Explanation Given Information: The provided linear programming problem is, Minimize 15 , 000 x + 12 , 000 y subject to the constraints { 1000 x + 2000 y ≥ 100 , 000 3000 x + 2000 y ≥ 180 , 000 2000 x + 1000 y ≥ 100 , 000 x ≥ 0 , y ≥ 0 Formula used: The dual of a linear programming problem: 1. If the original (primal) problems 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 . 2. If the original (primal) problems has the form Minimize CX subject to the constraints A X ≥ B , X ≥ 0 , then the dual problem is, Maximize B T U subject to the constraints A T U ≤ C T , U ≥ 0 . Calculation: Consider the provided linear programming problem, Minimize 15 , 000 x + 12 , 000 y subject to the constraints { 1000 x + 2000 y ≥ 100 , 000 3000 x + 2000 y ≥ 180 , 000 2000 x + 1000 y ≥ 100 , 000 x ≥ 0 , y ≥ 0 First write the provided linear programming problem into standard form. Minimize [ 15 , 000 12 , 000 ] [ x y ] subject to the constraints [ 1000 2000 3000 2000 2000 1000 ] [ x y ] ≥ [ 100 , 000 180 , 000 100 , 000 ] and [ x y ] ≥ [ 0 0 ] Suppose that one can buy smartphones, tablets and Bluetooth headphones from other supplier, instead of making them at a cost of u dollars per smartphone, v per tablets and w per Bluetooth headphones. To find the maximum cost at which this should be done, find the dual of original problem. Now, if the original (primal) problem has the form Minimize CX subject to the constraints A X ≥ B , X ≥ 0 , then the dual problem is Maximize B T U subject to the constraints A T U ≤ C T , U ≥ 0 . So, compare the statement, “Minimize [ 15 , 000 12 , 000 ] [ x y ] subject to the constraints [ 1000 2000 3000 2000 2000 1000 ] [ x y ] ≥ [ 100 , 000 180 , 000 100 , 000 ] and [ x y ] ≥ [ 0 0 ] ” with the statement “Minimize CX subject to the constraints A X ≥ B , X ≥ 0 ”. It can be seen that C = [ 15 , 000 12 , 000 ] , A = [ 1000 2000 3000 2000 2000 1000 ] , B = [ 100 , 000 180 , 000 100 , 000 ] and X = [ x y ]

Finite Mathematics & Its Applications (12th Edition)

12th Edition

ISBN: 9780134437767

Author: Larry J. Goldstein, David I. Schneider, Martha J. Siegel, Steven Hair

Publisher: PEARSON

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Chapter 4.5, Problem 16E

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To calculate: The economic interpretation to the dual of the linear programming problem,

Minimize 15,000x+12,000y subject to the constraints {1000x+2000y≥100,0003000x+2000y≥180,0002000x+1000y≥100,000x≥0,y≥0

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Chapter 4.5, Problem 15E

Chapter 4.5, Problem 17E

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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 economic interpretation to the dual of the linear programming problem, Minimize 15 , 000 x + 12 , 000 y subject to the constraints { 1000 x + 2000 y ≥ 100 , 000 3000 x + 2000 y ≥ 180 , 000 2000 x + 1000 y ≥ 100 , 000 x ≥ 0 , y ≥ 0

Solution Summary: The author calculates the economic interpretation to the dual of the linear programming problem: Minimize 15,000x+12,000y subject to l1000x+2000y

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To calculate: The economic interpretation to the dual of the linear programming problem,

Minimize 15,000x+12,000y subject to the constraints {1000x+2000y≥100,0003000x+2000y≥180,0002000x+1000y≥100,000x≥0,y≥0

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Chapter 4 Solutions

The economic interpretation to the dual of the linear programming problem, Minimize 15 , 000 x + 12 , 000 y subject to the constraints { 1000 x + 2000 y ≥ 100 , 000 3000 x + 2000 y ≥ 180 , 000 2000 x + 1000 y ≥ 100 , 000 x ≥ 0 , y ≥ 0

Solution Summary: The author calculates the economic interpretation to the dual of the linear programming problem: Minimize 15,000x+12,000y subject to l1000x+2000y

Videos

To calculate: The economic interpretation to the dual of the linear programming problem, Minimize 15,000x+12,000y subject to the constraints {1000x+2000y≥100,0003000x+2000y≥180,0002000x+1000y≥100,000x≥0,y≥0

Want to see the full answer?

Chapter 4 Solutions

To calculate: The economic interpretation to the dual of the linear programming problem,

Minimize 15,000x+12,000y subject to the constraints {1000x+2000y≥100,0003000x+2000y≥180,0002000x+1000y≥100,000x≥0,y≥0