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  1. MATH 527 Deterministic OR Graphical Solution Method for Linear Programs

  2. 30 20 10 4 12 20 MATH 327 - Mathematical Modeling

  3. 30 20 10 4 12 20 MATH 327 - Mathematical Modeling

  4. 30 20 10 4 12 20 MATH 327 - Mathematical Modeling

  5. 30 20 10 4 12 20 MATH 327 - Mathematical Modeling

  6. 30 20 10 4 12 20 MATH 327 - Mathematical Modeling

  7. 30 20 10 4 12 20 MATH 327 - Mathematical Modeling

  8. 30 20 10 4 12 20 MATH 327 - Mathematical Modeling

  9. 30 20 10 4 12 20 MATH 327 - Mathematical Modeling

  10. 30 20 Feasible region 10 4 12 20 The feasible region is a polygon!! MATH 327 - Mathematical Modeling

  11. We must graph the isoprofit line. Straight line All points on the line have the same objective value When problem is minimization, called an isocost line. How?? Choose any point in the feasible region Find its objective value (or z-value) Graph the line objective function = z-value. How do we find the optimal solution?? MATH 327 - Mathematical Modeling

  12. 30 Isoprofit line z = 300 20 10 4 12 20 MATH 327 - Mathematical Modeling

  13. 30 20 10 4 12 20 Isoprofit line MATH 327 - Mathematical Modeling

  14. 30 Isoprofit line 20 10 4 12 20 MATH 327 - Mathematical Modeling

  15. 30 20 10 4 12 20 Isoprofit line MATH 327 - Mathematical Modeling

  16. 30 20 10 4 12 20 Isoprofit line MATH 327 - Mathematical Modeling

  17. 30 20 10 4 12 20 Isoprofit line z = 433 1/3 optimal solution: (20/3, 40/3) z = 433 1/3 MATH 327 - Mathematical Modeling

  18. Binding vs. Nonbinding • A constraint is binding if the optimal solution satisfies that constraint at equality (left-hand side = right-hand side). Otherwise, it is nonbinding. • Binding constraints keep us from finding better solutions!! MATH 327 - Mathematical Modeling

  19. 30 20 10 4 12 20 optimal solution: (20/3, 40/3) z = 433 1/3 MATH 327 - Mathematical Modeling

  20. 30 20 10 4 12 20 binding optimal solution: (20/3, 40/3) z = 433 1/3 MATH 327 - Mathematical Modeling

  21. binding 30 20 10 binding 4 12 20 optimal solution: (20/3, 40/3) z = 433 1/3 MATH 327 - Mathematical Modeling

  22. Convex Sets • A set of points S is a convex set if the line segment joining any two points in S lies entirely in S Nonconvex Convex MATH 327 - Mathematical Modeling

  23. C A B D Extreme Points • A point P in a convex set S is an extreme point if, for any line segment containing P which lies entirely in S, P is an endpoint of that segment. MATH 327 - Mathematical Modeling

  24. C A B D Extreme Points • A point P in a convex set S is an extreme point if, for any line segment containing P which lies entirely in S, P is an endpoint of that segment. C and D are extreme points A and B are not MATH 327 - Mathematical Modeling

  25. Interesting Facts • The extreme points of a polygon are the corner points. • The feasible region for any linear program will be a convex set. MATH 327 - Mathematical Modeling

  26. Interesting Facts • The feasible region will have a finite number of extreme points • Extreme points are the intersections of constraints (including nonnegativity) • Any linear program that has an optimal solution has an extreme point that is optimal!! • What are the implications? MATH 327 - Mathematical Modeling

  27. 12 8 4 2 6 10 MATH 327 - Mathematical Modeling

  28. 12 8 4 2 6 10 MATH 327 - Mathematical Modeling

  29. 12 8 4 2 6 10 MATH 327 - Mathematical Modeling

  30. 12 8 4 2 6 10 MATH 327 - Mathematical Modeling

  31. 12 8 4 2 6 10 MATH 327 - Mathematical Modeling

  32. 12 8 4 2 6 10 MATH 327 - Mathematical Modeling

  33. 12 8 4 2 6 10 MATH 327 - Mathematical Modeling

  34. 12 8 4 2 6 10 MATH 327 - Mathematical Modeling

  35. 12 Feasible Region 8 4 2 6 10 MATH 327 - Mathematical Modeling

  36. 12 8 4 2 6 10 Isocost line z = 54 MATH 327 - Mathematical Modeling

  37. 12 8 4 2 6 10 Isocost line MATH 327 - Mathematical Modeling

  38. 12 8 4 2 6 10 Isocost line MATH 327 - Mathematical Modeling

  39. 12 8 4 2 6 10 Isocost line MATH 327 - Mathematical Modeling

  40. 12 8 optimal solution: (5/4, 21/4) z = 36 1/4 4 2 6 10 Isocost line z = 36 1/4 MATH 327 - Mathematical Modeling

  41. Special Cases • So far, our models have had • One optimal solution • A finite objective value • Does this always happen? • What if it doesn’t? MATH 327 - Mathematical Modeling

  42. Special Case # 1: Unbounded Linear Programs • If maximizing: there are points in the feasible region with arbitrarily large objective values. • If minimizing: there are points in the feasible region with arbitrarily small objective values. MATH 327 - Mathematical Modeling

  43. Special Case #1: Unbounded Linear Programs maximization minimization MATH 327 - Mathematical Modeling

  44. CAUTION!!! There is a difference between an unbounded linear program and an unbounded feasible region!!! MATH 327 - Mathematical Modeling

  45. Special Case #2: Infinite Number of Optimal Solutions • When isoprofit/isocost lie intersects an entire line segment corresponding to a binding constraint • Occurs when isoprofit/isocost line is parallel to one of the binding constraints MATH 327 - Mathematical Modeling

  46. Special Case #2: Infinite Number of Optimal Solutions MATH 327 - Mathematical Modeling

  47. Special Case # 3: Infeasible Linear Program • Feasible Region is empty MATH 327 - Mathematical Modeling

  48. Every Linear Program • Has a unique optimal solution, or….. • Has infinite optimal solutions, or….. • Is unbounded, or….. • Is infeasible. MATH 327 - Mathematical Modeling