Question

1. The Westchester 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	100000	18000	20000
   Cost per advertisement	$2000	$300	$600
   Maximum media usage	10	20	10

   To ensure a balanced use of advertising media, radio advertisements must not exceed 50% of the total number of advertisements authorized. In addition, television should account for at least 10% of the total number of advertisements authorized.

   1. If the promotional budget is limited to $29,300, how many commercial messages should be run on each medium to maximize total audience contact? What is the allocation of the budget among the three media? If required, round your answers to the nearest dollar.
      

      Let	T = number of television spot advertisements
      	R = number of radio advertisements
      	O = number of online advertisements

      	Budget ($)
      T =
      R =
      O =
      	Total Budget = $

      
      What is the maximum total audience that would be reached? Round your answer to the nearest whole number.
      
      
      
   2. By how much would audience contact increase if an extra $100 were allocated to the promotional budget? Round your answer to the nearest whole number.
      

Answer 1

Objective function:

Objective function is to maximize the audience reach

Z = max 100000T + 18000R + 20000O

Constraints:

2000T + 300R + 600O <= 29,300 (Budget)

Subject to,

2000T+300R+600O <= 29300 (Promotional budget constraint)

T <= 10 (Max media - Television)
R<= 20 (Max media - Radio)
O <= 10 (Max media - Online)

R<=(T+R+O)*0.5

or, -0.5T+0.5R-0.5O <= 0 (Radio <= 50% of total ads)

T >= 0.1*(T+R+O) (TV >= 10% of total ads)

or, 0.9T-0.1R-0.1O >= 0

T,R,O >= 0 (non-negativity constraint)

Solving in solver we get,

T = number of television spot advertisements = 10
R = number of radio advertisements = 17
O = number of online advertisements = 7

	Budget
T	20000
R	5100
O	4200

Maximum total audience can be reached = 1446000

b)

Shadow price = 42.22

So,

For each $100 increase in budget, the audience will increase by 100 * 42.22 = 4,222