Soham Uday Mehta

Linear Programming in 3 variables

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

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

Visualization  Each constraint becomes a polygon bounding the convex feasible region

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

Visualization  Each constraint may also create a new polygon

Visualization  Each constraint may also create a new polygon

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

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

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

Simplex Algorithm 1. Start with a random vertex

Simplex Algorithm 1. Start with a random vertex 2. Find extreme directions at current vertex 3. Pick maximum improvement direction 4. If no improvement in any direction, stop

Simplex Algorithm 1. Start with a random vertex 2. Find extreme directions at current vertex 3. Pick maximum improvement direction 4. If no improvement in any direction, stop 5. Find max. feasible step and move to next vertex 6. Go back to 2

Choosing the start vertex

Finding directions at any vertex

Finding the next vertex

Conclusion  Will be available for downloaddownload

Conclusion  Will be available for downloaddownload  Thanks for your attention  Comments / Questions?