Explanation Given Information: The provided linear programming problem is, Minimize 10 x + y Subject to the constraints { 3 x + y ≥ 16 x + 2 y ≥ 12 x ≥ 2 x ≥ 0 , y ≥ 0 . Formula used: The Simplex Method for Problems in Nonstandard Form 1 . If necessary, convert all inequalities (except x ≥ 0 , y ≥ 0 , … ) into the form. [Linear expression] ≤ [Constant]. 2 . If a negative number appears in the upper part of the last column of the simplex tableau, remove it by pivoting. (a) Select one of the negative entries in its row. The column containing the entry will be the pivot column. (b) Select the pivot element by determining the least of the positive ratios associated with entries in the pivot column (except the bottom entry). (c) Pivot. 3 . Repeat step 2 until there are no negative entries in the upper part of the right-hand column of the simplex tableau. (If this process will not remove all negative entries from the upper part of the right-hand column, then the linear programming problem does not have an optimal solution.). 4 . Proceed to apply the simplex method for tableaux in standard maximization form. Calculation: Consider the provided linear programming problem, Minimize 10 x + y Subject to the constraints { 3 x + y ≥ 16 x + 2 y ≥ 12 x ≥ 2 x ≥ 0 , y ≥ 0 First, transform the problem so that all constraints is in ≤ form: { − 3 x − y ≤ − 16 − x − 2 y ≤ − 12 − x ≤ − 2 x ≥ 0 , y ≥ 0 Instead of minimizing 10 x + y , let us maximize ( − 1 ) ( 10 x + y ) = − 10 x − y , Let M = − 10 x − y , From the above linear system, the simplex tableau is: x y u v w M u v w M [ − 3 − 1 1 0 0 0 − 16 − 1 − 2 0 1 0 0 − 12 − 1 0 0 0 1 0 − 2 10 1 0 0 0 1 0 ] The right-side column has a negative number − 50 which violates the condition that all variables be ≥ 0 . First put the tableau into standard maximization form by pivoting to remove the negative entry in the right column. To choose the pivot element - Look along the left side of the − 50 row of the tableau, and locate any negative entry. Use the column containing the − 3 as the pivot column (that is, use column 2). Now compute ratios corresponding to both positive and negative entries (except the last) in the pivot column, considering further only positive ratios

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ISBN: 9781323788707

Author: Goldstein

Publisher: PEARSON

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Chapter 4.3, Problem 16E

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To calculate: The solution of the provided linear programming problem using simplex method.

Minimize 10x+y Subject to the constraints {3x+ y≥16 x+2y≥12 x ≥2x≥0,y≥0.

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Chapter 4.3, Problem 15E

Chapter 4.3, Problem 17E

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Chapter 4 Solutions

FINITE MATHEMATICS & ITS APPLICATIONS

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The solution of the provided linear programming problem using simplex method. Minimize 10 x + y Subject to the constraints { 3 x + y ≥ 16 x + 2 y ≥ 12 x ≥ 2 x ≥ 0 , y ≥ 0 .

Solution Summary: The author calculates the solution of the provided linear programming problem using the simplex method.

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To calculate: The solution of the provided linear programming problem using simplex method.

Minimize 10x+y Subject to the constraints {3x+ y≥16 x+2y≥12 x ≥2x≥0,y≥0.

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Chapter 4 Solutions

The solution of the provided linear programming problem using simplex method. Minimize 10 x + y Subject to the constraints { 3 x + y ≥ 16 x + 2 y ≥ 12 x ≥ 2 x ≥ 0 , y ≥ 0 .

Solution Summary: The author calculates the solution of the provided linear programming problem using the simplex method.

Videos

To calculate: The solution of the provided linear programming problem using simplex method. Minimize 10x+y Subject to the constraints {3x+ y≥16 x+2y≥12 x ≥2x≥0,y≥0.

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Chapter 4 Solutions

To calculate: The solution of the provided linear programming problem using simplex method.

Minimize 10x+y Subject to the constraints {3x+ y≥16 x+2y≥12 x ≥2x≥0,y≥0.