(a) Explanation Given: A political party planned its fund-raising activities. It plans to raise money through large fund-raising parties, letters requesting funds, and dinner parties. Each large fund-raising party costs $ 3000 , each mail costs $ 1000 , and each dinner party costs $ 12 , 000 . The party can spend up to $ 102 , 000 for these activities. From experience, the planners know that each large party will raise $ 200 , 000 , each letter campaign will raise $ 100 , 000 , and each dinner party will raise $ 600 , 000 . They perform total 25 activities. Explanation: Consider The number of parties raising large fund is equal to x 1 . The number of letters requesting funds is equal to x 2 . The number of dinner parties is equal to x 3 . The constraints from the problem can be written as x 1 + x 2 + x 3 ≤ 25 3000 x 1 + 1000 x 2 + 12 , 000 x 3 ≤ 102 , 000 The objective function is z = 200 , 000 x 1 + 100 , 000 x 2 + 600 , 000 x 3 with x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 0 Now, apply the simplex method for maximization. Step 1: Add slack variables or basic variable to each constraint to convert it into linear equations. Basic variables are the variables that are nonzero. As a result, x 1 + x 2 + x 3 + s 1 = 25 3000 x 1 + 1000 x 2 + 12 , 000 x 3 + s 2 = 102 , 000 with x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 0 Here s 1 and s 2 are basic variables greater than or equal to zero and x 1 , x 2 , x 3 are nonbasic variables. Step 2: Represent the system of linear equations obtained above in simplex tableau. This is initial tableau. [ x 1 x 2 x 3 s 1 s 2 z s 1 1 1 1 1 0 0 25 s 2 3000 1000 12000 0 1 0 102 , 000 z − 200 , 000 − 100 , 000 − 600 , 000 0 0 1 0 ] _ This tableau leads to the solution x 1 , x 2 , x 3 ≥ 0 and s 1 = 25 , s 2 = 102 , 000 with z = 0 . Step 3: Select a pivot column. Select the negative entry in the bottom row with the largest numerical value. In initial tableau, the negative entry in the bottom row with the largest numerical value is − 600 , 000 .Hence, the column corresponding to − 600 , 000 is pivot column, which follows that x 3 is enter variable. Step 4: Select the pivot. Since the pivot must be a positive number, apply test ratio for the pivot. 25 1 = 25 , 102 , 000 12 , 000 = 8.5 Therefore, 12 , 000 is pivot because the smallest positive ratio is obtained by the element of R 2 , which follows that s 2 is depart variable. Step 5: Use the pivot to make other entries in the pivot column zero. For this, first, if pivot element is not 1 , make pivot element 1 by R 2 → R 2 12 , 000 [ x 1 x 2 x 3 s 1 s 2 z s 1 1 1 1 1 0 0 25 x 3 1 4 1 12 1 0 1 12 , 000 0 51 6 z − 200 , 000 − 100 , 000 − 600 , 000 0 0 1 0 ] _ Now, make other entries in the pivot column zero (b) To determine The reason that the obtained solution is the optimum solution while dinner parties are more expensive than letter campaigns.

11th Edition

ISBN: 9780321979438

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

Publisher: PEARSON

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Chapter 4.2, Problem 40E

(a)

To determine

The maximum amount.

(b)

To determine

The reason that the obtained solution is the optimum solution while dinner parties are more expensive than letter campaigns.

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Chapter 4.2, Problem 39E

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Chapter 4 Solutions

Finite Mathematics (11th Edition)

Ch. 4.1 - Convert each inequality into an equation by adding...Ch. 4.1 - Prob. 2E Ch. 4.1 - Convert each inequality into an equation by adding...Ch. 4.1 - Prob. 4E Ch. 4.1 - For Exercises 5-8. (a) determine the number of...Ch. 4.1 - Prob. 6E Ch. 4.1 - For Exercises 5-8, (a) determine the number of...Ch. 4.1 - For Exercises 5-8, (a) determine the number of...Ch. 4.1 - Introduce slack variables as necessary, then...Ch. 4.1 - Introduce slack variables as necessary, then write...

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The maximum amount.

Solution Summary: The author explains how a political party plans to raise money through large fund-raising parties, letters requesting funds, and dinner parties. They can spend up to 102,000 for these activities.

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The maximum amount.

The reason that the obtained solution is the optimum solution while dinner parties are more expensive than letter campaigns.

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