Explanation Given Information: The provided linear programming problem is, Minimize g = 12 y 1 + 8 y 2 + 10 y 3 subject to y 1 + 2 y 3 ≥ 10 y 1 + y 2 ≥ 12 2 y 1 + 2 y 2 + y 3 ≥ 8 Formula used: The Simplex Method for Problems in Standard Maximization Form 1. Introduce slack variables, and rewrite the given problem in terms of a system of linear equations. 2. Construct the simplex tableau corresponding to the system. 3. Determine if the left part of the lowest row contains negative entries. If there is no negative entry, the solution corresponding to the tableau gives a maximum and the problem is answered. 4. If the left part of the lowest row contains negative entries, construct a new simplex tableau. (a) Choose the pivot column by examining the entries of the lowest row of the current tableau, not including the right-hand entry. The pivot column is the one containing the most negative of these entries. (b) Choose the pivot element by computing ratios associated with the positive entries of the pivot column and the right column. The pivot element is the one corresponding to the smallest nonnegative ratio. (c) Construct the new simplex tableau by pivoting around the selected element. 5. Return to step 3 . Repeat steps 3 and 4 as many times as necessary to find a maximum. 6. Interpret the result. Calculation: Consider the provided linear programming problem, Minimize g = 12 y 1 + 8 y 2 + 10 y 3 subject to y 1 + 2 y 3 ≥ 10 y 1 + y 2 ≥ 12 2 y 1 + 2 y 2 + y 3 ≥ 8 The constraints can be written in matrix form as follows: [ 1 0 2 10 1 1 0 12 2 2 1 8 12 8 10 g ] Now transpose the given matrix. This will convert into: [ 1 1 2 12 0 1 2 8 2 0 1 10 10 12 8 g ] Introduce new slack variables x 1 , x 2 , x 3 to write the problem as: Maximize: f = 10 x 1 + 12 x 2 + 8 x 3 subject to constraints x 1 + x 2 + 2 x 3 ≤ 12 x 2 + 2 x 3 ≤ 8 2 x 1 + x 3 ≤ 10 Now the matrix with slack variables will become: [ 1 1 2 1 0 0 0 12 0 1 2 0 1 0 0 8 2 0 1 0 0 1 0 10 − 10 − 12 − 8 0 0 0 1 0 ] As − 12 is the most negative entry, column 2 is the pivot column. Now, find the quotient to select the pivot row. Value of quotient for the row 1 is, 12 1 = 12 Value of quotient for the row 2 is, 8 1 = 8 Value of quotient for row 3 is, 10 0 = not defined The minimum of 12 and 8 is 8, so, the pivot row is the row 2

Mathematical Applications for the Management, Life, and Social Sciences

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ISBN: 9781305108042

Author: Ronald J. Harshbarger, James J. Reynolds

Publisher: Cengage Learning

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Chapter 4.4, Problem 12E

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To calculate: The minimization of the function g=12y1+8y2+10y3 subject to

y1+2y3≥10y1+y2≥122y1+2y2+y3≥8

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Chapter 4.4, Problem 11E

Chapter 4.4, Problem 13E

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The minimization of the function g = 12 y 1 + 8 y 2 + 10 y 3 subject to y 1 + 2 y 3 ≥ 10 y 1 + y 2 ≥ 12 2 y 1 + 2 y 2 + y 3 ≥ 8

Solution Summary: The author explains the Simplex Method for Problems in Standard Maximization Form.

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To calculate: The minimization of the function g=12y1+8y2+10y3 subject to

y1+2y3≥10y1+y2≥122y1+2y2+y3≥8

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Chapter 4 Solutions

The minimization of the function g = 12 y 1 + 8 y 2 + 10 y 3 subject to y 1 + 2 y 3 ≥ 10 y 1 + y 2 ≥ 12 2 y 1 + 2 y 2 + y 3 ≥ 8

Solution Summary: The author explains the Simplex Method for Problems in Standard Maximization Form.

Videos

To calculate: The minimization of the function g=12y1+8y2+10y3 subject to y1+2y3≥10y1+y2≥122y1+2y2+y3≥8

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Chapter 4 Solutions

To calculate: The minimization of the function g=12y1+8y2+10y3 subject to

y1+2y3≥10y1+y2≥122y1+2y2+y3≥8