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

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10. MATH 327 - Mathematical Modeling 10 4 1220 10 20 30 Feasible region The feasible region is a polygon!!

11. MATH 327 - Mathematical Modeling 11 How do we find the optimal solution?? 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.

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

13. MATH 327 - Mathematical Modeling 13 4 1220 10 20 30 Isoprofit line

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15. MATH 327 - Mathematical Modeling 15 4 1220 10 20 30 Isoprofit line

16. MATH 327 - Mathematical Modeling 16 4 1220 10 20 30 Isoprofit line

17. MATH 327 - Mathematical Modeling 17 4 1220 10 20 30 Isoprofit line z = 433 1/3 optimal solution: (20/3, 40/3) z = 433 1/3

18. 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!!

19. MATH 327 - Mathematical Modeling 19 4 1220 10 20 30 optimal solution: (20/3, 40/3) z = 433 1/3

20. MATH 327 - Mathematical Modeling 20 4 1220 10 20 30 optimal solution: (20/3, 40/3) z = 433 1/3 binding

21. MATH 327 - Mathematical Modeling 21 4 1220 10 20 30 optimal solution: (20/3, 40/3) z = 433 1/3 binding

22. 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 Convex Nonconvex

23. MATH 327 - Mathematical Modeling 23 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. A B C D

24. MATH 327 - Mathematical Modeling 24 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. A B C D C and D are extreme points A and B are not

25. 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.

26. 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?

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35. MATH 327 - Mathematical Modeling 35 2 610 4 8 12 Feasible Region

36. MATH 327 - Mathematical Modeling 36 2 610 4 8 12 Isocost line z = 54

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40. MATH 327 - Mathematical Modeling 40 2 610 4 8 12 Isocost line z = 36 1/4 optimal solution: (5/4, 21/4) z = 36 1/4

41. 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?

42. 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.

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

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

45. 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

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

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

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