MATH 527 Deterministic OR Graphical Solution Method for Linear Programs.

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Presentation transcript:

MATH 527 Deterministic OR Graphical Solution Method for Linear Programs

MATH Mathematical Modeling

MATH Mathematical Modeling

MATH Mathematical Modeling

MATH Mathematical Modeling

MATH Mathematical Modeling

MATH Mathematical Modeling

MATH Mathematical Modeling

MATH Mathematical Modeling

MATH Mathematical Modeling Feasible region The feasible region is a polygon!!

MATH 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.

MATH Mathematical Modeling Isoprofit line z = 300

MATH Mathematical Modeling Isoprofit line

MATH Mathematical Modeling Isoprofit line

MATH Mathematical Modeling Isoprofit line

MATH Mathematical Modeling Isoprofit line

MATH Mathematical Modeling Isoprofit line z = 433 1/3 optimal solution: (20/3, 40/3) z = 433 1/3

MATH 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 Mathematical Modeling optimal solution: (20/3, 40/3) z = 433 1/3

MATH Mathematical Modeling optimal solution: (20/3, 40/3) z = 433 1/3 binding

MATH Mathematical Modeling optimal solution: (20/3, 40/3) z = 433 1/3 binding

MATH 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

MATH 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

MATH 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

MATH 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 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 Mathematical Modeling

MATH Mathematical Modeling

MATH Mathematical Modeling

MATH Mathematical Modeling

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MATH Mathematical Modeling Feasible Region

MATH Mathematical Modeling Isocost line z = 54

MATH Mathematical Modeling Isocost line

MATH Mathematical Modeling Isocost line

MATH Mathematical Modeling Isocost line

MATH Mathematical Modeling Isocost line z = 36 1/4 optimal solution: (5/4, 21/4) z = 36 1/4

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

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

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

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

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