Explanation Given Information: The provided linear programming problem is, Maximize 16 x + 14 y subject to the constraints, { 10 x + 8 y ≤ 3600 5 x + 3 y ≤ 1498 6 x + 8 y ≤ 2396 x ≥ 0 , y ≥ 0 Consider the provided linear programming problem, Maximize 16 x + 14 y subject to the constraints, { 10 x + 8 y ≤ 3600 5 x + 3 y ≤ 1498 6 x + 8 y ≤ 2396 x ≥ 0 , y ≥ 0 First write the provided linear programming problem into standard form. Maximize [ 16 14 ] [ x y ] subject to the constraints [ 10 8 5 3 6 8 ] [ x y ] ≤ [ 3600 1498 2396 ] and [ x y ] ≥ [ 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 [ 16 14 ] [ x y ] subject to the constraints [ 10 8 5 3 6 8 ] [ x y ] ≤ [ 3600 1498 2396 ] and [ x y ] ≥ [ 0 0 ] ” with the statement “Maximize CX subject to the constraints A X ≤ B , X ≥ 0 ”. It can be seen that C = [ 16 14 ] , A = [ 10 8 5 3 6 8 ] , B = [ 3600 1498 2396 ] and X = [ x y ] . Therefore, the dual problem of the provided linear programming problem is as follows, Minimize [ 3600 1498 2396 ] T [ u v w ] subject to the constraints [ 10 8 5 3 6 8 ] T [ u v w ] ≥ [ 16 14 ] T and [ u v w ] ≥ [ 0 0 0 ] . Simplify the above as below, Minimize [ 3600 1498 2396 ] [ u v w ] subject to the constraints [ 10 5 6 8 3 8 ] [ u v w ] ≥ [ 16 14 ] and [ u v w ] ≥ [ 0 0 0 ] . Rewrite the above statement, Minimize 3600 u + 1498 v + 2396 w subject to the constraints { 10 u + 5 v + 6 w ≥ 16 8 u + 3 v + 8 w ≥ 14 u ≥ 0 , v ≥ 0 , w ≥ 0 Hence, the dual problem of the linear programming problem, Maximize 16 x + 14 y subject to the constraints, { 10 x + 8 y ≤ 3600 5 x + 3 y ≤ 1498 6 x + 8 y ≤ 2396 x ≥ 0 , y ≥ 0 is Minimize 3600 u + 1498 v + 2396 w subject to the constraints, { 10 u + 5 v + 6 w ≥ 16 8 u + 3 v + 8 w ≥ 14 u ≥ 0 , v ≥ 0 , w ≥ 0 Now, find the final simplex tableau of linear programming problem, Maximize 16 x + 14 y subject to the constraints { 10 x + 8 y ≤ 3600 5 x + 3 y ≤ 1498 6 x + 8 y ≤ 2396 x ≥ 0 , y ≥ 0 Since, there are three non-trivial inequalities, the simplex method will have three slack variables. Denote the slack variables by u , v and w . So, the three inequalities become, 10 x + 8 y + u = 3600 5 x + 3 y + v = 1498 6 x + 8 y + w = 2396 Also, change the objective function ‘Maximize 16 x + 14 y ’ into an equation M = 16 x + 14 y or − 16 x − 14 y + M = 0 . Now, the above four equations in matrix form as below, x y u v w M u v w M [ 10 8 1 0 0 0 3600 5 3 0 1 0 0 1498 6 8 0 0 1 0 2396 − 16 − 14 0 0 0 1 0 ] The last row of the above table shows the coefficients of the objective function − 16 x − 14 y + M = 0 and the row 1 , row 2 and row 3 shows the coefficient of the constraints or system of inequality. Now, find the pivot column and pivot row of the table, Select most negative value from the last row for the pivot column. From the above table, the most negative value is − 16 and it lies in the column 1

Finite Mathematics & Its Applications (12th Edition)

12th Edition

ISBN: 9780134507125

Author: Goldstein

Publisher: PEARSON

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Chapter 4, Problem 6P

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The dual problem of the provided linear programming problem,

Maximize 16x+14y subject to the constraints,

{10x+8y≤36005x+3y≤14986x+8y≤2396x≥0,y≥0

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Chapter 4, Problem 5P

Chapter 5.1, Problem 1CYU

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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 dual problem of the provided linear programming problem, Maximize 16 x + 14 y subject to the constraints, { 10 x + 8 y ≤ 3600 5 x + 3 y ≤ 1498 6 x + 8 y ≤ 2396 x ≥ 0 , y ≥ 0

Solution Summary: The author explains that the provided linear programming problem is, Maximize 16x+14y subject to the constraints.

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The dual problem of the provided linear programming problem,

Maximize 16x+14y subject to the constraints,

{10x+8y≤36005x+3y≤14986x+8y≤2396x≥0,y≥0

Want to see the full answer?

Chapter 4 Solutions

The dual problem of the provided linear programming problem, Maximize 16 x + 14 y subject to the constraints, { 10 x + 8 y ≤ 3600 5 x + 3 y ≤ 1498 6 x + 8 y ≤ 2396 x ≥ 0 , y ≥ 0

Solution Summary: The author explains that the provided linear programming problem is, Maximize 16x+14y subject to the constraints.

Videos

The dual problem of the provided linear programming problem, Maximize 16x+14y subject to the constraints, {10x+8y≤36005x+3y≤14986x+8y≤2396x≥0,y≥0

Want to see the full answer?

Chapter 4 Solutions

The dual problem of the provided linear programming problem,

Maximize 16x+14y subject to the constraints,

{10x+8y≤36005x+3y≤14986x+8y≤2396x≥0,y≥0