Explanation Given Information: The provided linear programming problem is, Minimize 3 x + y subject to the constraints { x + y ≥ 3 2 x ≥ 5 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 3 x + y subject to the constraints { x + y ≥ 3 2 x ≥ 5 x ≥ 0 , y ≥ 0 . First write the provided linear programming problem into standard form. Minimize [ 3 1 ] [ x y ] subject to the constraints [ 1 1 2 0 ] [ x y ] ≥ [ 3 5 ] and [ x y ] ≥ [ 0 0 ] . 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 [ 3 1 ] [ x y ] subject to the constraints [ 1 1 2 0 ] [ x y ] ≥ [ 3 5 ] 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 = [ 3 1 ] , A = [ 1 1 2 0 ] , B = [ 3 5 ] and X = [ x y ] . Therefore, the dual problem of the provided linear programming problem is as follows, Maximize [ 3 5 ] T [ u v ] subject to the constraints [ 1 1 2 0 ] T [ u v ] ≤ [ 3 1 ] T and [ u v ] ≥ [ 0 0 ] . Simplify the above as below, Maximize [ 3 5 ] [ u v ] subject to the constraints [ 1 2 1 0 ] [ u v ] ≤ [ 3 1 ] and [ u v ] ≥ [ 0 0 ] . Rewrite the above statement, Maximize 3 u + 5 v subject to the constraints { u + 2 v ≤ 3 u ≤ 1 u ≥ 0 , v ≥ 0 Hence, the dual problem of the linear programming problem, Minimize 3 x + y subject to the constraints { x + y ≥ 3 2 x ≥ 5 x ≥ 0 , y ≥ 0 is Maximize 3 u + 5 v subject to the constraints { u + 2 v ≤ 3 u ≤ 1 u ≥ 0 , v ≥ 0 Now, solve the dual problem by simplex method. Since, there are two non-trivial inequities, the simplex method will have two slack variables. Denote the slack variables by x , y . Consider M = 3 u + 5 v or − 3 u − 5 v + M = 0 and apply the simplex method. u v x y M x y M [ 1 2 1 0 0 3 1 0 0 1 0 1 − 3 − 5 0 0 1 0 ] The last row of the above table shows the coefficients of the objective function and the row 1 , row 2 show 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 11E

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

Minimize 3x+y subject to the constraints,

{x+y≥32x≥5x≥0,y≥0

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Chapter 4.5, Problem 10E

Chapter 4.5, Problem 12E

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Finite Mathematics & Its Applications (12th Edition)

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The solution of the linear programming problem written below and its dual using simplex method. Minimize 3 x + y subject to the constraints, { x + y ≥ 3 2 x ≥ 5 x ≥ 0 , y ≥ 0

Solution Summary: The author calculates the solution of the linear programming problem written below and its dual using simplex method.

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

Minimize 3x+y subject to the constraints,

{x+y≥32x≥5x≥0,y≥0

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Chapter 4 Solutions

The solution of the linear programming problem written below and its dual using simplex method. Minimize 3 x + y subject to the constraints, { x + y ≥ 3 2 x ≥ 5 x ≥ 0 , y ≥ 0

Solution Summary: The author calculates the solution of the linear programming problem written below and its dual using simplex method.

Videos

To calculate: The solution of the linear programming problem written below and its dual using simplex method. Minimize 3x+y subject to the constraints, {x+y≥32x≥5x≥0,y≥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.

Minimize 3x+y subject to the constraints,

{x+y≥32x≥5x≥0,y≥0