Explanation Given Information: The provided linear programming problem is, Maximize 10 x + 12 y + 10 z subject to the constraints { x − 2 y ≤ 6 3 x + z ≤ 9 y + 3 z ≤ 12 x ≥ 0 , y ≥ 0 , z ≥ 0 Calculation: Consider the provided linear programming problem, Maximize 10 x + 12 y + 10 z subject to the constraints { x − 2 y ≤ 6 3 x + z ≤ 9 y + 3 z ≤ 12 x ≥ 0 , y ≥ 0 , z ≥ 0 First write the provided linear programming problem into standard form. Maximize [ 10 12 10 ] [ x y z ] subject to the constraints [ 1 − 2 0 3 0 1 0 1 3 ] [ x y z ] ≤ [ 6 9 12 ] and [ x y z ] ≥ [ 0 0 0 ] Now, if the original (primal) problem has the form Maximize CX subject to the constraints A X ≤ B , X ≥ 0 , then the dual problem is Minimize B T U subject to the constraints A T U ≥ C T , U ≥ 0 . So, compare the statement “Maximize [ 10 12 10 ] [ x y z ] subject to the constraints [ 1 − 2 0 3 0 1 0 1 3 ] [ x y z ] ≤ [ 6 9 12 ] and [ x y z ] ≥ [ 0 0 0 ] ” with the statement “Maximize CX subject to the constraints A X ≤ B , X ≥ 0 ”. It can be seen that C = [ 10 12 10 ] , A = [ 1 − 2 0 3 0 1 0 1 3 ] , B = [ 6 9 12 ] and X = [ x y z ] . Therefore, the dual problem of the provided linear programming problem is as follows, Minimize [ 6 9 12 ] T [ u v w ] subject to the constraints [ 1 − 2 0 3 0 1 0 1 3 ] T [ u v w ] ≥ [ 10 12 10 ] T and [ u v w ] ≥ [ 0 0 0 ] Simplify the above as below, Minimize [ 6 9 12 ] [ u v w ] subject to the constraints [ 1 3 0 − 2 0 1 0 1 3 ] [ u v w ] ≥ [ 10 12 10 ] and [ u v w ] ≥ [ 0 0 0 ] . Rewrite the above statement, Minimize 6 u + 9 v + 12 w subject to the constraints { u + 3 v ≥ 10 − 2 u + w ≥ 12 v + 3 w ≥ 10 u ≥ 0 , v ≥ 0 , w ≥ 0 . Hence, the dual problem of the linear programming problem, Maximize 10 x + 12 y + 10 z subject to the constraints { x − 2 y ≤ 6 3 x + z ≤ 9 y + 3 z ≤ 12 x ≥ 0 , y ≥ 0 , z ≥ 0 is Minimize 6 u + 9 v + 12 w subject to the constraints { u + 3 v ≥ 10 − 2 u + w ≥ 12 v + 3 w ≥ 10 u ≥ 0 , v ≥ 0 , w ≥ 0 . Now, solve the original problem by simplex method. Since, there are three non-trivial inequalities, the simplex method will have three slack variables. Denote the slack variables by x , y and z . Consider M = 10 x + 12 y + 10 z or − 10 x − 12 y − 10 z + M = 0 and apply the simplex method. x y z u v w M u v w M [ 1 − 2 0 1 0 0 0 6 3 0 1 0 1 0 0 9 0 1 3 0 0 1 0 12 − 10 − 12 − 10 0 0 0 1 0 ] The last row of the above table shows the coefficients of the objective function − 10 x − 12 y − 10 z + M = 0 and the row 1 , row 2 and row 3 shows the coefficient of the constraints or system of inequality

Finite Mathematics & Its Applications (12th Edition)

12th Edition

ISBN: 9780134507125

Author: Goldstein

Publisher: PEARSON

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Chapter 4.5, Problem 13E

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To calculate: The solution of the linear programming problem written below and its dual using simplex method.

Maximize 10x+12y+10z subject to the constraints {x−2y≤63x+z≤9y+3z≤12x≥0,y≥0,z≥0

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Chapter 4.5, Problem 12E

Chapter 4.5, Problem 14E

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Chapter 4 Solutions

Finite Mathematics & Its Applications (12th Edition)

Ch. 4.1 - 1. Determine by inspection a particular solution...Ch. 4.1 - Prob. 2CYU Ch. 4.1 - For each of the following linear programming...Ch. 4.1 - For each of the following linear programming...Ch. 4.1 - For each of the following linear programming...Ch. 4.1 - For each of the following linear programming...Ch. 4.1 - For each of the following linear programming...Ch. 4.1 - For each of the following linear programming...Ch. 4.1 - 712For each of the linear programming problems in...Ch. 4.1 - 7–12 For each of the linear programming problems...

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The solution of the linear programming problem written below and its dual using simplex method. Maximize 10 x + 12 y + 10 z subject to the constraints { x − 2 y ≤ 6 3 x + z ≤ 9 y + 3 z ≤ 12 x ≥ 0 , y ≥ 0 , z ≥ 0

Solution Summary: The author calculates the linear programming problem and its dual using simplex method. Maximize 10x+12y+10z subject to the constraints.

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To calculate: The solution of the linear programming problem written below and its dual using simplex method.

Maximize 10x+12y+10z subject to the constraints {x−2y≤63x+z≤9y+3z≤12x≥0,y≥0,z≥0

Want to see the full answer?

Chapter 4 Solutions

The solution of the linear programming problem written below and its dual using simplex method. Maximize 10 x + 12 y + 10 z subject to the constraints { x − 2 y ≤ 6 3 x + z ≤ 9 y + 3 z ≤ 12 x ≥ 0 , y ≥ 0 , z ≥ 0

Solution Summary: The author calculates the linear programming problem and its dual using simplex method. Maximize 10x+12y+10z subject to the constraints.

Videos

To calculate: The solution of the linear programming problem written below and its dual using simplex method. Maximize 10x+12y+10z subject to the constraints {x−2y≤63x+z≤9y+3z≤12x≥0,y≥0,z≥0

Want to see the full answer?

Chapter 4 Solutions

To calculate: The solution of the linear programming problem written below and its dual using simplex method.

Maximize 10x+12y+10z subject to the constraints {x−2y≤63x+z≤9y+3z≤12x≥0,y≥0,z≥0