Explanation Given information: The objective function which has to maximize is: M = 2 x + 3 y which is subjected to the constraints: { x + 2 y ≤ 14 x + y ≤ 9 3 x + 2 y ≤ 2 x ≥ 0 , y ≥ 0 Formula used: The duality of primal problem can be determined as below: If the primal problem is in the form: Maximize: z = a x + b y , which is subjected to constraints: { a 1 x + b 1 y ≤ c 1 a 2 x + b 2 y ≤ c 2 x ≥ 0 y ≥ 0 In matrix form, the objective function can be written as: Maximize: z = C X . And, constraint can be written as: A X ≤ B , and X ≥ O . Where, A = [ a 1 b 1 a 2 b 2 ] , B = [ c 1 c 2 ] , C = [ a b ] and X = [ x y ] . Then its dual will be: Minimize: z = B T U And, the constraint is: A T U ≥ C T and U ≥ O . Where, A T = [ a 1 a 2 b 1 b 2 ] , B T = [ c 1 c 2 ] , U = [ u v ] and C T = [ a b ] . And, if the primal problem is in the form: Minimize: z = a x + b y , which is subjected to constraints: { a 1 x + b 1 y ≥ c 1 a 2 x + b 2 y ≥ c 2 x ≥ 0 y ≥ 0 In matrix form, the objective function can be written as: Minimize: z = C X . And, constraint can be written as: A X ≥ B , and X ≥ O . Where, A = [ a 1 b 1 a 2 b 2 ] , B = [ c 1 c 2 ] , C = [ a b ] and X = [ x y ] . Then its dual is: Minimize: z = B T U And, the constraint is: A T U ≤ C T and U ≥ O . Where, A T = [ a 1 a 2 b 1 b 2 ] , B T = [ c 1 c 2 ] , U = [ u v ] and C T = [ a b ] . Calculation: The provided primal problem is in the form: Maximize: M = 2 x + 3 y , subjected to constraints: { x + 2 y ≤ 14 x + y ≤ 9 3 x + 2 y ≤ 24 x ≥ 0 , y ≥ 0 In matrix form, the objective function is: Maximize: z = C X And, constraint can be written as: A X ≤ B , and X ≥ O

Finite Mathematics & Its Applications (12th Edition)

12th Edition

ISBN: 9780134507125

Author: Goldstein

Publisher: PEARSON

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Chapter 4, Problem 15RE

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The matrix A, B, C, X and U of maximization problem which has objective function: M=2x+3y which is subjected to the constraints:

{x+2y≤14x+y≤93x+2y≤24x≥0,y≥0

Also, state the problem of the given primal and its dual.

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Chapter 4, Problem 14RE

Chapter 4, Problem 16RE

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Finite Mathematics & Its Applications (12th Edition)

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The matrix A , B , C , X and U of maximization problem which has objective function: M = 2 x + 3 y which is subjected to the constraints: { x + 2 y ≤ 14 x + y ≤ 9 3 x + 2 y ≤ 24 x ≥ 0 , y ≥ 0 Also, state the problem of the given primal and its dual.

Solution Summary: The author states the problem of the given primal and its dual.

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The matrix A, B, C, X and U of maximization problem which has objective function: M=2x+3y which is subjected to the constraints:

{x+2y≤14x+y≤93x+2y≤24x≥0,y≥0

Also, state the problem of the given primal and its dual.

Want to see the full answer?

Chapter 4 Solutions

The matrix A , B , C , X and U of maximization problem which has objective function: M = 2 x + 3 y which is subjected to the constraints: { x + 2 y ≤ 14 x + y ≤ 9 3 x + 2 y ≤ 24 x ≥ 0 , y ≥ 0 Also, state the problem of the given primal and its dual.

Solution Summary: The author states the problem of the given primal and its dual.

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The matrix A, B, C, X and U of maximization problem which has objective function: M=2x+3y which is subjected to the constraints: {x+2y≤14x+y≤93x+2y≤24x≥0,y≥0 Also, state the problem of the given primal and its dual.

Want to see the full answer?

Chapter 4 Solutions

The matrix A, B, C, X and U of maximization problem which has objective function: M=2x+3y which is subjected to the constraints:

{x+2y≤14x+y≤93x+2y≤24x≥0,y≥0

Also, state the problem of the given primal and its dual.