An advertising agency wishes to reach to two types of audiences- customers with annual income greater than Rs. 15000 (target audience A) and customers with annual income less than Rs. 15000 (target audience B). The total advertising budget is Rs. 2,00,000. One program of TV advertising costs Rs. 50,000 and one program of radio advertising costs Rs. 20,000. For contract reasons, atleast 3 programs ought to be on TV and number of radio programs must be limited to 5. Surveys indicate that a single TV program reaches 4,50,000 customers in target audience A and 50,000 in target audience B. One radio program reaches 20,000 in target audience A and 80,000 in target audience B. Formulate the LP problem to maximize the total reach. -

(Note: Please read the question carefully at the beginning itself. Even though it’s 10 marks, the question merely says to formulate the LP, not solve it.)

We have to form the LP to maximize the total reach.

The advertising agency reaches out to the people via TV and radio programs.

A single TV program reaches out to 4,50,000 + 50,000 = 5,00,000 customers.

A single radio program reaches out to 20,000 + 80,000 = 1,00,000 customers.

(Note: The question does not specify that via a TV program, a certain target number of customers must be reached in a PARTICULAR type of audience. Hence, we consider the total number of customers that a TV program can reach out to. There is no need to consider audiences A & B separately. Same goes for the radio program.)

Suppose X number of TV programs were made, and Y number of TV programs.

So the total number of customers these programs can reach out to is:

5,00,000 X + 1,00,000 Y

This becomes the function which has to be maximized, since the total reach has to be maximized.

So the maximization function is:

Maximize Z = 5,00,000 X + 1,00,000 Y

To set the constraints:

• Budget constraint:

  Total advertising budget available is Rs. 2,00,000

  One TV program costs Rs. 50,000; so X TV programs will cost 50,000 X

  One radio program costs Rs. 20,000; so Y radio programs will cost 20,000 Y

  So the total cost should be limited as follows:

  50,000 X + 20,000 Y ≤ 2,00,000

• Programs constraint:

  Atleast 3 programs ought to be on TV:

  X ≥ 3

  Radio programs must be limited to 5:

  Y ≤ 5

• Non-negativity constraint:

  We can’t have a negative number of programs. That makes no sense. Hence:

  X, Y ≥ 0

  So the LPP formed is as follows:

  Maximize Z = 5,00,000 X + 1,00,000 Y

  Subject to $50,000 X + 20,000 Y ≤ 2,00,000 \\ X ≥ 3 \\ Y ≤ 5 \\ X, Y ≥ 0$

  Where X, Y are the number of TV and radio programs respectively.