Explanation Given: The First Street branch of Capitol Bank has a sum of $ 60 million for home loans and commercial-development loans. 8 % Annual rate of return on the home loans and a 6 % annual rate of return on the commercial-development loans. Total amount of home loans is to be greater than or equal to 3 times the total amount of commercial-development loans. At least $ 10 million of the funds have been designated for commercial-development loans. Calculations: Let the amount of home loan extents represented by x and amount of commercial development loans represented by y . Since, it is given that bank estimates 8 % annual rate of return on the home loans and a 6 % annual rate of return on the commercial-development loans. So, the maximization objective function is, P = 0.08 x + 0.06 y It is given that a bank has a sum of $ 60 million for home loans and commercial-development loans. x + y = 60 Total amount of home loans is to be greater than or equal to 3 times the total amount of commercial-development loans and at least $ 10 million of the funds have been designated for commercial-development loans. Consider these given conditions, y ≥ 10 x ≥ 3 y Now the maximization problem is, Maximize P = 0.08 x + 0.06 y Subjected to x + y = 60 x − 3 y ≥ 0 y ≥ 10 The constraint x + y = 60 has two constraints. x + y ≥ 60 x + y ≤ 60 Change all the signs as ≤ . So, now problem is, Maximize P = 0.08 x + 0.06 y Subjected to x + y ≤ 60 − x − y ≤ − 60 − x + 3 y ≤ 0 − y ≤ − 10 x ≥ 0 Introduce the slack variable u and v in the given linear programming problem, x + y + u = 60 − x − y + v = − 60 − x + 3 y + w = 0 − y + r = − 10 − 0.08 x − 0.06 y + P = 0 The initial simplex tableau is, x y u v w r P Constant 1 1 1 0 0 0 0 60 − 1 − 1 0 1 0 0 0 − 60 ( − 1 ) 3 0 0 1 0 0 0 0 − 1 0 0 0 1 0 − 10 − 0.08 − 0.06 0 0 0 0 1 0 Table ( 1 ) At x = 0 , y = 0 we have u = 60 , v = − 60 , w = 0 , r = − 10. P = 0 In the initial simplex tableau the minimum value for bottom row is − 0.08 so pivot this column. The pivoted element is − 1 and now apply − R 3 . x y u v w r P Constant 1 1 1 0 0 0 0 60 − 1 − 1 0 1 0 0 0 − 60 1 − 3 0 0 − 1 0 0 0 0 − 1 0 0 0 1 0 − 10 − 0

Bundle: Finite Mathematics For The Managerial, Life, And Social Sciences, 12th + Student Solutions Manual

12th Edition

ISBN: 9781337762182

Author: Soo T. Tan

Publisher: Cengage Learning

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Chapter 4.3, Problem 23E

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Chapter 4 Solutions

Bundle: Finite Mathematics For The Managerial, Life, And Social Sciences, 12th + Student Solutions Manual

Ch. 4.1 - Give the three characteristics of a standard...Ch. 4.1 - a. When the initial simplex tableau is set up, how...Ch. 4.1 - In the simplex method, how is a pivoted column...Ch. 4.1 - In Exercises 1-6, a write the linear programming...Ch. 4.1 - In Exercises 1-6, a write the linear programming...Ch. 4.1 - In Exercises 1-6, a write the linear programming...Ch. 4.1 - In Exercises 1-6, a write the linear programming...Ch. 4.1 - In Exercises 1-6, a write the linear programming...Ch. 4.1 - Prob. 6E Ch. 4.1 - In Exercises 7-16, determine whether the given...

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The amount if each type of loan for maximum return and amount of maximum return.

Solution Summary: The author calculates the maximum return for home loans and commercial-development loans. The maximization objective function is P=0.08x+0.06y.

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The amount if each type of loan for maximum return and amount of maximum return.

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