## Dual Problem: Linear Programming

Solving the dual problem. ## Example: Duality

Maximize z = 40w 1 + 50w 2

2w 1 + 3w 2 ≤ 3 4w 1 + 2w 2 ≤ 3

w 1 , w 2 ≥ 0

After adding slack variables, we have

Maximize z = 40w 1 + 50w 2 + 0x 3 + 0x 4

2w 1 + 3w 2 + x 3 = 3 4w 1 + 2w2 + x 4 = 3 w 1 , w 2 , x 3 , x 4 ≥ 0

Where x 3 and x 4 are slack variables.

## Initial basic feasible solution

w 1 = 0, w 2 = 0, z = 0 x 3 = 3, x 4 = 3 ## Table 1: Simplex Method

On small screens, scroll horizontally to view full calculation

Use Horizontal Scrollbar to View Full Table Calculation

The optimal solution is: w 1 = 3/8, w 2 = 3/4 z = 40 X 3/8 + 50 X 3/4= 105/2.

In case of primal problem , you noted that the values of z j -c j under the surplus variables x 3 and x 4 were 3/8 and 3/4. In case of dual problem , these values are the optimal values of dual variables w 1 and w 2 .

The optimal values of the dual variables are often called shadow prices.

Further, the values of the objective functions in both the problems are same (i.e., 105/2)

## Important Primal-Dual Results

• If one problem has an unbounded optimal solution, then the other problem cannot have a feasible solution.
• Optimal solution of either problem gives complete information about the optimal solution of the other.
• If either the primal or dual problem has a finite optimal solution, then the other has also a finite optimal solution.
• The dual of a dual is primal.

Operations Research Simplified Back Next

Goal programming Linear programming Transportation Problem Assignment Problem #### IMAGES

1. Write the dual for the following linear program 2. Duality in Linear Programming 3. Duality in Linear Programming 4. Duality In Linear Programming Problems 5. Solved Using the Strong Duality for Linear Programming, 6. Duality for Linear programming problem #### VIDEO

1. Linear Programming Classic Problems 10

2. SLST Mathematics: Linear Programming Problem

3. Linear programming (duality in linear programming video 1)

4. Lecture 14 10/16 Linear Programming

5. Duality Problem 4

6. 19

1. How Is Linear Programming Used in the Real World?

Linear programming is used daily in the real world to optimize the allocation of resources or activities to generate the most benefit or profit. Linear programming can take multiple factors into account into the thousands and is used extens...

Advantages of linear programming include that it can be used to analyze all different areas of life, it is a good solution for complex problems, it allows for better solution, it unifies disparate areas and it is flexible.

3. What Are Some Real World Uses for Linear Functions?

Real world uses for linear functions include solving problems and finding unknowns in engineering, economics and finances. A linear function describes a gradual rate of change, either positive or negative. When drawn, it presents a straight...

4. Duality in Linear Programming

Thus, a greater-than-or-equal-to constraint in a minimization problem has an associated nonnegative dual variable. The primal and dual problems that we have

5. Lecture 12 Linear programming : Duality in LPP

If either the primal or dual problem has a solution then the other also has a solution and their optimum values are equal. 3. If any of the two problems has an

6. Linear Programming Duality 1 Taking a dual

If we transform the dual linear program into an equivalent minimization problem, and take its.

7. Writing the Dual Programming of a Linear Programming Problem

Make sure you watch part 1 and 2 before part 3. In this video, we go over another example on how to write the dual programming of a LP.

8. Linear Program Duality Example

Linear Program Duality Example. 25K views · 6 ... Primal to Dual conversion | Duality in LPP | linear programming problem | Method-1| Lecture.16.

9. Duality in Linear Programming

There is always a corresponding. Linear Programming Problem (LPP) associated to every LPP, which is called as dual problem of the original. LPP(Primal Problem).

10. Concept with Examples

This video tutorial provides a basic concept of Duality in LPP and explains the Duality in Linear Programming with the help of plenty of

11. Dual problem: Linear Programming, Duality Examples, Primal

z = 40 X 3/8 + 50 X 3/4= 105/2. In case of primal problem, you noted that the values of zj-cj under the surplus variables x3 and x4

12. Linear Programming Duality

Worked examples · Write down the dual problem of the following: min 4 x 1 − 2 x 2 s.t. x 1 + 2 x 2 ≥ 3 3 x 1 − 4 x 2 = 0 x 2 ≥ 0. · Write down the dual

13. Duality in Linear Programming

LP Duality Theorem: Consider the linear programming problem: maximize cT x.

14. Examples 4.2: Dual problems

solved like in sec 4.1. • LPP: Linear Programming Problem, one of these “find the optimal value of a linear function subject to linear constraints” problems.