Explanation Given: The objective function to be minimized is C = 2 x + 3 y . The constraints are, x + y ≤ 10 x + 2 y ≥ 12 2 x + y ≥ 12 x ≥ 0 , y ≥ 0 Approach: (1) If all the entries are nonnegative, the optimal solution has been reached. (2) If there are one or more negative entries, the optimal solution has not been reached. Calculation: Convert the given problem into dual problem. Convert the given problem into dual problem Write the given constraints in tabular form. x y Constant − 1 − 1 − 10 1 2 12 2 1 12 2 3 Now interchange the columns and rows of the above table and head the two columns of the resulting array with the two variables u and v . u v w Constant − 1 1 2 2 − 1 2 1 3 − 10 12 12 The dual problem for the given objective function and the constraints is, Maximize P = − 10 u + 12 v + 12 w Subjected to, − u + v + 2 w ≤ 2 − u + 2 v + w ≤ 3 u ≥ 0 , v ≥ 0 , w ≥ 0 Introducing slack variable x and y Construct the initial simplex table. u v w x y P Constant − 1 1 2 1 0 0 2 − 1 2 1 0 1 0 3 10 − 12 − 12 0 0 1 0 Calculate the pivot element. The most negative element in the last row is − 12 which is in column 2 and 3 . Selecting column 2 as the pivot column. Calculate the pivot row. u v w x y P Constant Ratio − 1 1 2 1 0 0 2 2 1 = 2 − 1 2 1 0 1 0 3 3 2 = 1.5 10 − 12 − 12 0 0 1 0 The least ratio is 1.5 that lies in second row. So select the pivot row as the second row. Hence, the element corresponding to pivot column and the pivot row is 2 . Apply the row operation R 2 → 1 2 R 2 in the above matrix. u v w x y P Constant − 1 1 2 1 0 0 2 − 1 2 1 1 2 0 1 2 0 3 2 10 − 12 − 12 0 0 1 0 Apply operation R 1 → R 1 − R 2 and R 3 → R 3 + 12 R 2 u v w x y P Constant − 1 2 0 3 2 1 − 1 2 0 1 2 − 1 2 1 1 2 0 1 2 0 3 2 4 0 − 6 0 6 1 18 The most negative element in the last row is − 6 which is in column 3

Finite Mathematics for the Managerial, Life, and Social Sciences, 11th Edition

11th Edition

ISBN: 9781285464657

Author: Soo T. Tan

Publisher: Brooks Cole

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Chapter 4.CRE, Problem 16CRE

To determine

To find:

The solution to the minimization problem.

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Chapter 4 Solutions

Finite Mathematics for the Managerial, Life, and Social Sciences, 11th Edition

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 solution to the minimization problem.

Solution Summary: The author explains the solution to the minimization problem. The objective function is C=2x+3y.

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To find: The solution to the minimization problem.

Chapter 4 Solutions

To find:

The solution to the minimization problem.