Question

The Chamber of Commerce periodically sponsors public service seminars and programs. Currently, promotional plans are under way for this year's program. Advertising alternatives include television, radio, and online. Audience estimates, costs, and maximum media usage limitations are as shown:

Constraint	Television	Radio	Online
Audience per advertisement	140,000	18,000	30,000
Cost per advertisement	$1,500	$200	$550
Maximum media usage	15	19	12

To ensure a balanced use of advertising media, radio advertisements must not exceed 45% of the total number of advertisements authorized. In addition, television should account for at least 15% of the total number of advertisements authorized.

(a)	If the promotional budget is limited to $24,100, how many commercial messages should be run on each medium to maximize total audience contact?
	Advertisement Alternatives No of commercial messages Television Radio Online
	What is the allocation of the budget among the three media?
	Advertisement Alternatives Budget ($) Television $ Radio $ Online $
	What is the total audience reached?
(b)	By how much would audience contact increase if an extra $100 were allocated to the promotional budget?
	Increase in audience coverage of approximately

Answer 1

Let the no. of television advertisements be “t”, radio be “r” and online be “o”.

Here we have to maximize total audience contact. So objective function = 140,000t + 18,000r + 30,000o

This has to be maximized.

Constraints:

(i): t<= 15, r <= 19 and o <=12 (media usage constraints)

(ii): 1500t + 200r + 550o <=24,100 (amount of promotional budget)

(iii): r<=0.45*(t+r+o) [radio advertisements must not exceed 45% of the total number of advertisements authorized]

(iv): t>=0.15*(t+r+o) [radio advertisements must not exceed 45% of the total number of advertisements authorized]

Solving in excel, using the solver function, we get the following solution:

(a):

Advertisement alternatives	No. of commercial messages
Television	15.00
Radio	8.00
Online	-
Advertisement alternatives	Budget ($)
Television	22,500.00
Radio	1,600.00
Online	0

Total audience reached = 2,244,000

(b):If an extra $100 were allocated to the promotional budget then the new budget will be = 24100+100 = 24,200. In that case the scenario will become:

Advertisement alternatives	No. of commercial messages
Television	15.00
Radio	8.50
Online	-
Advertisement alternatives	Budget ($)
Television	22,500.00
Radio	1,700.00
Online	0
Objective function	2,253,000.00
Constraints
15.00	<=	15.00
8.50	<=	19.00
-	<=	12.00
24,200.00	<=	24,200.00
8.50	<=	10.58
15.00	>=	3.53

Thus increase in audience coverage = 2,253,000 (now) - 2,244,000 (earlier) = 9,000

This is considering that the variables can be in decimals. If we put additional condition that the variables has to be integers then there will be no change in the number of audience coverage.

Excel output when budget is increased by $100 and the variables need not be integers:

Advertisement alternatives	No. of commercial messages
Television	15.00
Radio	8.50
Online	-
Advertisement alternatives	Budget ($)
Television	22,500.00
Radio	1,700.00
Online	0
Objective function	2,253,000.00
Constraints
15.00	<=	15.00
8.50	<=	19.00
-	<=	12.00
24,200.00	<=	24,200.00
8.50	<=	10.58
15.00	>=	3.53