INVESTIGACION DE OPERACIONES

APICACION DEL METODO SIMPLEX

INGENIERIA INDUSTRIAL 4°

ISRAEL RIVAS ALVAREZ

ASESESOR:

ING. IND MIGUEL ANGEL RUBIO DAMAZO

Aplicacion del metodo simplex .......................................................................................... 3

Ejercicio #1 ............................................................................................................................... 3

Ejercicio #2 ............................................................................................................................... 4

3.- ZMAX = 4X1+3X2+6X3 4X1+3X2+6X3+0S1+0S2 ................................................... 5

EJERCICIO #4.- ....................................................................................................................... 7

EJERCICIO #5.- ....................................................................................................................... 8

EJERCICIO #6 .......................................................................................................................... 9

EJERCICO #7 ......................................................................................................................... 10

EJERCICIO #8 ........................................................................................................................ 11

EJERCICIO #9 ........................................................................................................................ 12

EJERCICIO #10 ..................................................................................................................... 13

EJERCICIO #11 ..................................................................................................................... 14

EJERCICIO #12 ..................................................................................................................... 15

EJERCICIO #13 ..................................................................................................................... 16

Aplicación del método simplex

Ejercicio #1

Zmax

= 2x1+3x

2………….. Z-2x

1-3x

2=0

-3x1+x2

2≤1……………… -3x

1+x

2+S

1=1

4x1+2x

2≤20……………..4x

1+2x

2+S

2=20

4x1-4x

2≤10………………4x

1-4x

2+S

3=10

-x1+2x

2≤5………………..-x

1+2x

2+S

4=5

x1,x

2≥0

z X1 X

2 S

1 S

2 S

3 S

4 R

S1 0 -3 1 1 0 0 0 1

S2 0 4 2 0 1 0 0 20

S3 0 4 -1 0 0 1 0 10

S4 0 -1 2 0 0 0 1 5

X2

1 -2 -3 0 0 0 0 0

0 -3 1 1 0 0 0 1

S2 10 0 -2 1 0 0 18

S3 0 1 0 0 0 1 0 11

S4 0 5 0 -2 0 0 1 3

z 1 -11 0 3 0 0 0 3

X2

0 0 1 -6/5 0 0 3/5 14/5

S2

0 0 0 2 1 0 -2 12

S3

0 0 0 2/5 0 1 -1/5 52/5

X1

0 1 0 -2/5 0 0 1/5 3/5

z 1 0 0 -7/5 0 0 11/5 48/5

Z=48/5 X1=3/5 X

2=14/5

Por lo tanto Zmax

= 2x1+3x

2…………. Zmax= 2(3/5)+3(14/5)

