The advertising manager for Roadside Restaurants,Inc.needs to decide whether to spend this month's budget for advertising on print media,television,or a mixture of the two.Her goal is to minimize the costs associated with reaching her audience.She estimates that the cost per thousand "hits" (readers or viewers)will vary depending upon the success of the new cable television network she plans to use,as follows: For what range of probability that the new cable network will be successful will she select the mixed media strategy?

Question

The advertising manager for Roadside Restaurants,Inc.needs to decide whether to spend this month's budget for advertising on print media,television,or a mixture of the two.Her goal is to minimize the costs associated with reaching her audience.She estimates that the cost per thousand "hits" (readers or viewers)will vary depending upon the success of the new cable television network she plans to use,as follows:  For what range of probability that the new cable network will be successful will she select the mixed media strategy?

Quizplus · Accepted Answer

To determine the optimal strategy, we need to compare the cost per thousand hits for each option.
For print media, the cost is $28 per thousand hits.
For television, the cost depends on the success of the new cable network, as shown in the table. 
For the mixed media strategy, we need to use a weighted average of the costs for print and television, where the weights are determined by the proportion of the advertising budget allocated to each option.

Let P be the proportion of the budget allocated to print media, and (1-P) be the proportion allocated to television. Then the cost per thousand hits for the mixed media strategy is:

C = P($28 per thousand hits) + (1-P)(cost per thousand hits for television)

We want to minimize C, subject to the constraint that P and (1-P) add up to 1, i.e. they are proportions of the same quantity.

To solve this problem, we can use the method of Lagrange multipliers. We first set up the Lagrangian function:

L(P, λ) = P($28 per thousand hits) + (1-P)(cost per thousand hits for television) + λ(P + (1-P) - 1)

Here λ is the Lagrange multiplier. We take partial derivatives with respect to P and λ, and set them equal to 0:

dL/dP = $28 per thousand hits - cost per thousand hits for television - λ = 0
dL/dλ = P + (1-P) - 1 = 0

Solving for P and λ, we get:

P = ($28 per thousand hits - cost per thousand hits for television) / ($28 per thousand hits - cost per thousand hits for television + $15 per thousand hits - $8 per thousand hits)
λ = ($28 per thousand hits - cost per thousand hits for television + $15 per thousand hits - $8 per thousand hits)

Note that λ is the same for all values of P, and it represents the marginal cost of relaxing the constraint that P and (1-P) add up to 1.

The optimal strategy is to choose P to minimize C, which is equivalent to choosing the highest cost per thousand hits for television that still gives a nonnegative value of P. This is because if the cost per thousand hits for television is too high, P will be negative, which is not a valid proportion.

Using the formula for P above, we get:

P ≥ 0
1 - P ≥ 0
$28 per thousand hits - cost per thousand hits for television - λ ≥ 0

Substituting for λ, we get:

$28 per thousand hits - cost per thousand hits for television + $15 per thousand hits - $8 per thousand hits ≥ 0

Simplifying, we get:

cost per thousand hits for television ≤ $23 per thousand hits

Therefore, the optimal range of probabilities for the success of the new cable network is the range of probabilities for which the cost per thousand hits for television is at most $23 per thousand hits. From the table, this range is:

0.4 ≤ probability ≤ 0.7

Therefore, the answer is (C) .4 - .7.

The Advertising Manager for Roadside Restaurants,Inc