Explanation Given: The minimization function is: w = y 1 + 2 y 2 + y 3 + 5 y 4 Subjected to the system of inequality or constraints: y 1 + y 2 + y 3 + y 4 ≥ 5 3 y 1 + y 2 + 2 y 3 + y 4 ≥ 50 3 y 1 + y 2 + 2 y 3 + y 4 ≥ 100 where, y 1 ≥ 0 , y 2 ≥ 0 , y 3 ≥ 0 . Calculation: Now, the dual problem for the above function: First, find the dual problem with maximization. Find the argument matrix with a system of inequality and the coefficient of the objective function. But the negative sign is not considered then, 1 1 1 1 5 3 1 2 1 50 3 1 2 1 100 1 2 1 5 0 The above argument matrix contains first two rows as constraints and the last row shows the coefficient of the objective function. Now, find the transpose matrix of the argument matrix to find the dual problem. 1 3 3 1 1 1 1 2 1 2 2 1 1 1 1 5 5 50 100 0 Now, use ( x ) instead of ( y ) for dual problem, and the above transpose matrix shows that the first two rows as constraints and the last row is an objective function. Then, The maximization function is: z = 5 x 1 + 50 x 2 + 100 x 3 Subjected to the system of inequality or constraints: x 1 + 3 x 2 + 3 x 3 ≤ 1 x 1 + x 2 + x 3 ≤ 2 x 1 + 2 x 2 + 2 x 3 ≤ 1 x 1 + x 2 + x 3 ≤ 5 x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 0 Now, use simplex method to solve the maximization problem. The maximization function is: z = 5 x 1 + 50 x 2 + 100 x 3 Subjected to the system of inequality or constraints. x 1 + 3 x 2 + 3 x 3 ≤ 1 x 1 + x 2 + x 3 ≤ 2 x 1 + 2 x 2 + 2 x 3 ≤ 1 x 1 + x 2 + x 3 ≤ 5 x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 0 Then, use slack variables for converting the inequalities into equalities equations, then the equations are: x 1 + 3 x 2 + 3 x 3 + s 1 = 1 x 1 + x 2 + x 3 + s 2 = 2 x 1 + 2 x 2 + 2 x 3 + s 3 = 1 x 1 + x 2 + x 3 + s 4 = 5 Form the simplex table, x 1 x 2 x 3 s 1 s 2 s 3 s 4 1 3 3 1 0 0 0 1 1 1 1 0 1 0 0 2 1 2 2 0 0 1 0 1 1 1 1 0 0 0 1 5 − 5 − 50 − 100 0 0 0 0 0 The last row of the above table shows the coefficient 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

12th Edition

ISBN: 8220106844557

Author: MULLINS

Publisher: PEARSON

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Chapter 7.6, Problem 16E

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To calculate: The minimum value of the 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 15E

Chapter 7.6, Problem 17E

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