Explanation Given: Total cost of each newspaper ads, internet banners and TV ads to the boutique is $400, $20 and $2000, respectively. Total number of newspaper ads, internet banners and TV ads per month is 30, 60 and 10, respectively. Total number of women read the newspaper ads, internet banners and TV ads is 4000, 3000 and 10,000, respectively. The boutique can spend only $8000 into the advertisement. Calculations: Assume the number of advertisements be represented by x 1 , x 2 and x 3 , respectively. Construct a table in order to fetch the data from the given information as follows. Newspaper Ads Internet Banners TV Ads Money available Cost per Ad $400 $20 $2000 $8000 Maximum Number 30 60 10 Women seeing Ad 4000 3000 10,000 Obtain the first constraint as 400 x 1 + 20 x 2 + 2000 x 3 ≤ 8000 since the newspaper ads produced is 400, internet banner produced is 20 and TV ads produced is 2000 as well as total of advertisement produced by the boutique is $8000. Obtain the second constraint as x 1 ≤ 30 since 30 newspapers ads are produced per month. Obtain the third constraint as x 2 ≤ 60 since 60 internet banners are produced per month. Obtain the fourth constraint as x 4 ≤ 10 since 10 TV ads are produced per month. The objective function is, z = 4000 x 1 + 3000 x 2 + 10000 x 3 . The linear programming problem can be stated as follows. Maximize z = 4000 x 1 + 3000 x 2 + 10 , 000 x 3 subject to 400 x 1 + 20 x 2 + 2000 x 3 ≤ 8000 x 1 ≤ 30 x 2 ≤ 600 x 3 ≤ 10 with x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 0 Add slack variables to the above constraints and obtain the linear programming problem as follows. Maximize z = 4000 x 1 + 3000 x 2 + 10000 x 3 subject to 400 x 1 + 20 x 2 + 2000 x 3 + s 1 = 8000 x 1 + s 2 = 30 x 2 + s 3 = 60 x 3 + s 4 = 10 with x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 0 , s 1 ≥ 0 , s 2 ≥ 0 , s 3 ≥ 0 , s 4 ≥ 0 Rewrite the objective function as, z − 4000 x 1 − 3000 x 2 − 10 , 000 x 3 = 0

Finite Mathematics and Calculus with Applications

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ISBN: 9781323188361

Author: Margaret Lial

Publisher: Pearson Education

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Finite Mathematics and Calculus with Applications

Ch. 4.1 - Prob. 1YT Ch. 4.1 - Prob. 2YT Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Convert each inequality into an equation by adding...Ch. 4.1 - For Exercises 58, (a) determine the number of...Ch. 4.1 - Prob. 6E Ch. 4.1 - Prob. 7E Ch. 4.1 - Prob. 8E

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The linear constraints, objective function and initial simplex tableau.

Solution Summary: The author explains the linear programming problem that includes the objective function, linear constraints, and slack variables is, lmaximizez=4000x

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