Explanation Given Information: The provided matrix is: Minimize c = 2 x + y Subject to 3 x + 2 y ≥ 60 , 2 x + y ≥ 60 , x + 3 y ≥ 60 and x ≥ 0 , y ≥ 0 Formula used: Solving the given Dual problem by the following steps: Step 1) Represent the problem as a matrix. Step 2) Take the transpose of the matrix. Step 3) Write down the dual, which is the LP problem corresponding to the new matrix. If the original problem was a maximization problem, its dual will be a minimization problem, and vice versa. Fundamental Theorem of duality as: If an LP problem has a optimal solution, then so does its dual. Moreover, the primal problem and the dual problem have the same optimal value for their objective functions. Contained in the final tableau of the simplex method applied to an LP problem is the solution to its dual problem: It is given by the bottom entries in the column associated with the slack variables, divided by the entry under the objective variable. Calculation: Consider the provided matrix is: Minimize c = 2 x + y Subject to 3 x + 2 y ≥ 60 , 2 x + y ≥ 60 , x + 3 y ≥ 60 and x ≥ 0 , y ≥ 0 Step 1) Represent the problem as a matrix as: [ 3 2 60 2 1 60 1 3 60 2 1 0 ] Step 2) Take the transpose of the matrix as: [ 3 2 1 2 1 3 60 60 60 2 1 0 ] Step 3) Write down the dual, which is the LP problem corresponding to the new matrix. If the original problem was a maximization problem, its dual will be a minimization problem, and vice versa. Maximize p = 60 x + 60 y + 60 z Subject to 3 x + 2 y ≥ 60 , 2 x + y ≥ 60 , x + 3 y ≥ 60 and x ≥ 0 , y ≥ 0 Solve the dual problem as: x y z s t p s 3 2 1 1 0 0 2 t 2 1 3 0 1 0 1 p − 60 − 60 − 60 0 0 1 0 Apply the row operation 2 R 1 − 3 R 2 and R 3 +

7th Edition

ISBN: 9781337515542

Author: Waner

Publisher: Cengage

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Chapter 5, Problem 19RE

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To calculate: The linear programming problem by the method of duality. Where,

Minimize $c=2x+y$

Subject to $3x+2y≥60,2x+y≥60,x+3y≥60 and x≥0,y≥0$

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Chapter 5, Problem 18RE

Chapter 5, Problem 20RE

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The linear programming problem by the method of duality. Where, Minimize c = 2 x + y Subject to 3 x + 2 y ≥ 60 , 2 x + y ≥ 60 , x + 3 y ≥ 60 and x ≥ 0 , y ≥ 0

Solution Summary: The author explains the formula used to solve the linear programming problem by the method of duality.

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To calculate: The linear programming problem by the method of duality. Where,

Minimize $c=2x+y$

Subject to $3x+2y≥60,2x+y≥60,x+3y≥60 and x≥0,y≥0$

Want to see the full answer?

Chapter 5 Solutions