Explanation Given Information: The provided matrix is: Minimize c = 2 x + y + 2 z Subject to 3 x + 2 y + z ≥ 100 , 2 x + y + 3 z ≥ 200 and x ≥ 0 , y ≥ 0 , z ≥ 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 + 2 z Subject to 3 x + 2 y + z ≥ 100 , 2 x + y + 3 z ≥ 200 and x ≥ 0 , y ≥ 0 , z ≥ 0 Step 1) Represent the problem as a matrix. [ 3 2 1 2 1 3 2 1 2 100 200 0 ] Step 2) Take the transpose of the matrix. [ 3 2 2 2 1 1 1 3 2 100 200 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 = 100 x + 200 y Subject to 3 x + 2 y + z ≥ 100 , 2 x + y + 3 z ≥ 200 and x ≥ 0 , y ≥ 0 , z ≥ 0 Solve the dual problem as: x y s t u p s 3 2 1 0 0 0 2 t 2 1 0 1 0 0 1 u 1 3 0 0 1

7th Edition

ISBN: 9781337280426

Author: Stefan Waner, Steven Costenoble

Publisher: Cengage Learning

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Chapter 5, Problem 20RE

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

Minimize c=2x+y+2z

Subject to 3x+2y+z≥100,2x+y+3z≥200 and x≥0,y≥0,z≥0

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Chapter 5, Problem 19RE

Chapter 5, Problem 21RE

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The linear programming problem by the method of duality. Minimize c = 2 x + y + 2 z Subject to 3 x + 2 y + z ≥ 100 , 2 x + y + 3 z ≥ 200 and x ≥ 0 , y ≥ 0 , z ≥ 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.

Minimize c=2x+y+2z

Subject to 3x+2y+z≥100,2x+y+3z≥200 and x≥0,y≥0,z≥0

Want to see the full answer?

Chapter 5 Solutions

The linear programming problem by the method of duality. Minimize c = 2 x + y + 2 z Subject to 3 x + 2 y + z ≥ 100 , 2 x + y + 3 z ≥ 200 and x ≥ 0 , y ≥ 0 , z ≥ 0

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

Videos

To calculate: The linear programming problem by the method of duality. Minimize c=2x+y+2z Subject to 3x+2y+z≥100,2x+y+3z≥200 and x≥0,y≥0,z≥0

Want to see the full answer?

Chapter 5 Solutions

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

Minimize c=2x+y+2z

Subject to 3x+2y+z≥100,2x+y+3z≥200 and x≥0,y≥0,z≥0