(a) Explanation Given: A political party is planning its fund-raising activities which include fund raising parties that costs $ 3 , 000 , dinner party that cost $ 12 , 000 and each mailing costs $ 1 , 000 . Also, the party can spend up to $ 102 , 000 . From experience it is also known that each party will raise $ 200 , 000 , each dinner party will raise $ 600 , 000 and each letter campaign will raise $ 100 , 000 . And, the party is able to carry out as many as 25 of these activities. Calculation: Assume y 1 be the number of fund raising parties, y 2 be the number of mailings and y 3 be the number of dinner parties. So, maximize 200 , 000 y 1 + 100 , 000 y 2 + 600 , 000 y 3 to obtain maximum amount of money raised. Since, the party is able to carry out 25 of these activities: y 1 + y 2 + y 3 ≤ 25 Also, the party can spend up to $ 102 , 000 . Therefore, 3 , 000 y 1 + 1 , 000 y 2 + 12 , 000 y 3 ≤ 102 , 000 As the number of any type of party organized can never be negative. Therefore, y 1 ≥ 0 , y 2 ≥ 0 , and y 3 ≥ 0 Now, write the above system linear inequalities as a system of linear equations. For this add slack or subtract surplus variables as required: y 1 + y 2 + y 3 + s 1 = 25 3 , 000 y 1 + 1 , 000 y 2 + 12 , 000 y 3 + s 2 = 102 , 000 And, the equation to be maximized is: z = 200 , 000 y 1 + 100 , 000 y 2 + 600 , 000 y 3 Now, form the initial simplex tableau from the formed equation: y 1 y 2 y 3 s 1 s 2 [ 1 1 1 1 0 25 3 , 000 1 , 000 12 , 000 0 1 102 , 000 − 200 , 000 − 100 , 000 − 600 , 000 0 0 0 ] Further, take 12 , 000 as pivot in row 2 of column 3 . Perform the shown operation to convert the pivot element to 1 : y 1 y 2 y 3 s 1 s 2 [ 1 1 1 1 0 25 1 4 1 12 1 0 1 12 , 000 17 2 − 200 , 000 − 100 , 000 − 600 , 000 0 0 0 ] 1 12 , 000 R 2 Perform the shown operation to convert all elements of that column to zero except pivot element: y 1 y 2 y 3 s 1 s 2 [ 3 4 11 12 0 1 − 1 12 , 000 33 2 1 4 1 12 1 0 1 12 , 000 17 2 − 50 , 000 − 50 , 000 0 0 50 5 , 100 , 000 ] R 1 − R 2 R 3 + 600 , 000 R 2 Now, take 11 12 as pivot in row 1 of column 2 (b) To determine The optimal solution of provided problem has dinner parties but no letter campaigns even though dinner parties are more expensive than letter campaigns.

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ISBN: 8220106844557

Author: MULLINS

Publisher: PEARSON

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Chapter 7.5, Problem 13E

(a)

To determine

To calculate: Number of each type of party planned to raise the maximum amount of money. Also, the maximum amount of money raised.

(b)

To determine

The optimal solution of provided problem has dinner parties but no letter campaigns even though dinner parties are more expensive than letter campaigns.

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Chapter 7.5, Problem 12E

Chapter 7.5, Problem 14E

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