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L15 LP Problems

L15 LP Problems. Homework Review Why bother studying LP methods History N design variables, m equations Summary. H14 part 1. H14 Part 1. H14 Part 1. H14. Curve fitting. Curve Fitting. Need to find the parameters a i Another way? Especially for non-linear curve fits?.

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L15 LP Problems

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  1. L15 LP Problems • Homework • Review • Why bother studying LP methods • History • N design variables, m equations • Summary

  2. H14 part 1

  3. H14 Part 1

  4. H14 Part 1

  5. H14

  6. Curve fitting

  7. Curve Fitting Need to find the parameters ai Another way? Especially for non-linear curve fits?

  8. Curve Fit example

  9. Goodness of fit? • R2 = coefficient of determination 0≤R2≤1. • R = correlation coefficien

  10. Curve Fit example

  11. Linear Programming Prob.s Linear

  12. Why study LP methods • LP problems are “convex” If there is a solution…it’s global optimum • Many real problems are LP Transportation, petroleum refining, stock portfolio, airline crew scheduling, communication networks • Some NL problems can be transformed into LP • Most widely used method in industry

  13. Std Form LP Problem Matrix form All “=“ All “≥0” i.e. non-neg. How do we transform an given LP problem into a Standard LP Prob.?

  14. Recall LaGrange/KKT method Add slack variable Subtract surplus variable

  15. Handling negative xi When x is unrestricted in sign:

  16. Transformation example

  17. Transformation example pg2

  18. Trans pg3

  19. Solving systems of linear equations n equations in n unknowns Produces a unique solution, for example

  20. Elimination methods Gaussian Elimination

  21. Elimination methods cont’d Gauss-Jordan Elimination

  22. Can we find unique solutions forn unknowns with m equations? 5 unknowns and 2 equations! What’s the best you can do? MUST set 3 xi to zero! Solve for remaining 2. Just like us=0 in LaGrange Method!

  23. m equations= m unknowns Most we can do is to solve for m unknowns, e.g. we can “solve” for 2 xi but which 2?

  24. Combinations? m=2, n=5

  25. Combinations from m=2, n=2 m=2, n=4

  26. Example 8.2 Figure 8.1 Solution to the profit maximization problem. Optimum point = (4, 12). Optimum cost = -8800. 5 unknowns, n=5 3 equations, m=3 10 combinations

  27. Example 8.2 cont’d Solutions are vertexes (i.e. extreme points, corners) of polyhedron formed by the constraints

  28. Example 8.2 cont’d • Ten solutions created by setting (n-m) variables to zero, they are called basic solutions • Some of them were basic feasible solutions • Any solution in polygon is a feasible solution • Variables not set to zero are basic variables • Variables set to zero = non-basic variables

  29. Canonical form Ex 8.4 & TABLEAU basis

  30. Ex 8.4 cont’d Pivot row Pivot column

  31. Method? • Set up LP prob in “tableau” • Select variable to leave basis • Select variable to enter basis (replace the one that is leaving) • Use Gauss-Jordan elimination to form identity sub-matrix, (i.e. new basis, identity columns) • Repeat steps 2-4 until opt sol’n is found!

  32. Can we be efficient? • Do we need to calculate all the combinations? • Is there a more efficient way to move from one vertex to another? • How do we know if we have found the opt solution, or need to calculate another tableau? SIMPLEX METHOD! (Next class)

  33. Summary • Curve fit = min Sum Squared Errors Min SSE, check R • Many important LP problems • LP probs are “convex progprobs” • Need to transform into Std LP format slack, surplus variables, non-negative b and x • Polygon surrounds infinite # of sol’ns • Opt solution is on a vertex • Must find combinations of basic variables

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