1010 optimization project

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Optimizing an Advertising Campaign Math 1010 Intermediate Algebra Group Project Background Information: Linear Programming is a technique used for optimization of a real-world situation. Examples of optimization include maximizing the number of items that can be manufactured or minimizing the cost of production. The equation that represents the quantity to be optimized is called the objective function, since the objective of the process is to optimize the value. In this project the ob
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  Optimizing an Advertising Campaign Math 1010 Intermediate Algebra Group Project Background Information: Linear Programming is a techniue used !or optimization o! a real #orld situation$ %&amples o! optimization include ma&imizing the number o! items that can be manu!actured or minimizing the cost o! production$ 'he euation that represents the uantit( to be optimized is called the objective !unction) since the objective o! the process is to optimize the value$ In this project the objective is to ma&imize the number o! people #ho #ill be reached b( an advertising campaign$'he objective is subject to limitations or constraints that are represented b( ineualities$ Limitations on the number o! items that can be produced) the number o! hours that #or*ers are available) and the amount o! land a !armer has !or crops are e&amples o! constraints that can be represented using ineualities$ +roadcasting an in!inite number o! advertisements is not a realistic goal$ In this project one o! the constraints #ill be based on an advertising budget$Graphing the s(stem o! ineualities based on the constraints provides a visual representation o! the possible solutions to the problem$ I! the graph is a closed region) it can be sho#n that the values that optimize the objective !unction #ill occur at one o! the ,corners, o! the region$ The Problem: In this project (our group #ill solve the !ollo#ing situation-A local business plans on advertising their ne# product b( purchasing advertisements on the radio and on '.$ 'he business plans to purchase at least /0 total ads and the( #ant to have at least t#ice as man( '. ads as radio ads$ adio ads cost 20 each and '. ads cost 30 each$ 'he advertising budget is 4520$ It is estimated that each radio ad #ill be heard b( 2000 listeners and each '. ad #ill be seen b( 1600 people$ 7o# man( o! each t(pe o! ad should be  purchased to ma&imize the number o! people #ho #ill be reached b( the advertisements8 Modeling the Problem: Let 9 be the number o! radio ads that are purchased and : be the number o! '. ads$1$ ;rite do#n a linear ineualit( !or the total number o! desired ads$r < adio Ads) t < '. Ads=r > t ?< /0  2$;rite do#n a linear ineualit( !or the cost o! the ads$20r > 30t @< 45205$ecall that the business #ants at least t#ice as man( '. ads as radio ads$ ;rite do#n a linear ineualit( that e&presses this !act$r @< 2t4$'here are t#o more constraints that must be met$ 'hese relate to the !act that there cannot be negative numbers o! advertisements$ ;rite the t#o ineualities that model these constraints-r ?< 0) t ?< 06$e&t) #rite do#n the !unction !or the number o! people that #ill be e&posed to the advertisements$ 'his is the Objective Bunction !or the problem$   P   = 2000r + 1500t :ou no# have !ive linear ineualities and an objective !unction$ 'hese together describe the situation$ 'his combined set o! ineualities and objective !unction ma*e up #hat is *no#n mathematicall( as a linear programming  problem$ ;rite all o! the ineualities and the objective !unction together belo#$ 'his is t(picall( #ritten as a list o! constraints) #ith the objective !unction last$ r + t >= 6020r + 80t <= 4320r <= 2tr >= 0t >= 0  P   = 2000r + 1500t  /$'o solve this problem) (ou #ill need to graph the intersection  o! all !ive ineualities onone common 9: plane$ o this on the grid belo#$ 7ave the bottom le!t be the srcin) #ith the horizontal a&is representing 9 and the vertical a&is representing :$ Label the a&es #ith #hat the( represent and label (our lines as (ou graph them$ (56, 40)(60, 30)(40, 20)    Y Axis = t (TV Ads)X Axis = r (Radio Ads)  D$'he shaded region in the above graph is called the !easible region$ An( E&) (F point in the region corresponds to a possible number o! radio and '. ads that #ill meet all the reuirements o! the  problem$ 7o#ever) the values that #ill ma&imize the number o! people e&posed to the ads #ill occur at one o! the vertices or corners o! the region$ :our region should have three corners$ Bind the coordinates o! these corners b( solving the appropriate s(stem o! linear euations$ +e sure to  show your work   and label   the E&) (F coordinates o! the corners in (our graph$ r + t >= 60 | t = 20 | r >= 60  t | r >= 60  20 | r >= 40!or #r $1 = (40, 20)20r + 80t <= 4320 | 20r <= 4320  80t | r <= (4320  80t)20 | r <= 216  4t | t = 40 | r <= 216  4(40) | r <= 216  160 | r <= 56!or #r $2 = (56, 40) %o#s &t 'orr <= 2t | t = 30 | r <= 2(30) | r <= 60!or #r $3 = (60, 30) 3$'o !ind #hich number o! radio and '. ads #ill ma&imize the number o! people #ho are e&posed to the business advertisements) evaluate the objective !unction P !or each o! the vertices (ou !ound$ ho# (our #or*$ * = 2000r + 1500t(40, 20) | * = 2000(40) + 1500(20) | * = 80,000 + 30,000 | * = 110,000(60, 30) | * = 2000(60) + 1500(30) | * = 120,000 + 45,000 | * = 165,000 H$;rite a sentence describing ho# man( o! each t(pe o! advertisement should be purchased and #hat is the ma&imum number o! people #ho #ill be e&posed to the ad$  According to my calculations, 60 radio ads, and 30 TV ads should be purchased for a maximum audience os 6!,000 people #owe$er, my linear ine%uality for the cost of the ads did not work with the other criteria pro$ided
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