfy this restriction.

Now after drawing all the diagrams described above, you will a get a common area PQRS, which will satisfy all these restrictions. Your optimum solution must be at any of these four corner points.

So next activity is to identify corner points. They can be identified by solving equations of two lines intersecting at the corner point. Calculations are shown below:

Here you have to solve following two equations:

X1=X2 and

X1+X2 = 30.

Solution is X1= 15; X2 = 15

It can be ascertained by solving following two equations:

300X1+1200X2 =25,500

X1 = X2

Answer is X1= 17, X2 = 17

It is ascertained by solving following two equations:

300X1+1200X2 =25,500

X1 = 25

At X1 = 25 , X2 is (25,500 – 300 x 25)/1200 = 15

Finally this point can be ascertained by solving equations

X1 +X2 = 30 and

X1 = 25.

At X2 = 25 the value of X1 is 5.

Now substitute these points value in the objective function to identify highest Z value. It is shown below:

At point P : Z = 200 x 15 +600 x 15 = 3,000 + 9,000 = 12,000

At point Q: Z = 200 x 17 + 600 x 17 = 3,400 + 10,200 = 13,600

At point R: Z = 200 x 25 + 600 x 15 = 5,000 + 9,000 = 14,000

At point S: Z = 200 x 25 + 600 x 5 = 5,000 + 3,000 = 8,000

Answer:

Number of radio announcement = X1 = 25

Number of Television announcement = X2 = 15

Maximum value of Z = 800

Diagram is shown below:

In the above diagram, area PQRS will satisfy all restrictions. Point R is optimum solution

2. Financial worthiness will mean total rate contributed by each medium. Here contribution of Radio announcement will be 200 x 25 = 5,000 and of TV announcement will be 600 x 15 = 9,000 . Total is 14,000 as mentioned above.

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