Explanation Given: The authors have $2200 for a maximum of 400 figures. The cost of a simple figure, a figure with additions, and a computer-drawn sketch is $20, $35, and $40, respectively. Computer-drawn sketches should not be more than the other two kinds combined and the simple figures should be at least double the figures with additions. Every simple figure, figure with additions, and computer-drawn sketch add $95, $200, and $325 to the royalty, respectively. The objective is to maximize the royalties. Explanation: Let the number of simple figures, figure with additions, and computer-drawn sketches be x 1 , x 2 , and x 3 , respectively. The provided information can be represented as linear inequalities as follows: Total number of figures is limited to 400: x 1 + x 2 + x 3 ≤ 400 Total cost of project is up to $2200: 20 x 1 + 35 x 2 + 60 x 3 ≤ 2200 The number of computer-drawn sketches is not more than other two types combined: x 3 ≤ x 1 + x 2 Figures with additions are at least twice as many as simple figures: 2 x 2 ≤ x 1 To maximize royalties, each simple figure, figure with additions, and computer-drawn sketch increase the royalties by 95, 200, and 325, respectively. Objective function is 95 x 1 + 200 x 2 + 325 x 3 = z Linear programming problem can now be stated as follows: Maximize z = 95 x 1 + 200 x 2 + 325 x 3 Subject to: x 1 + x 2 + x 3 ≤ 400 20 x 1 + 35 x 2 + 60 x 3 ≤ 2200 x 3 ≤ x 1 + x 2 2 x 2 ≤ x 1 with x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 0 Four slack variables are needed as there are four constraints. Convert each constraint into linear equation by introducing slack variables as follows: x 1 + x 2 + x 3 + s 1 = 400 20 x 1 + 35 x 2 + 60 x 3 + s 2 = 2200 x 3 + s 3 = x 1 + x 2 2 x 2 + s 4 = x 1 Convert the objective function and respective constraints into standard form as follows: − 95 x 1 − 200 x 2 − 325

11th Edition

ISBN: 9780321979438

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

Publisher: PEARSON

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Chapter 4.1, Problem 27E

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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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