CS 270, Spring 2013 Project A Visual TOOL FOR THE simplex method in LP

1. SohamUday Mehta CS 270, Spring 2013 ProjectA Visual TOOL FOR THE simplex method in LP

2. Linear Programming in 3 variables • Subject to

3. Goals • Visualize the convex feasible region specified by constraints (in 3D)

4. Goals • Visualize the convex feasible region specified by constraints (in 3D) • Visualize Simplex Algorithm to solve the LP (simple version)

5. Visualization • Each constraint becomes a polygon bounding the convex feasible region

6. Visualization • Each constraint becomes a polygon bounding the convex feasible region • Start with 3 Quads for • Assume we have all polygons for current convex region and we want to add a new constraint

7. Visualization • Each constraint becomes a polygon bounding the convex feasible region • Start with 3 Quads for • Assume we have all polygons for current convex region and we want to add a new constraint • Need to clip all polygons by new constraint

8. Visualization • Find the ‘side’ of plane each vertex of polygon is on • If an edge of poly cuts plane, add new vertex and remove the ‘wrong’ side vertex

9. Visualization • Each constraint may also create a new polygon

10. Visualization • Each constraint may also create a new polygon

11. Visualization • Each constraint may also create a new polygon • Store ‘new’ vertices created by clipping existing polygons

12. Visualization • Each constraint may also create a new polygon • Store ‘new’ vertices created by clipping existing polygons • Remove duplicates

13. Visualization • Each constraint may also create a new polygon • Store ‘new’ vertices created by clipping existing polygons • Remove duplicates, re-order vertices, and create new poly

14. Simplex Algorithm • Start with a random vertex

15. Simplex Algorithm • Start with a random vertex • Find extreme directions at current vertex • Pick maximum improvement direction • If no improvement in any direction, stop

16. Simplex Algorithm • Start with a random vertex • Find extreme directions at current vertex • Pick maximum improvement direction • If no improvement in any direction, stop • Find max. feasible step and move to next vertex • Go back to 2

17. Choosing the start vertex • First add 1 auxiliary variable to each constraint to convert it to an equation, create matrix • Randomly choose from the and set to , solve the equations for rest If matrix is not invertible or some , solve with different choice The ones set to zero are called non-basic, rest are basic – store this in a boolean vector “B”

18. Choosing the start vertex • First add 1 auxiliary variable to each constraint to convert it to an equation, create matrix • Randomly choose from the and set to , solve the equations for rest • If matrix is not invertible or some , solve with different choice • The ones set to zero are called non-basic, rest are basic – store this in a boolean vector “B”

19. Finding directions at any vertex • Each non-basic variable gives a direction For , set 1 non-basic value to 1, other two to 0 and solve for rest (free upto a constant) Pick direction which gives maximum If all directions give , reached maximum, STOP

20. Finding directions at any vertex • Each non-basic variable gives a direction • For , set 1 non-basic value to 1, other two to 0 and solve for rest (free upto a constant) • Pick direction which gives maximum • If all directions give , reached maximum, STOP

21. Finding the next vertex • Choose step size • Move to next vertex

22. Finding the next vertex • Choose step size • Move to next vertex • Update “B” vector to correspond to new vertex: non-basic generating the max direction becomes basic, the basic fixing becomes non-basic

24. Conclusion • Will be available for download

25. Conclusion • Will be available for download • Thanks for your attention • Comments / Questions?