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LP—Graphical Solution Method MSci331—Week 2-3 1

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Page 1: LP—Graphical Solution Method MSci331—Week 2-3 1. Convex Set and Extreme Points 2

LP—Graphical Solution Method

MSci331—Week 2-3

1

Page 2: LP—Graphical Solution Method MSci331—Week 2-3 1. Convex Set and Extreme Points 2

Convex Set and Extreme Points

2

Page 3: LP—Graphical Solution Method MSci331—Week 2-3 1. Convex Set and Extreme Points 2

LP: Example (Papa Louis )

• Papa Louis manufacturers wooden tables and chairs for small kids. Each "table" built: Sells for $27 and uses $10 worth of raw materials, increases Papa Louis’s variable labor/overhead costs by $14. Requires 2 hours of finishing labor and 1 hour of carpentry labor. Each "chair" built: Sells for $21 and uses $9 worth of raw materials, increase Papa Louis’s variable labor/overhead costs by $10. Requires 1 hours of finishing labor AND 1 hour of carpentry labor. Each week Papa Louis can obtain only 100 finishing hours and only 80 carpentry hours. Also demand for the chairs is unlimited. However, at most 40 tables are bought each week. Papa Louis wants to maximize weekly profit (revenues - expenses).

3

Page 4: LP—Graphical Solution Method MSci331—Week 2-3 1. Convex Set and Extreme Points 2

LP: Example (Papa Louis )

4

Page 5: LP—Graphical Solution Method MSci331—Week 2-3 1. Convex Set and Extreme Points 2

LP: Example (Papa Louis )

5

Page 6: LP—Graphical Solution Method MSci331—Week 2-3 1. Convex Set and Extreme Points 2

LP: Example (Papa Louis )

6

Page 7: LP—Graphical Solution Method MSci331—Week 2-3 1. Convex Set and Extreme Points 2

LP: Example (Papa Louis )

7

Page 8: LP—Graphical Solution Method MSci331—Week 2-3 1. Convex Set and Extreme Points 2

8

100

90

80

70

60

50

40

30

20

10

10 20 30 40 50 60 70 80 90 100 BP100

BP200N

umbe

r of

Bod

yPlu

s 20

0

Number of BodyPlus 100

Example: 2

Machining and Welding

Painting and Finishing

Assembly, Test and Packaging

BodyPlus 200 Requirement

Feasible region

(0,0)

(0,45), Z= 20,745

(30,30), Z=24,960

(50,50/ 3), Z=26,233

Max 371BP100 + 461BP200 s.t.

8BP100 + 12BP200 600 Machining and Welding 5BP100 + 10BP200 450 Painting and Finishing 2BP100 + 2BP200 140 Assembly, Test, and Packaging -0.25BP100 + 0.75BP200 0 BodyPlus 200 Requirement

BP100, BP200 > 0

Page 9: LP—Graphical Solution Method MSci331—Week 2-3 1. Convex Set and Extreme Points 2

Example 2: Multiple Optimal Solutions

• Consider the LP model

9

Page 10: LP—Graphical Solution Method MSci331—Week 2-3 1. Convex Set and Extreme Points 2

Example 2: Multiple Optimal Solutions

10

Page 11: LP—Graphical Solution Method MSci331—Week 2-3 1. Convex Set and Extreme Points 2

Infeasible Solution

11

Page 12: LP—Graphical Solution Method MSci331—Week 2-3 1. Convex Set and Extreme Points 2

Unbounded Solution

12

Page 13: LP—Graphical Solution Method MSci331—Week 2-3 1. Convex Set and Extreme Points 2

LP Model to Standard Form

• Convert LP format to a standard form1 21 2

11

2 2

1 2 1 2

1 2 1 2

1 2 51 2 3 4 6

3

4

5

6

, ,

max 12 9max 12 9s.t.s.t.

10001000

1500 1500

1750 1750

4 2 4800 4 2 4800

, 0 , , , 0

x xx x

xx

x x

x x x x

x x x x

x x x

x

x

x x

x

x

x x x

1 21 2

1 21 2

1 41

1 21 2

51 21 2

1 21 2

51 2 3 4 6

3

6

, ,

min 9 6min 9 6 s.t.s.t. 2 10

2 1050

5040

402 100

100 2 152 15

, 0, , , 0

w ww w

w ww w

w ww

w ww w

w w ww w

w ww w

w w w w w w

w

w

13

Page 14: LP—Graphical Solution Method MSci331—Week 2-3 1. Convex Set and Extreme Points 2

Insight from the Geometric Procedure

1 2

1 2

1 2

1

1 2

Max 2 3

. .

5

3 35

20

, 0

Z x x

s t

x x

x x

x

x x

For a constraint to be reasonable, all terms in the constraints must have the same units.

