Explanation Given: The minimization function: w = 6 y 1 + 4 y 2 + 2 y 3 Subjected to the system of inequality or constraints. 2 y 1 + 2 y 2 + y 3 ≥ 2 y 1 + 3 y 2 + 2 y 3 ≥ 3 y 1 + y 2 + 2 y 3 ≥ 4 And, y 1 ≥ 0 , y 2 ≥ 0 , y 3 ≥ 0 , y ≥ 0 . Calculation: First, find the dual problem with maximization. Construct the argument matrix with coefficients of system of inequality and the coefficient of the objective function.But the negative sign is not considered then, 2 2 1 2 1 3 2 3 1 1 2 4 6 4 2 0 The above argument matrix contains first three rows as the coefficients of constraints and the last row shows the coefficient of the minimization function. Now, find the transpose matrix of the argument matrix to find the dual problem. 2 1 1 6 2 3 1 4 1 2 2 2 2 3 4 0 Now, use ( x ) instead of ( y ) for dual problem, and the above transpose matrix shows that the first three as the coefficients constraints and the last row as the coefficients of the dual maximization problem. Then, The maximization function is z = 2 x 1 + 3 x 2 + 4 x 4 , And inequalities equations are, 2 x 1 + x 2 + x 3 ≤ 6 2 x 1 + 3 x 2 + x 3 ≤ 4 x 1 + 2 x 2 + 2 x 3 ≤ 2 And, x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 0 . Now, use simplex method to solve the maximization problem. The maximization function is: z = 2 x 1 + 3 x 2 + 4 x 4 Subjected to the system of inequality or constraints. 2 x 1 + x 2 + x 3 ≤ 6 2 x 1 + 3 x 2 + x 3 ≤ 4 x 1 + 2 x 2 + 2 x 3 ≤ 2 And, x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 0 . Now, use slack variables for converting the inequalities into equalities equations then the equations are, 2 x 1 + x 2 + x 3 + s 1 = 6 2 x 1 + 3 x 2 + x 3 + s 2 = 4 x 1 + 2 x 2 + 2 x 3 + s 3 = 2 Form the simplex table, x 1 x 2 x 3 s 1 s 2 s 3 2 1 1 1 0 0 6 2 3 1 0 1 0 4 1 2 2 0 0 1 2 − 2 − 3 − 4 0 0 0 0 The last row of the above table shows the coefficients of the objective function and the row-1, row-2, row-3 and row-4 show the coefficient of the constraints or system of inequality

Mathematics With Applications, Loose-leaf Edition Plus Mylab Math With Pearson Etext -- 18-week Access Card Package (12th Edition)

12th Edition

ISBN: 9780135998304

Author: Margaret L. Lial, Tom Hungerford, John P. Holcomb, Bernadette Mullins

Publisher: PEARSON

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Chapter 7.6, Problem 21E

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To calculate: The minimum value of the given function and the point at which the value of the function is minimum by using duality theorem.

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Chapter 7.6, Problem 20E

Chapter 7.6, Problem 22E

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Mathematics With Applications, Loose-leaf Edition Plus Mylab Math With Pearson Etext -- 18-week Access Card Package (12th Edition)

Ch. 7.1 - Checkpoint 1 Graph the given...Ch. 7.1 - Prob. 2CP Ch. 7.1 - Prob. 3CP Ch. 7.1 - Prob. 4CP Ch. 7.1 - Prob. 5CP Ch. 7.1 - Checkpoint 6 Graph the feasible region of the...Ch. 7.1 - Prob. 7CP Ch. 7.1 - Match the inequality with its graph, which is one...Ch. 7.1 - Match the inequality with its graph, which is one...Ch. 7.1 - Match the inequality with its graph, which is one...

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