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MATH 527 Deterministic OR

Graphical Solution Method for Linear Programs

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Feasible region

The feasible region is a polygon!!

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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.

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Isoprofit linez = 300

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Isoprofit line

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Isoprofit line

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Isoprofit line

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Isoprofit line

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Isoprofit linez = 433 1/3

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

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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!!

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

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

binding

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

binding

binding

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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

ConvexNonconvex

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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

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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 pointsA and B are not

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Interesting Facts

The extreme points of a polygon are the corner points.

The feasible region for any linear program will be a convex set.

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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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Isocost linez = 54

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Isocost line

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Isocost line

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Isocost linez = 36 1/4

optimal solution: (5/4, 21/4)z = 36 1/4

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Special Cases

So far, our models have had– One optimal solution– A finite objective value

Does this always happen?

What if it doesn’t?

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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.

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Special Case #1: Unbounded Linear Programs

maximization minimization

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CAUTION!!!

There is a difference between an unbounded linear program and an unbounded feasible region!!!

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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

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Special Case #2: Infinite Number of Optimal Solutions

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Special Case # 3: Infeasible Linear Program Feasible Region is empty

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Every Linear Program

Has a unique optimal solution, or…..

Has infinite optimal solutions, or…..

Is unbounded, or…..

Is infeasible.