1 / 34

Simplex Method

Simplex Method. MSci331 — Week 3~4. Simplex Algorithm. Consider the following LP, solve using Simplex:. Step 1: Preparing the LP. Step 2: Express the LP in a tableau form. Step 3: Obtain the initial basic feasible solution (if available). a) Set n - m variables equal to 0.

Download Presentation

Simplex Method

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Simplex Method MSci331—Week 3~4

  2. Simplex Algorithm • Consider the following LP, solve using Simplex:

  3. Step 1: Preparing the LP

  4. Step 2: Express the LP in a tableau form

  5. Step 3: Obtain the initial basic feasible solution (if available) a) Set n-m variables equal to 0 These n-m variables the NBV b) Check if the remaining m variables satisfy the condition of BV = If yes, the initial feasible basic solution (bfs) is readily a available = else, carry on some ERO to obtain the initial bfs

  6. Step 4: Apply the Simplex Algorithm a) Is the initial bfs optimal? (Will bringing a NBV improve the value of Z?) b) If yes, which variable from the set of NBV to bring into the set of BV? - The entering NBV defines the pivot column c) Which variable from the set of BV has to become NBV? - The exiting BV defines the pivot row Exits Pivot cell Enters

  7. Summary of Simplex Algorithm for Papa Louis Set: n-m=0 m≠0 BFS (intial) BFS (1) BFS (2) BFS (3) 1 The optimal solution is x1=20, x2=60 The optimal value is Z=180 The BFS at optimality x1=20, x2=60, s3=20

  8. Geometric Interpretation of Simplex Algorithm

  9. Class activity • Consider the following LP: This is a maximizing LP, in normal form. So an initial BFS exists.

  10. Class activity

  11. Class activity

  12. Class activity Make this coefficient equal 1 and pivot all other rows relative to it Exits Enters 4/1 6/1 2/2* ---

  13. Class activity

  14. Class activity Make this coefficient equal 1 and pivot all other rows relative to it Enters 3/2.5* 5/3.5 --- 5/0.5

  15. Class activity

  16. Example: LP model with Minimization Objective • Solve the following LP model: • Initial Tableau

  17. Example: LP model with Minimization Objective • Iteration 0 • Iteration 1 • Optimality test:

  18. > Constraint x2 40 35 30 25 20 15 10 5 5 10 15 20 25 30 35 40 Constraint 1 Constraint 3 Z Constraint 2 x1 Constraint 4 New feasible region

  19. Equality Constraint x2 40 35 30 25 20 15 10 5 5 10 15 20 25 30 35 40 Constraint 1 Constraint 3 Z Constraint 2 x1 New feasible region

  20. The Problem of Finding an Initial Feasible BV An LP Model Standard Form Cannot find an initial basic variable that is feasible.

  21. Example: Solve Using the Big M Method Write in standard form

  22. Example: Solve Using the Big M Method Adding artificial variables

  23. Example: Solve Using the Big M Method Put in tableau form

  24. Example: Solve Using the Big M Method Eliminating a2 from row 0 by operations: new Row 0 = old Row 0 -M*old Row 2

  25. Example: Solve Using the Big M Method Eliminating a3 from the new row 0 by operations: new Row=old Row-M*old Row 3

  26. Example: Solve Using the Big M Method The initial basic variables are s1=25, a2=12, and a3=0. Now ready to proceed for the simplex algorithm. The initial Tableau

  27. Example: Solve Using the Big M Method Using EROs change the column of x1 into a unity vector. Iteration 1

  28. Example: Solve Using the Big M Method Using EROs change the column of z into a unity vector. Iteration 2 Students to try more iterations. The solution is infeasible. See the attached solution.

  29. Special case 1: Alternative Optima . See Notes on this slide (below) for more information

  30. Special case 1: Alternative Optima

  31. Special case 2: Unbounded LPs s1 0 1 x1 See Notes on this slide (below) for more information

  32. Special Case 3: Degeneracy

  33. Special Case 5: Degeneracy Iteration 0 Iteration 1 Iteration 2

  34. Special Case 5: Degeneracy Degeneracy reveals from practical standpoint that the model has at least one redundant constraint.

More Related