# MATH 527 Deterministic OR - PowerPoint PPT Presentation

MATH 527 Deterministic OR

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MATH 527 Deterministic OR

## MATH 527 Deterministic OR

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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

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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

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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