Explanation Given: The minimization function: w = 12 y 1 + 10 y 2 + 7 y 3 Subjected to the system of inequality or constraints, 2 y 1 + y 2 + y 3 ≥ 7 y 1 + 2 y 2 + y 3 ≥ 4 And, y 1 ≥ 0 , y 2 ≥ 0 , y 3 ≥ 0 . Formula used: Calculation: First, find the dual problem with maximization. Construct the argument matrix with coefficients of system of inequality and the coefficient of the minimization function. But the negative sign is not considered then, 2 1 1 7 1 2 1 4 12 10 7 0 The above argument matrix contains first two-row 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 12 1 2 10 1 1 7 7 4 0 Now, use ( x ) instead of ( y ) for dual problem, and the above transpose matrix shows that the first three rows as coefficients of constraints and the last row show the coefficients ofdual maximization function. The objective function is z = 7 x 1 + 4 x 2 , And inequalities equations are, 2 x 1 + x 2 ≤ 12 x 1 + 2 x 2 ≤ 10 x 1 + x 2 ≤ 7 And, x 1 ≥ 0 , x 2 ≥ 0 . Now, use simplex method to solve the problem. The maximization function is: z = 7 x 1 + 4 x 2 Subjected to the system of inequality or constraints. 2 x 1 + x 2 ≤ 12 x 1 + 2 x 2 ≤ 10 x 1 + x 2 ≤ 7 And, x 1 ≥ 0 , x 2 ≥ 0 . Now, use slack variables for converting the inequalities into equalities equations then the equations are, 2 x 1 + x 2 + s 1 = 12 x 1 + 2 x 2 + s 2 = 10 x 1 + x 2 + s 3 = 7 Form the simplex table, x 1 x 2 s 1 s 2 s 3 2 1 1 0 0 12 1 2 0 1 0 10 1 1 0 0 1 7 − 7 − 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 and row-3 show the coefficient of the constraints or system of inequality. Now, find the pivot column and pivot row of the table. Select most negative value column for the pivot column. From the table most, negative value is − 7 and it lies in the column-1. Now, find the quotient to select the pivot row. Then, the value of quotient for the row-1 is, 12 2 = 6 Value of quotient for the row-2 is, 10 1 = 10 Value of quotient for the row-3 is, 7 1 = 7 Select the minimum value of the quotient to select the pivot row, Since, the minimum value of the quotient is 6 so the pivot row is row-1 and the pivot point is 2 as shown in the below table: x 1 x 2 s 1 s 2 s 3 2 1 1 0 0 12 1 2 0 1 0 10 1 1 0 0 1 7 − 7 − 4 0 0 0 0 Now, the pivot point leads the table as follow then use the operation to find the pivot point as a unit. R 1 → 1 2 R 1 , Then the table is, x 1 x 2 s 1 s 2 s 3 1 0.5 0.5 0 0 6 1 2 0 1 0 10 1 1 0 0 1 7 − 7 − 4 0 0 0 0 Now, the pivot point 1 leads the table as follow, Then, use the following operations to find the simplex table for the pivot point

Mathematics with Applications In the Management, Natural, and Social Sciences (12th Edition)

12th Edition

ISBN: 9780134767628

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

Publisher: PEARSON

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Chapter 7.6, Problem 22E

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To calculate: The minimum value of the given function w=12y1+10y2+7y3 and the point at which the value of the function is minimum by using duality theorem.

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Chapter 7.6, Problem 21E

Chapter 7.6, Problem 23E

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Chapter 7 Solutions

Mathematics with Applications In the Management, Natural, and Social Sciences (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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The minimum value of the given function w = 12 y 1 + 10 y 2 + 7 y 3 and the point at which the value of the function is minimum by using duality theorem.

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To calculate: The minimum value of the given function w=12y1+10y2+7y3 and the point at which the value of the function is minimum by using duality theorem.

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Chapter 7 Solutions