Simplex Method, Duality Problem

In summary, the conversation discusses an LPP problem and its dual form, and the question is to prove that the given solution is optimal without using the simplex method. The correct form of the problem is provided and the next steps involve using the properties of the relation between the primal and dual solutions at optimality.
  • #1
IsaacStats
11
0
Hello everyone, I have the following question:

Show without using the simplex method that
x1=5/26, x2=5/2, x3=27/26
is an optimal solution to the following LPP.

Maximize z=9x1+14x2+7x3 subject to
2x1+x2+3x3<= 6
5x1+4x2+x3<= 12
12x2 <= 5
x1,x2,x3 unrestricted.

=>
Dual is the following:

Minimize z'=6w1+12w2+6w3 subject to
2w1+5w2 >= 9
w1+4w2+2w3>= 14
3w1+w2 >= 7
w1,w2,w3 >= 0

I am lost regarding where I should proceed next. Looking for your guidance.
 
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  • #2
As this appears to be a homework question, I have moved it to the Homework & Coursework section.
 
  • #3
IsaacStats said:
Hello everyone, I have the following question:

Show without using the simplex method that
x1=5/26, x2=5/2, x3=27/26
is an optimal solution to the following LPP.

Maximize z=9x1+14x2+7x3 subject to
2x1+x2+3x3<= 6
5x1+4x2+x3<= 12
12x2 <= 5
x1,x2,x3 unrestricted.

=>
Dual is the following:

Minimize z'=6w1+12w2+6w3 subject to
2w1+5w2 >= 9
w1+4w2+2w3>= 14
3w1+w2 >= 7
w1,w2,w3 >= 0

I am lost regarding where I should proceed next. Looking for your guidance.

If the third primal right-hand-side is 5 (as written) the third dual objective coefficient is wrong. If the coefficient of x2 on the left of the third primal constraint is 12 (as written) the coefficient of w3 in the second dual constraint is wrong.

After deciding on correct statements of both the primal and dual problems, use the known properties of the relation between the primal and dual solution at optimality. For example, if a primal variable ##x_j## is ##> 0##, what can you say about the ##j##th dual constraint, etc.?
 

FAQ: Simplex Method, Duality Problem

What is the Simplex Method?

The Simplex Method is an algorithm used to solve linear programming problems. It is an iterative process that starts with an initial feasible solution and then moves towards the optimal solution by improving it at each iteration. The method involves finding the optimal solution by moving from one feasible solution to another along the edges of the feasible region until the best solution is reached.

How does the Simplex Method work?

The Simplex Method works by first converting a linear programming problem into a standard form. The algorithm then begins with an initial feasible solution and checks if it is optimal. If not, it moves towards the optimal solution by improving the objective function value at each iteration. This is done by selecting a pivot element and performing row operations to update the basic variables until the optimal solution is reached.

What is the Duality Problem in the Simplex Method?

The Duality Problem in the Simplex Method is a mathematical concept where a linear programming problem can be transformed into another problem, known as the dual problem, with the same optimal solution. The dual problem involves maximizing the minimum of a set of constraints, while the primal problem involves minimizing a linear objective function subject to a set of constraints.

How is the Duality Problem used in the Simplex Method?

The Duality Problem plays an important role in the Simplex Method by providing additional information about the primal problem. It helps in identifying the optimal solution and checking the feasibility of the primal problem. The dual problem can also be solved using the Simplex Method, and the optimal solution of the dual problem can be used to obtain the optimal solution of the primal problem.

What are the advantages of using the Simplex Method over other methods?

The Simplex Method has several advantages over other methods for solving linear programming problems. It is a relatively simple algorithm that is easy to understand and implement. It also guarantees finding the optimal solution if one exists. Additionally, the duality concept of the Simplex Method provides additional information about the problem and can be used to check the accuracy of the solution. Moreover, the Simplex Method can handle a large number of variables and constraints efficiently, making it suitable for real-world applications.

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