Question

For each of the following questions, explain how you reached the selected answer.

	Exposure (people/ad or commercial)	cost
television commercial	20,000	15,000
radio commercial	2,000	6,000
Newspaper ad	9,000	4,000

Budget limit $100,000, Television time for four commercials, Radio time for 10 commercials, Newspaper space for 7 ads, Resources for no more than 15 commercials and/or ads, write down your mathematical formulation and solve the problem.

Decision variables

Objective function

Constraints (please label each constraint)

Provide the values of each decision variable and provide an explanation about the result.

Answer 1

Let the no. of TV commercial be x, Radio commercial be y, and Newspaper commercial be z

Hence, Decision variables are x, y, z

Total number of people exposed = 20000*x + 2000*y + 9000*z

We have to maximize this exposure

Hence, we get the objective function as:

Maximize E = 20000*x + 2000*y + 9000*z

The constraints based on conditions given are as below:

Total cost = 15000*x + 6000*y + 4000*z

Total Budget = 100,000

Hence, 15000*x + 6000*y + 4000*z <= 100,000.............Constraint for budget

x <= 4...............................................................................Constraint for no. of TV commercials

y <= 10.............................................................................Constraint for no. of radio commercials

z <= 7................................................................................Constraint for no. of newspaper commercials

x + y + z <= 15..................................................................Constraint for available resources

We solve the given problem using excel solver as shown below:

Variables | у z 7 Objective 1,47,000 Condition !! 1,00,000 Constraints 1 100000 24 3 2 !! 10 !! !! 5 13 15 !

The above tables in the form of formulas along with Excel Solver extract are shown below for better understanding and reference:

C D Solver Parameters A 1 Variables 2 x 3 y 4 Z 4 2 7 Set objective: $B$7 Nm7ON To: O Max O Min O Value of: Objective E =20000*B2+2000*B3+9000*B4 By Changing Variable Cells: $B$2:$B$4 Condition 100000 Subject to the Constraints: $B$10:$B$14 <= $D$10:$D$14 Add 9 Constraints 10 1 11 2 12 3 13 4 14 5 = 15000*B2+6000*B3+4000*B4 =B2 =B3 =B4 =B2+B3+B4 <= 10 Change <= 15 Delete Reset All Load/Save Make Unconstrained variables Non-Negative Select a Solving Method: Simplex LP Options Solving Method Select the GRG Nonlinear engine for Solver Problems that are smooth nonlinear. Select the LP Simplex engine for linear Solver Problems, and select the Evolutionary engine for Solver problems that are non-smooth. Help Solve Close

As seen from above,

No. of TV ads = 4 commercials

No. of Radio ads = 2 commercials

No. of Newspaper ads = 7 commercials

Total No. of ads = 4 + 2 + 7 = 13 commercials

Still, the no. of Radio ads can be increased and this constraint is not fully exploited as the budget of $100,000 is reached. If the budget is increased further, no. of radio ads will be increased. Constraints for TV ads and Newspaper ads is already reached to maximum value.