Explanation Given Information: The provided function is, g = 2 y 1 + y 2 And the given inequalities are, 3 y 1 + y 2 ≥ 8 y 1 + y 2 ≥ 6 2 y 1 + 5 y 2 ≥ 18 Formula used: The following method is used, Step 1. Form a dual matrix by writing the matrix A without slack variables and for the last row with positive coefficients. Step 2. Transpose matrix A to yield matrix B. Step 3. Then write the dual maximization problem by the renaming of the variables and the function. Step 4. Solve this problem with simplex method by introducing slack variables. Step 5. Use different slack variables to write each constraint inequality as an equation, on the right side with positive constants. Step 6. Set the simplex matrix. Step 7. Find out the pivot entry. Step 8. Use row operations with only the row contains the pivot entry to make the pivot entry 1 and the all other entries in the pivot column zero. Step 9. a. If there is negative indicator then return to step 7. b. If all the indicators are positive or 0, an optimum value has been obtained. Step 10. Set the non-basic variables equal to zero. Step 11. The values of the variables that will solve the dual minimization problem are the last entries in the columns which corresponds to the slack variables. Calculation: Consider the provided function and inequalities, g = 2 y 1 + y 2 subject to: 3 y 1 + y 2 ≥ 8 y 1 + y 2 ≥ 6 2 y 1 + 5 y 2 ≥ 18 The dual of this minimization problem will be the maximization problem and to form the dual, first write the matrix A for the variables in the last row without slack variables and with positive coefficients as, A = [ 3 1 8 1 1 6 2 5 18 2 1 g ] Now, transpose matrix A to yield matrix B as, B = [ 3 1 2 2 1 1 5 1 8 6 18 g ] Now write the dual maximization problem, renaming the variables and the function. Maximize f = 8 x 1 + 6 x 2 + 18 x 3 subject to: 3 x 1 + x 2 + 2 x 3 ≤ 2 x 1 + x 2 + 5 x 3 ≤ 1 Now solve this problem with simplex method by introducing the slack variables and rewriting the objective function, 3 x 1 + x 2 + 2 x 3 + s 1 = 2 x 1 + x 2 + 5 x 3 + s 2 = 1 − 8 x 1 − 6 x 2 − 18 x 3 + f = 0 Write this in a matrix with the objective function in the last row, x 1 x 2 x 3 s 1 s 2 f [ 3 1 2 1 0 0 2 1 1 5 0 1 0 1 − 8 − 6 − 18 0 0 1 0 ] Most negative value in the last row is − 18 . So, third column is the pivot column. Now divide the constants of pivot column from the last row as, 2 / 2 = 1 1 / 5 = 0.2 Here 0.2 is the smallest non-negative quotient. So, 5 is the pivot entry. The pivot entry is shown in the matrix inside brackets. Multiply the row 2 with 1/5 to get pivot entry as 1, [ 3 1 2 1 0 0 2 1 5 1 5 1 0 1 5 0 1 5 − 8 − 6 − 18 0 0 1 0 ] Now use operations with other rows to make other entries in the pivot column equal to zero. So, use the following operations, Add − 2 times row 2 to row 1, Add 18 times row 2 to row 3, [ 13 5 3 5 0 1 − 2 5 0 8 5 1 5 1 5 1 0 1 5 0 1 5 − 22 5 − 12 5 0 0 18 5 1 18 5 ] Most negative value in the last row is − 22 5

Mathematical Applications for the Management, Life, and Social Sciences

11th Edition

ISBN: 9781305108042

Author: Ronald J. Harshbarger, James J. Reynolds

Publisher: Cengage Learning

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Chapter 4, Problem 24RE

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To calculate: The minimization problem by forming the dual and using simplex method assuming that all variables are non-negative.

Minimize g=2y1+y2 subject to:

3y1+y2≥8y1+y2≥62y1+5y2≥18

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Chapter 4, Problem 23RE

Chapter 4, Problem 25RE

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Let R be the region bounded by the following curves. Let S be the solid generated when R is revolved about the given axis. If possible, find the volume of S by both the disk/washer and shell methods. Check that your results agree y = (x-4) - 1 x = 0, y = 63 revolved about the y-axis Set up the integral(s) that gives the volume of the solid as a single integral if possible using the disk/washer method. Use increasing limits of integration. Select the correct choice below and fill in any answer boxes within your choice DA SOD dy + SO UB. dy (Type exact answers.) [ C) dy (Type exact answers ) Set up the integral(s) that gives the volume of the solid as a single integral if possible using the shell method. Use increasing limits of integration. Select the correct choice below and fill in any answer boxes within your choice. OA. dx (Type exact answers.) OB. dx+ SO dx (Type exact answers.) The volume of the solid is (Type an exact answer.)

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The minimization problem by forming the dual and using simplex method assuming that all variables are non-negative. Minimize g = 2 y 1 + y 2 subject to: 3 y 1 + y 2 ≥ 8 y 1 + y 2 ≥ 6 2 y 1 + 5 y 2 ≥ 18

Solution Summary: The author explains how to calculate the minimization problem by forming the dual and using simplex method assuming that all variables are non-negative.

Videos

To calculate: The minimization problem by forming the dual and using simplex method assuming that all variables are non-negative.

Minimize g=2y1+y2 subject to:

3y1+y2≥8y1+y2≥62y1+5y2≥18

Want to see the full answer?

Chapter 4 Solutions

The minimization problem by forming the dual and using simplex method assuming that all variables are non-negative. Minimize g = 2 y 1 + y 2 subject to: 3 y 1 + y 2 ≥ 8 y 1 + y 2 ≥ 6 2 y 1 + 5 y 2 ≥ 18

Solution Summary: The author explains how to calculate the minimization problem by forming the dual and using simplex method assuming that all variables are non-negative.

Videos

To calculate: The minimization problem by forming the dual and using simplex method assuming that all variables are non-negative. Minimize g=2y1+y2 subject to: 3y1+y2≥8y1+y2≥62y1+5y2≥18

Want to see the full answer?

Chapter 4 Solutions

To calculate: The minimization problem by forming the dual and using simplex method assuming that all variables are non-negative.

Minimize g=2y1+y2 subject to:

3y1+y2≥8y1+y2≥62y1+5y2≥18