Zmax= 6/5+42/5

Zmax=48/5 o.k

Ejercicio #2

Zmin= 5x1+7x

2………5x

1+7x

2-0S

1-0S

2-0S

3+MA

1+MA

2+MA

3

2x1+3x

2 ≥ 147………..2x

1+3x

2-S

1+A

1=147

3x1+3x

2≥210………….3x

1+3x

2-S

2+A

2=210

X1+x

2≥63……………….x

1+x

2-S

3+A

3=63

X2,x

1≥0

5 7 0 0 0 M M M

Cj X1 X

2 S

1 S

2 S

3 A

1 A

2 A

3 b

MA1 2 3 -1 0 0 1 0 0 147

MA2 3 3 0 -1 0 0 1 0 210

MA3 1 1 0 0 -1 0 0 1 63

Zj 6M 7M -M -M M M M M 420

Cj-Zj 5-6M 7-7M -M M -M 0 0 0

3.- ZMAX = 4X1+3X2+6X3 4X1+3X2+6X3+0S1+0S2 3X1+X2+3X3≤30 3X1+X2+3X3+S1=30

2X1+2X2+3X3≤40 2X1+2X2+3X3+S1=40

CJ 4 3 6 0 0

X1 X2 X3 S1 S2 B

0S1 3 1 3 1 0 30

0S2 2 2 3 0 1 40

ZJ 0 0 0 0 0 0

CJ-ZJ 4 3 6 0 0 ---

CJ 4 3 6 0 0

X1 X2 X3 S1 S2 B

6X1 1 1/3 1 1/3 0 10

-3R1+R2 0S2 -1 1 0 -1 0 10

ZJ 6 2 6 0 0 60

CJ-ZJ -2 1 0 0 0 ---

CJ 4 3 6 0 0

X1 X2 X3 S1 S2 B

-1/3 R2+R1 6X3 4/3 0 6 1/3 0 20/3

3X2 -1 1 0 -1 0 10

ZJ 5 3 6 -1 0 70

CJ-ZJ -1 0 0 1 0 ---

CJ

4 3 6 0 0

X1 X2 X3 S1 S2 B

R2/1/3 0S1 4 0 18 1 0 20

R1+R2 3X2 3 1 0 0 0 30

ZJ 9 3 0 0 0 90

CJ-ZJ -5 0 6 0 0 ---

CJ

4 3 6 0 0

X1 X2 X3 S1 S2 B

6X3 4/18 0 1 1/18 0 20/18

3X2 3 1 0 0 0 30

ZJ 10 1/3 3 6 1/3 0 96 2/3

CJ-Z -6 1/3 6 0 -1/3 0 ---

X3=10

X2=10

S1=20

Zj=96 2/3

EJERCICIO #4.-

ZMAX =3X1+4X2+5X3 3X1+4X2+5X3+0S1+0S2+0S3

3X1+X2+5X3≤150 3X1+X2+5X3+S1=150

X1+4X2+X3≤120 X1+4X2+X3+S2=120

2X1+2X3≤105 2X1+2X3+S3=105

3 4 5 0 0 0 B

CJ X1 X2 X3 S1 S2 S3

0S1 3 1 5 1 0 0 150

0S2 1 4 1 0 1 0 120

0S3 2 0 2 0 0 1 105

ZJ 0 0 0 0 0 0 0

CJ-ZJ 3 4 5 0 0 0 -

3 4 5 0 0 0 B

CJ X1 X2 X3 S1 S2 S3

-5R2+R1

0S1

-2 -19 0 1 -5 0 -450

5X3 1 4 1 0 1 0 120

-2R2+R3

0S3

0 -8 0 0 -2 1 -135

ZJ 5 20 5 0 5 0 600

CJ-ZJ -2 -16 0 0 -5 0 -

X1=0

X2=0

X3=120

ZJ=600

EJERCICIO #5.-

Z MAX= 5X1+3X2+4X3 5X1+3X2+4X3+0S1+0S2

2X1+X2+X3≤120 2X1+X2+X3+S1=120

3X1+X2+2X3≤30 3X1+X2+2X3+S2=30

5 3 4 0 0

CJ X1 X2 X3 S1 S2 B

0S1 2 1 1 1 0 120

0S2 3 1 2 0 1 30

ZJ 0 0 0 0 0 0

CJ-ZJ 5 3 4 0 0 -

5 3 4 0 0

CJ X1 X2 X3 S1 S2 B

-2R2+R1

0S1

0 1/3 -1/3 1 -2/3 100

5X1 0 1/3 2/3 0 1/3 10

ZJ 5 5/3 10/3 0 5/3 50

CJ-ZJ 0 4/3 2/3 0 -5/3 -

5 3 4 0 0

CJ X1 X2 X3 S1 S2 B

-1/3R2+R1

3X2

-1 0 -1 1 -1 90

SX1 3 1 2 0 1 30

ZJ 12 5 7 3 2 420

CJ-ZJ -7 -2 -3 -3 -2 -

X1=10

X2=30

ZJ=420

EJERCICIO #6

ZMAX=2X1+4X2 +3X3…………2X1+4X2 +3X3+0S1+0S2+0S3

X1 +3X2+2X3 ≤ 30………… X1 +3X2+2X3+S1= 30

X1+X2+X3 ≤ 24………… X1+X2+X3+0s 2 =24

3X1+5X2+3X3 ≤ 60………… 3X1+5X2+3X3 +0S3 = 60

X1, X2, X3 ≤ 0

X1 X2 X3 S1 S2 S3 b Cj 2 4 3 0 0 0

0S1 1 3 2 1 0 0 30 30/10=10 0S2 1 1 1 0 1 0 24 24/1=24 0S3 1 5 3 0 0 1 60 60/5=12 Zj 0 0 0 0 0 0 0

Cj-Zj 2 4 3 0 0 0

X1 X2 X3 S1 S2 S3 b Cj 2 4 3 0 0 0

R1(1/3) 4X1 1/3 1 2/3 1/3 0 0 10 -R1+R2 S2 2/3 0 1/3 -1/3 1 0 14

-5R1+R3 S3 -2/3 0 -1/3 -5/3 0 1 10 Zj 4/3 4 8/3 4/3 0 0 40 Cj-Zj 2/3 0 1/3 -4/3 0 0

X1=0 X2=10 X3=0 Zj=40

EJERCICO #7

ZMIN=5000X1+7000X2 …………5000X1+7000X2 -0S1-0S2+MA1+MA2

-2X1 +X2≥1………… -2X1 +X2 -S1+A1= 1

X1 -2X2≥1………… X1 -2X2 –S2+A2= 1

X1, X2, X3 ≥0

X1 X2 X3 S1 S2 S3 b Cj 5000 7000 0 0 M M

MA1 -2 1 -1 0 1 0 1 1/-2=-1/2 MA2 1 -2 0 -1 0 1 1 1/1=1

Zj -M -M -M -M M M M Cj-Zj 5OOO-M 7000-M M M 0 0

X1 X2 X3 S1 S2 S3 b Cj 5000 7000 0 0 M M

2R2+R1 MA1 0 -3 -1 0 1 0 1 5000X2 1 -2 0 -1 0 1 1 Zj 5000 -3M-10000 -M -2M-5000 M 2M+5000 3M+5000 Cj-Zj 0 3M+17000 M 2M+5000 0 -M-5000

