# Math Help - Linear Programming question

1. ## Linear Programming question

Hi There,

I've been given this linear programming question, but I am a little stuck on where to go from here.

Your local Chamber of Commerce periodically sponsors public service seminars. In your role as the marketing manager, you are putting together an advertising plan for this year’s program of events. The promotional budget is $18,200. Advertising alternatives include television, radio and newspaper. You have obtained the following estimates for the size of an audience and cost for each type of advertisement: Television Radio Newspaper Audience per ad 100,000 18,000 40,000 Cost per ad$2,000 $300$600

Past experience suggests that at most ten TV advertisements should be aired. Similarly, no more than 10 newspaper ads will be placed. For radio, maximum usage is set at twenty advertisements. You must find the mix of advertisements that will reach a maximum audience. You have decided to use linear programming to obtain a solution to the problem before you.
3 marks) What are the decision variables for this problem?

b. (4 marks) Using decision variables identified in part (a), formulate the objective function for this problem. Is the quantity of interest to be maximized or minimized? Explain briefly.

c. (3 marks) What is the limited resource in this problem? Using decision variables from part (a), formulate the constraint for this resource.

d. (4 marks) What other constraints are relevant to this problem? Using the decision variables from part (a), formulate those constraints.

e. (4 marks) Suppose that to ensure a balanced use of advertising media, radio advertisements must not exceed 50% of the total number of advertisements authorized. Using decision variables from part (a), formulate a constraint based on this requirement. Use algebra to rewrite this constraint in the standard linear format. (Hint: A constraint like , can be transformed into , which is in the standard linear format.)

f. (4 marks) Suppose that, in addition, television should account for at least 10% of the total number of advertisements authorized. Using decision variables from part (a), formulate a constraint based on this requirement in the standard linear format. (See the Hint given in part (e).)

g. (2 marks) Give the full mathematical model for this problem.

h. (10 marks) Use EXCEL to obtain a solution to the linear programming problem from part (g), together with an Answer Report and a Sensitivity Report. Provide a printout of your EXCEL spreadsheet and of the two Excel reports with your assignment Solver submission.

EXCEL Instructions: All EXCEL output should have your e-mail ID on it. To ensure this, you will need to save your EXCEL file as ‘E-mail ID Advertising Plan.xls’. In addition, your constraint names should begin with your initials, e.g. JB_Budget.

Please refer to the Linear Programming Using Solver (supplement to the EXCEL Booklet) or the linear programming supplement to the textbook (pdf file posted on the course website) for detailed instructions on how to set up your spreadsheet and use Solver.

i. (5 marks) Suppose that you are not convinced that the estimated audience per television ad is accurate. You think it could be 10% higher or 10% lower than the initially obtained figure of 100,000 viewers per TV ad. Would the solution obtained in part (h) still be optimal? Which of the EXCEL reports helps you answer this question? Justify your answer carefully. What would be the total audience reached if the TV audience per ad were 10% lower or 10% higher?
That is the questions.

Here is what I have come up with so far.

b. Objective function: maximise audience; Audience = 10xT+10xN+20xR.

c. The limited resource is budget(money)

e. -0.5xT+0.5xR+0.5xN<=0.

from here on i am completely lost/stuck. any help would be greatly appreciated.

2. Originally Posted by Storm20
Hi There,

I've been given this linear programming question, but I am a little stuck on where to go from here.

That is the questions.

Here is what I have come up with so far.