(a) Explanation Given: The large fund-raising party costs $3000, mailing costs $1000, and dinner party costs $12000. Total money spend by the party is $102000. By past experience, each large party will raise $200000, each letter campaign will raise $100000, and each dinner party will raise $600000. The total number of activities is 25. Method used: “Simplex method: 1. Determine the objective function. 2. Write all the necessary constraints. 3. Convert each constraint into an equation by adding a slack variable in each. 4. Set up the initial simplex tableau. 5. Locate the most negative indicator. If there are two such indicators, choose the one farther to the left. 6. Form the necessary quotients to find the pivot. Disregard any quotients with 0 or a negative number in the denominator. The smallest nonnegative quotient gives the location of the pivot. If all quotients must be disregarded, no maximum solution exists. If two quotients are both equal and smallest, choose the pivot in the row nearest the top of the matrix. 7. Use row operations to change all other numbers in the pivot column to zero by adding a suitable multiple of the pivot row to a positive multiple of each row. 8. If the indicators are all positive or 0, this is the final tableau. If not, go back to step 5 and repeat the process until a tableau with no negative indicators is obtained. 9. Read the solution from the final tableau.” Calculation: The political party is planning its fund-raising activities for a coming election is listed below in table. Large Party Letter Party Dinner Part Total Available Fund Cost in ($) $3000 $1000 $12000 $102,000 1 1 1 25 Amount Raise $200,000 $100,000 $600,000 Let us assume that x 1 be the number of large fund-raising parties, x 2 be the number of letter requesting funds, and x 3 be the number of dinner parties. Observe that each large party will raise $200,000, each letter campaign will raise $100,000, and each dinner party will raise $600,000. Thus, the objective function that raises the maximum amount of money for the political party is, Maximum : z = 200 , 000 x 1 + 100 , 000 x 2 + 600 , 000 x 3 . Construct the constraints for the given model as follows. The political party are able to carry out a total of 25 of these activities. That is the first constraint is x 1 + x 2 + x 3 ≤ 25 . Observe that each large fund-raising party costs $3000, each mailing costs $1000, and each dinner party costs $12000, and the party can spend up to $102,000 for these activities. Hence, the constraint is 3000 x 1 + 1000 x 2 + 12 , 000 x 3 ≤ 102 , 000 . Combine the objective function and constraints and obtain the mathematical form of the linear programming problem is as follows. Maximum : z = 200 , 000 x 1 + 100 , 000 x 2 + 600 , 000 x 3 subject t o : x 1 + x 2 + x 3 ≤ 25 3000 x 1 + 1000 x 2 + 12 , 000 x 3 ≤ 102 , 000 with x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 0. Use the simplex algorithm and obtain the solution of the linear programming problem as follows. Add the slack variable s 1 in the first constraint, s 2 in the second constraint and rewrite the original problem in equality constraints linear programming problem as follows. Maximum : z = 200 , 000 x 1 + 100 , 000 x 2 + 600 , 000 x 3 subject t o : 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 , s 1 ≥ 0 , s 2 ≥ 0. (1) The system (1) can be written in the system of linear equation as, x 1 + x 2 + x 3 + s 1 = 25 3000 x 1 + 1000 x 2 + 12 , 000 x 3 + s 2 = 102 , 000 − 200 , 000 x 1 − 100 , 000 x 2 − 600 , 000 x 3 + z = 0 Construct the initial simplex tableau form the above linear system as follows. [ x 1 x 2 x 3 s 1 s 2 z 1 1 1 1 0 0 25 3000 1000 12 , 000 0 1 0 102 , 000 − 200 , 000 − 100 , 000 − 600 , 000 0 0 1 0 ] Look the 3 × 3 identity matrix (b) To determine To explain: The possibility that dinner parties are more expensive than letter campaigns, yet the optimum solution found in part (a) includes dinner parties but no letter campaigns.

Finite Mathematics and Calculus with Applications

1st Edition

ISBN: 9781323188361

Author: Margaret Lial

Publisher: Pearson Education

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Chapter 4.2, Problem 40E

(a)

To determine

To find: The number of each party plan in order to raise the maximum amount of money, and also determine the maximum amount.

(b)

To determine

To explain: The possibility that dinner parties are more expensive than letter campaigns, yet the optimum solution found in part (a) includes dinner parties but no letter campaigns.

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Chapter 4.2, Problem 39E

Chapter 4.3, Problem 1YT

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Chapter 4 Solutions

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 number of each party plan in order to raise the maximum amount of money, and also determine the maximum amount.

Solution Summary: The author calculates the total amount of money raised by a political party.