X1=1 X2=0 Zj=3M+5000

EJERCICIO #8

ZMAX=5X1+9X2 +7X3…………5X1+9X2 +7X3+0S1+0S2+0S3

X1 +3X2+2X3 ≤ 10………… X1 +3X2+2X3+S1= 10

3X1+4X2+2X3 ≤ 12………… 3X1+4X2+2X3+0S2 =12

2X1+X2+2X3 ≤ 8………… 2X1+X2+2X3+0S3 = 8

X1, X2, X3 ≤ 0

X1 X2 X3 S1 S2 S3 b Cj 5 9 7 0 0 0

0S1 1 3 2 1 0 0 10 10/3=3.3 0S2 3 4 2 0 1 0 12 12/4=3 0S3 2 1 2 0 0 1 8 8/1=8 Zj 0 0 0 0 0 0 0

Cj-Zj 5 0 7 0 0 0

X1 X2 X3 S1 S2 S3 b Cj 5 9 7 0 0 0

-3R2+R1 0S1 -5/4 0 1/2 1 -3/4 0 1 1/1/2= 2 1/4R2 9X2 3/4 1 1/2 0 1/4 0 3 3/1/2 = 6 -R1+R3 0S3 5/4 0 3/2 0 -1/4 1 5 5/ ½ = 10

Zj 27/4 9 9/2 0 9/4 0 27 Cj-Zj -7/4 0 5/2 0 -9/4 0

X1 X2 X3 S1 S2 S3 b Cj 5 9 7 0 0 0

2R1 7X3 -5/2 0 1 2 -6/4 0 2 -1/2R1 +R2 9X2 2 1 0 -1 1 0 2 -3/2R1 +R3 0S3 5 0 0 -3 2 1 2

Zj 1/2 9 7 5 -3/2 0 32 Cj-Zj 9/2 9 0 -5 -3/2 0

X1=0 X2=2 X3=2 Zj=32

EJERCICIO #9

Z MAX= 3x1+5x2+6x3 3x1+5x2+6x3

2x1+x2+x3 ≤ 4 2x1+x2+x3+s1 = 4

X1+2x2+x3 ≤ 4 X1+2x2+x3 + s2= 4

X1+x2+2x3 ≤ 4 X1+x2+2x3+ s3 = 4

X1+x2+x3 ≤ 3 X1+x2+x3+s4 = 3

CJ-ZJ 3 5 6 0 0 0 0 -

CJ

3 X1

5 X2

6 X3

0 S1

0 S2

0 S3

0 S4

b/R

3/2 1/2 0 1 0 -1/2 0 2

½ ½ 0 0 1 -1/2 0 0

½ ½ 1 0 0 ½ 0 2

½ ½ 0 0 0 -1/2 1 1

ZJ 3 3 6 0 0 3 0 12

CJ-ZJ 0 2 0 0 0 0 0 1

1 0 0 1 0 0 -2 1

0 0 0 0 1 0 -2 1

0 0 1 0 0 1 -2 1

1 1 0 0 0 -1 2 2

ZJ 5 5 6 0 0 1 -2 16

CJ-ZJ -2 0 0 0 0 -1 2 -

CJ

3 X1

5 X2

6 X3

0 S1

0 S2

0 S3

0 S4

b/R

S1 2 1 1 1 0 0 0 4/1= 4

S2 1 2 1 0 1 0 0 4/1=4

S3 1 1 2 0 0 1 0 4/2=2

1 1 1 0 0 0 1 3/1=3

ZJ 0 0 0 0 0 0 0 0

ZJ=16

X3 =2

X2 =2

X1=0

S4

EJERCICIO #10

Z MIN= 2X1+3X2+X3 2X1+3X2+X3 -0S1 -0S 2 +0S3+MA1+MA2

X1+4X2+2X3 ≥ 8 X1+4X2+2X3-S1+A1 = 8

3X1+2X2 ≥ 6 3X1+2X2 - S2+A2 = 6

X1, X2, X3 ≥ 0

CJ

3 X1

5 X2

6 X3

0 S1

0 S2

0 A1

0 A2

b/R

MA1 1 4 2 -1 0 1 0 8

MA2 3 2 0 0 -1 0 1 6

ZJ 4M 6M 2M -M -M M M 14M

CJ-ZJ 2-4M

3-6M 1-2M M M -M -M -

-R2+R1

MA1

0 10/3 2 -1 1/3 1 -1/3 6

2X1 1 2/3 0 0 -1/3 1 -1/3 6

ZJ 2 4/3+10/3M 2M -M -2/3+1/3M 4 2/3+(-1/3M)

