Explanation Given: An investor makes 3 types of investments. The details can be reproduced in the following table: Types and Return of investments Oil leases Stocks Safe bond Variables x 1 x 2 x 3 50,000 % of investment combined 1 1 ≤ 30 % of investment combined 1 1 ≤ 50 Return on investment (%) 15 9 5 z Explanation The table summarizes the information of the company. Types and return of investments Oil leases Stocks Safe bond Variables x 1 x 2 x 3 50,000 % of investment combined 1 1 ≤ 30 % of investment combined 1 1 ≤ 50 Return on investment (%) 15 9 5 z The maximization problem can be stated as follows: Maximize z = 15 x 1 + 9 x 2 + 5 x 3 Subject to x 1 + x 2 + x 3 ≤ 50 , 000 x 1 + x 2 ≤ 15 , 000 x 1 + x 3 ≤ 25 , 000 with x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 0 The problem has 3 constraints, with “at most” ( ≤ ) values, for the objective function. We need 3 slack variables to make them linear equation. Adding slack variable s 1 to 15 x 1 + 10 x 2 + 8 x 3 ≤ 1500 : 15 x 1 + 10 x 2 + 8 x 3 + s 1 = 1500 Adding slack variable s 2 to 5 x 1 + 4 x 2 + 4 x 3 ≤ 2700 , 5 x 1 + 4 x 2 + 4 x 3 + s 2 = 2700 Adding slack variable s 3 to 5 x 1 + 4 x 2 + 4 x 3 ≤ 2700 : 6 x 1 + 5 x 2 + 5 x 3 + s 3 = 1200 Rewriting the objective function: − 15 x 1 − 9 x 2 − 5 x 3 + z = 0 Initial simplex tableau is as follows: [ x 1 x 2 x 3 s 1 s 2 s 3 z 1 1 1 1 0 0 0 50 , 000 1 1 0 0 1 0 0 15 , 000 1 0 1 0 0 1 0 25 , 000 − 15 − 9 − 5 0 0 0 1 0 ] The solutions from the simplex tableau are not optimum. To get maximum of the function, pivot procedure is applied till all the indictors are nonnegative and solutions are read from the tableau. Step 1: Find the largest negative indicator (number in the last row) because it indicates the variable with the largest coefficient in the objective function. It becomes the pivot column. From the above tableau, − 15 is the largest negative number. Step 2: The smallest quotient formed by dividing each number in the pivot column from the right side of the tableau of the corresponding row becomes the pivot row. Corresponding quotients: 50 , 000 1 = 50 , 000 15 , 000 1 = 15 , 000 25 , 000 1 = 25 , 000 Quotient 15,000 is the smallest, specifying 1 as the pivot in the 2nd row, 1st column: [ x 1 x 2 x 3 s 1 s 2 s 3 z 1 1 1 1 0 0 0 50 , 000 1 1 0 0 1 0 0 15 , 000 1 0 1 0 0 1 0 25 , 000 − 15 − 9 − 5 0 0 0 1 0 ] Perform row operations to make all zeros except pivot in the x 1 column

Finite Mathematics, Books a la Carte Plus MyLab Math Access Card Package (11th Edition)

11th Edition

ISBN: 9780133886818

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

Publisher: PEARSON

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Chapter 4, Problem 42RE

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Chapter 4 Solutions

Finite Mathematics, Books a la Carte Plus MyLab Math Access Card Package (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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Solve the maximization problem for the investment problem in Exercise 38.

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