Explanation Given: Let x 1 be the number of church groups and x 2 the number of unions groups contacted by Donna. From the given description, following table can be formed: Church group ( x 1 ) Labor union( x 2 ) Total Letter writing 2 hours 2 hours ≤ 16 hours Follow-up 1 hour 3 hour ≤ 12 hours Amount raised $100 $200 Explanation: The objective is to determine the number of churches and labor unions at the most money Donna De Paul can raise in a month.From the table, it can be seen that she can raise $100 from each church group and $200 from each union group. First, write the objective function as Maximize z = 100 x 1 + 200 x 2 Here x 1 and x 2 are nonnegative. Now write the constraint equation with respect to the objective function and express the constraint pertaining to the number of hours spent on letter writing as 2 x 1 + 2 x 2 ≤ 16 Add the slack variable s 1 to get 2 x 1 + 2 x 2 + s 1 = 16 Express the constraint pertaining to the number of hours spent on follow-up as 1 x 1 + 3 x 2 ≤ 12 Now add the slack variable s 2 to get x 1 + 3 x 2 + s 2 = 12 With x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 0 Therefore, the problem becomes Maximize z = 100 x 1 + 200 x 2 With respect to x 1 + x 2 + s 1 = 8 x 1 + 3 x 2 + s 2 = 12 x 1 , x 2 , s 1 , s 2 ≥ 0 Then, write the objective function − 100 x 1 − 200 x 2 + z = 0 So, the initial tableau becomes [ x 1 x 2 s 1 s 2 z 2 2 1 0 0 16 1 3 0 1 0 12 − 100 − 200 0 0 0 0 ] _ Here s 1 and s 2 are basic variables because the columns for these variables can have all zero except for one no-zero entry. Now the objective is to make x 1 and x 2 as basic variables and find the solution. [ x 1 x 2 s 1 s 2 z 2 2 1 0 0 16 1 3 0 1 0 12 − 100 − 200 0 0 0 0 ] _ Find the ratio: 16 2 = 8 , 12 3 = 4 Observe that the smallest ratio is 4 and the pivot element is 3, which lies in the second column and in the second row

11th Edition

ISBN: 9780321979438

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Chapter 4.2, Problem 23E

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The most profitable mixture of groups and the most money that can be raised in a month.

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Chapter 4.2, Problem 22E

Chapter 4.2, Problem 24E

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The most profitable mixture of groups and the most money that can be raised in a month.

Solution Summary: The author explains how Donna De Paul could contact six churches and two labor unions for a maximum of 1000 per month.

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