4+6M

CJ-ZJ 0 3-(2/3+10/3M)

1-2M M -(-2/3+1/3M) -4 -(2/3(-1/3M)

-

R1/ 10/3 3X2

0 1 -3/10 -3/10 1/10 3/10 -1/10 9/5

-2/3R1+R2

2X2

1 0 -2/5 1/5 -2/5 -1/5 1/15 4/5

ZJ 2 3 -1/2 -1/2 -1/2 1/2 -1/6 7

CJ-ZJ 0 0 -1/2 1/2 1/2 -1/2 1/6 -

R1/2/3 X3 0 5/3 1 -1/2 1/6 ½ -1/6 3

2/5R1+R2

2X1

1 2/3 0 0 -1/3 0 -2/15 2

ZJ 2 3 1 -1/2 -1/2 1/2 -13/30 7

CJ-ZJ 0 0 0 1/2 1/2 -1/2 13/30 -

X1=2

X2=9/5

X3=3

ZJ=7

EJERCICIO #11

Zmin= 2X1+ X2+ 3X3------------------------------2X1+ X2+ 3X3+0S1-0S2+MA1

5X1+2X2+7X3=420----------5X1+2X2+7X3+0S1=420

3X1+2X2+5X3≥280---------3X1+2X2+5X3-0S2+A1=280

X1,X2,X3≥0

Zj=168

X1=60

X2=0

X3=56

CJ X1 X2 X3 0S1 0S2 A1 R

0S1 5 2 7 1 0 0 420

MA1 3 2 5 0 -1 1 280

Zj 3M 2M 5M 0 -M M 280M

CJ-Zj 2-3M 1-2M 3-5M 0 +M 0 --------

CJ X1 X2 X3 0S1 0S2 A1 R

0S1 4/5 -4/5 0 1 7/5 -7/5 60

3x3 3/5 2/5 1 0 -1/5 1/5 56

Zj 9/5 6/5 3 0 -3/5 3/5 168

CJ-Zj 1/5 -1/5 0 0 +3/5 m-3/5 --------

2 1 3 0 0 M

420/7=60

280/5=56

-7(R2)+R1

R2(1/5)

2 1 3 0 0 M

EJERCICIO #12

Zmin= 3X1+ 2X2+ 4X3------------------------------3X1+ 2X2+ 4X3+0S1-0S2+MA1

2X1+X2+3X3=60----------2X1+X2+3X3+S1=60

3X1+5X2+5X3=120---------3X1+3X2+5X3-S2+A1=120

X1,X2,X3≥0

Zj=96

X3=24

X2=0

X1=0

CJ X1 X2 X3 0S1 0S2 A1 R

0S1 2 1 3 1 0 0 60

MA1 3 3 5 0 1 1 120

Zj 3M 3M 5M 0 M M 120M

CJ-Zj 3-3M 2-3M 4-5M 0 -M -M --------

CJ X1 X2 X3 0S1 0S2 A1 R

0S1 2/3 1/3 1 1/3 0 0 20

4x3 -1/3 1/3 0 -5/3 1 1 24

Zj -4/3 16/3 0 -20/3 4 4 96

CJ-Zj 13 -10/3 4 20/3 4 4M --------

60/3=20

120/5=24

3 2 4 0 0 M

R1(1/3)

-5R1+R2

3 2 4 0 0 M

EJERCICIO #13

Zmax= -X1+ 2X2+ X3------------------------------3X1+ 2X2+ 4X3+0S1-0S2+MA1

+3X2+X3≤120----------3X2+X3-S1 =120

X1-X2-4X3≤120--------- X1-X2-4X3-S2=80

-3X1+X2+2X3≤100-------3X1+X2+2X3-S3=100

X1,X2,X3≥0

CJ X1 -1

X2 2

X3 1

-S1 0

-S2 0

-S3 0

R

0S1 0 3 1 -1 0 0 120

0S2 1 -1 -4 0 -1 0 80

0S2 -3 1 2 0 0 -1 100

Zj 0 0 0 0 0 0

Cj-Zj -1 2 1 0 0 0

CJ X1 -1

X2 2

X3 1

-S1 0

-S2 0

-S3 0

R

2X2 0 1 1/3 -1/3 0 0 40

0S2 1 0 -11/3 -1/3 -1 1 120

0S2 -3 0 5/3 1/3 0 -1

Zj 0 2 2/3 -2/3 0 0 80

Cj-Zj -1 0 1/3 2/3 0 0

ZJ=80

X1=0

X2=40

X3=0

180/-1=-180

120/3=40

100/-1=-100

R1+R2

R1(1/3)

-1R1+R3