-;J. ~>rrU>\J cl~ I 'Optimizing an Advertising CampaignMath 1010 Intermediate AlgeQ'raGroup Project. Group size is 3 to 4 students. One person isnot a group and any projJcts s~bmitted by individual students will not be graded.

Background Information:

Linear Programming is alteCi' que used for optimization of a real-world situation. Examples ofoptimization include maximiz ng the number of items that can be manufactured or minimizingthe cost of production. The eq ation that represents the quantity to be optimized is called theobjective function, since the 0 jective of the process is to optimize the value. In this project theobjective is to maximize the nhmber of people who will be reached by an advertising campaign.

The objective is subject 10 Jtations or constraints that are represented by inequalities.Limitations on the number of items that can be produced, the number of hours that workers areavailable, and the amount of td a farmer has for crops are examples of constraints that canbe represented using ine9uali es. Broadcasting an infinite number of advertisements is not arealistic goal. In this project 0 e of the constraints will be based on an advertising budget.

Graphing the system of ineqU~ties based on the constraints provides a visual representation ofthe possible solutions to the p oblem. If the graph is a closed region, it can be shown that thevalues that optimize the objeq ive function will occur at one of the "comers" of the region.

The Problem:

In this project your group will solve the following situation:

A local business plans on ad,ertising their new product by purchasing advertisements on theradio and on TV. The busine~s plans to purchase at least 70 total ads and they want to have atleast as many TV ads as radiladS. Radio ads cost $25 each and TV ads cost $100 each. Theadvertising budget is $5500. t is estimated that each radio ad will be heard by 1800 listenersand each TV ad will be seen y 2000 people. How many of each type of ad should bepurchased to maximize the number of people who will be reached by the advertisements?

1. Write down a linear-inequality for the total number of desired ads.

Modeling the Problem:

Let X be the number of radiol ads that are purchased and Y be the number of TV ads.

2. Write down a linear inequality for the cost of the ads.

1003 <: S50020X

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3. Recall that the busine s wants at lleastas many TV ads as radio ads. Write down a linearinequality that expres es this fact

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4. There are two more censtrainrs that must be met. These relate to the fact that therecannot be s negative numbers of advertisements. Write the two inequalities that modelh . IIt ese constramts:

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5. Next, write down the function for the number of people that will be exposed to theadvertisements. This is the Objective Function for the problem.

x -1- ,)OOO~

You now have four liner in qualities and an objective function. These together describe thesituation. This combined set pf inequalities and objective function make up what is knownmathematically as a linear programming problem. Write all of the inequalities and theobjective function together b~low. This is typically written as a list of constraints, with theobjective function last.

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Zsx + I 0,Y ::; S~O

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?= /lSaJ t -t- 2000 J

1-0 '1'/ -'1+'16>,~

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Z S Y, +-100 j ~S SOO

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6. To solve this pmblem~Yo u will need 10 grap~ the intersection of all five inequalitieson one common ~ pane. Do this on the grid below. Have the bottom left be the origin,with the horizonf<tI .s representing X and the vertical axis repre senting Y. Label the

.th aoorooriate umbers and verbal descriptions. and label vour lines as vou zranhL L L L J J ~ L

them. You may use our own graph paper or graphing software if you prefer.

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I t , tl I r ,I I I I I I.1 _L . L II I~ I ...••..... 1 I I I t= - I

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7. The shaded region' : the above graph is called the feasible region. Any (x, y) point in theregion corresponds 0 a possible number of radio and TV ads that will meet all therequirements pf the roblem. However, the values that will maximize the number of peopleexposed to the ads ill occur at one of the vertices or comers of the region. Your regionshould have three c rners. Find the coordinates of these comers by solving the appropriatesystem of line.ar equations. Be sure to show your work and label the (x, y) coordinates of the

. Ihcomers III your grap .

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@D,lsO) 6'?/?>sj ,(1 :!\.[1f4>1!Lj

~~ -X+--:rD

~~ --lql'l( -H5S

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f -z: 3(P,'bW + 10

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y,;::: \ ~()O (~q"j +-1-l)lP(JIt.t)

r~I"2.00 + q 9 1000& :" M l) k 1<b/'6~O

f:r I '3:"b(,b r ~\01-, ZOO

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10. Write a sentence describing how many of each type of advertisement should be purchased andwhat is the maximum number of people who will be exposed to the ad.

~Y'N~ qq tv AJS /It'\J 44 ~iC) Ms w&JJ 6e;r /'& 7;OOD

p~~1 cvsMeJ<s_ Mhd-{ ;s 7lIe &?51 l4:;sv/-f-

Your group will submit o~e project. Be sure that all of your names are on it, and be sure thateach member of the groupl approves of the final draft. Your work should be neat and easy tofollow.

Each group member will al~o need a scanned copy of the project to put in their e-Portfolio.

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