14

Page 15: LP—Graphical Solution Method MSci331—Week 2-3 1. Convex Set and Extreme Points 2

40

35

30

25

20

15

10

5

0 5 10 15 20 25 30 35 40

Insights from the Geometric Procedure

x111

22

33

99

55

x2

66

Constraint 1

Constraint 2

Constraint 3

88

1010

Z

77

11 (x1=0,x2=0,Z=0)

22 (x1=0,x2=5,Z=15)

88 (x1=5,x2=10,Z=40)

1010 (x1=20,x2=5,Z=55)

77 (x1=20,x2=0,Z=40)

1 2Max 2 3Z x x

15

Page 16: LP—Graphical Solution Method MSci331—Week 2-3 1. Convex Set and Extreme Points 2

Basic and Nonbasic Variables

4040

3535

3030

2525

2020

1515

1010

55

0 5 10 15 20 0 5 10 15 20 25 30 35 40 25 30 35 40 xx11111

222

333

999

555

xx22

666

Constraint 1

Constraint 2

Constraint 3

888

101010

Z

777

1 2Max 2 3Z x x

4040

3535

3030

2525

2020

1515

1010

55

0 5 10 15 20 0 5 10 15 20 25 30 35 40 25 30 35 40 xx11111

222

333

999

555

xx22

666

Constraint 1

Constraint 2

Constraint 3

888

101010

Z

777

1 2Max 2 3Z x x

Point Basic VariableNonbasic Variable

Solution (x1,x2,s1,s2,s3)

FeasibilityObjective

Function Value

1 (s1,s2,s3) (x1,x2) (0,0,5,35,20) Yes 0

2 (x2,s2,s3) (x1,s1) (0,5,0,20,20) Yes 15

3 (x2,s1,s3) (x1,s2) (0,11.7,-6.7,0,20) No

4 (x2,s1,s2) (x1,s3) No solution n/a n/a

5 (x1,s2,s3) (x2,s1) (-5,0,0,40,25) No

6 (x1,s1,s3) (x2,s2) (35,0,40,0,-15) No

7 (x1,s1,s2) (x2,s3) (20,0,25,15,0) Yes 40

8 (x1,x2,s3) (s1,s2) (5,10,0,0,10) Yes 40

9 (x1,x2,s2) (s1,s3) (20,25,0,-60,0) No

10 (x1,x2,s1) (s2,s3) (20,5,20,0,0) Yes 55

16

Page 17: LP—Graphical Solution Method MSci331—Week 2-3 1. Convex Set and Extreme Points 2

Point Basic VariableNonbasic Variable

Solution (x1,x2,s1,s2,s3)

FeasibilityObjective

Function Value

123456789

10

Point Basic VariableNonbasic Variable

Solution (x1,x2,s1,s2,s3)

FeasibilityObjective

Function Value

1 (s1,s2,s3) (x1,x2) (0,0,5,35,20) Yes 0

23456789

10

Point Basic VariableNonbasic Variable

Solution (x1,x2,s1,s2,s3)

FeasibilityObjective

Function Value

1 (s1,s2,s3) (x1,x2) (0,0,5,35,20) Yes 0

2 (x2,s2,s3) (x1,s1) (0,5,0,20,20) Yes 15

3456789

10

Basic and Nonbasic Variables

4040

3535

3030

2525

2020

1515

1010

55

0 5 10 15 20 0 5 10 15 20 25 30 35 40 25 30 35 40 xx11111

222

333

999

555

xx22

666

Constraint 1

Constraint 2

Constraint 3

888

101010

Z

777

1 2Max 2 3Z x x

4040

3535

3030

2525

2020

1515

1010

55

0 5 10 15 20 0 5 10 15 20 25 30 35 40 25 30 35 40 xx11111

222

333

999

555

xx22

666

Constraint 1

Constraint 2

Constraint 3

888

101010

Z

777

1 2Max 2 3Z x x

Point Basic VariableNonbasic Variable

Solution (x1,x2,s1,s2,s3)

FeasibilityObjective

Function Value

1 (s1,s2,s3) (x1,x2) (0,0,5,35,20) Yes 0

2 (x2,s2,s3) (x1,s1) (0,5,0,20,20) Yes 15

3 (x2,s1,s3) (x1,s2) (0,11.7,-6.7,0,20) No

456789

10

Point Basic VariableNonbasic Variable

Solution (x1,x2,s1,s2,s3)

FeasibilityObjective

Function Value

1 (s1,s2,s3) (x1,x2) (0,0,5,35,20) Yes 0

2 (x2,s2,s3) (x1,s1) (0,5,0,20,20) Yes 15

3 (x2,s1,s3) (x1,s2) (0,11.7,-6.7,0,20) No

4 (x2,s1,s2) (x1,s3) No solution n/a n/a

5 (x1,s2,s3) (x2,s1) (-5,0,0,40,25) No

6 (x1,s1,s3) (x2,s2) (35,0,40,0,-15) No

7 (x1,s1,s2) (x2,s3) (20,0,25,15,0) Yes 40

8 (x1,x2,s3) (s1,s2) (5,10,0,0,10) Yes 40

9 (x1,x2,s2) (s1,s3) (20,25,0,-60,0) No

10 (x1,x2,s1) (s2,s3) (20,5,20,0,0) Yes 55

17


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