math 527 deterministic or

MATH 527 Deterministic OR

Graphical Solution Method for Linear Programs

MATH 327 - Mathematical Modeling

2

4 12 20

10

20

30

6025 21 xx


3

4 12 20

10

20

30

6025 21 xx


4

4 12 20

10

20

30

6025 21 xx


5

4 12 20

10

20

30

6025 21 xx

2021 xx


6

4 12 20

10

20

30

6025 21 xx

2021 xx


7

4 12 20

10

20

30

6025 21 xx

2021 xx


8

4 12 20

10

20

30

6025 21 xx

2021 xx

01 x


9

4 12 20

10

20

30

6025 21 xx

2021 xx

01 x

02 x


10

4 12 20

10

20

30

6025 21 xx

2021 xx

01 x

02 x

Feasible region

The feasible region is a polygon!!


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

4 12 20

10

20

30

Isoprofit linez = 300


13

4 12 20

10

20

30

Isoprofit line


14

4 12 20

10

20

30

Isoprofit line


15

4 12 20

10

20

30

Isoprofit line


16

4 12 20

10

20

30

Isoprofit line


17

4 12 20

10

20

30

Isoprofit linez = 433 1/3

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


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

4 12 20

10

20

30



20

4 12 20

10

20

30


binding


21

4 12 20

10

20

30


binding

binding


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

ConvexNonconvex


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

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


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

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?


27

2 6 10

4

8

12

1826 21 xx


28

2 6 10

4

8

12

1826 21 xx


29

2 6 10

4

8

12

1826 21 xx


30

2 6 10

4

8

12

1826 21 xx

3053 21 xx


31

2 6 10

4

8

12

1826 21 xx

3053 21 xx


32

2 6 10

4

8

12

1826 21 xx

3053 21 xx


33

2 6 10

4

8

12

1826 21 xx

3053 21 xx

01 x


34

2 6 10

4

8

12

1826 21 xx

3053 21 xx

01 x02 x


35

2 6 10

4

8

12

1826 21 xx

3053 21 xx

01 x02 x

Feasible Region


36

2 6 10

4

8

12

Isocost linez = 54


37

2 6 10

4

8

12

Isocost line


38

2 6 10

4

8

12

Isocost line


39

2 6 10

4

8

12

Isocost line


40

2 6 10

4

8

12

Isocost linez = 36 1/4



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

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

Special Case #1: Unbounded Linear Programs

maximization minimization


44

CAUTION!!!

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


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

Special Case #2: Infinite Number of Optimal Solutions


47

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


48

Every Linear Program

Has a unique optimal solution, or…..

Has infinite optimal solutions, or…..

Is unbounded, or…..

Is infeasible.

math 527 deterministic or

Documents