Simplex

MAX: Z = 4 x1 + 3 x2

Sujeto a:6 x1 + 16 x2 48000

12 x1 + 6 x2 420009 x1 + 9 x2 36000

siendo: x1, x2 0 y continuas

6 x1 + 16 x2 + x3 = 4800012 x1 + 6 x2 + x4 = 420009 x1 + 9 x2 + x5 = 36000

Z = 4 x1 + 3 x2 + 0 x3 + 0 x4 + 0 x5

cjck xk B A1 A2 A3 A4 A5 bi/aij

Z

6 x1 + 16 x2 + x3 = 4800012 x1 + 6 x2 + x4 = 420009 x1 + 9 x2 + x5 = 36000

Z = 4 x1 + 3 x2 + 0 x3 + 0 x4 + 0 x5

cj 4 3 0 0 0ck xk B A1 A2 A3 A4 A5 bi/aij

Z

6 x1 + 16 x2 + x3 = 4800012 x1 + 6 x2 + x4 = 420009 x1 + 9 x2 + x5 = 36000

Z = 4 x1 + 3 x2 + 0 x3 + 0 x4 + 0 x5

cj 4 3 0 0 0ck xk B A1 A2 A3 A4 A5 bi/aij

x3x4x5

Z

6 x1 + 16 x2 + x3 = 4800012 x1 + 6 x2 + x4 = 420009 x1 + 9 x2 + x5 = 36000

Z = 4 x1 + 3 x2 + 0 x3 + 0 x4 + 0 x5

cj 4 3 0 0 0ck xk B A1 A2 A3 A4 A5 bi/aij0 x3 48.0000 x4 42.0000 x5 36.000

Z

6 x1 + 16 x2 + x3 = 4800012 x1 + 6 x2 + x4 = 420009 x1 + 9 x2 + x5 = 36000

Z = 4 x1 + 3 x2 + 0 x3 + 0 x4 + 0 x5

cj 4 3 0 0 0ck xk B A1 A2 A3 A4 A5 bi/aij0 x3 48.000 6 16 10 x4 42.000 12 6 10 x5 36.000 9 9 1

Z

6 x1 + 16 x2 + x3 = 4800012 x1 + 6 x2 + x4 = 420009 x1 + 9 x2 + x5 = 36000

Z = 4 x1 + 3 x2 + 0 x3 + 0 x4 + 0 x5


Z = 0

= kk xcZ

6 x1 + 16 x2 + x3 = 4800012 x1 + 6 x2 + x4 = 420009 x1 + 9 x2 + x5 = 36000

Z = 4 x1 + 3 x2 + 0 x3 + 0 x4 + 0 x5


Z = 0 - 4 - 3 0 0 0

jijkjj caccz =

x2

x1

1

2

3

1 32

x3 = 0

x4 = 0

x5 = 0

4

4


Z = 0 - 4 - 3 0 0 0

x2

x1

x3=0

x4=0

x5=0


Z = 0 - 4 - 3 0 0 0

cj 4 3 0 0 0ck xk B A1 A2 A3 A4 A5 bi/aij0 x3 48.000 6 16 1 8.0000 x4 42.000 12 6 1 3.5000 x5 36.000 9 9 1 4.000

Z = 0 - 4 - 3 0 0 0


Z = 0 - 4 - 3 0 0 0

ij

i

abmin=


Z = 0 - 4 - 3 0 0 0


Z = 0 - 4 - 3 0 0 0

x3x1x5


Z = 0 - 4 - 3 0 0 0

0 x34 x10 x5


Z = 0 - 4 - 3 0 0 0

0 x3 14 x1 10 x5 1

0 0 0


Z = 0 - 4 - 3 0 0 0

0 x3 14 x1 3.500 1 1/2 1/120 x5 1

0 0 0


Z = 0 - 4 - 3 0 0 0

0 x3 27.000 14 x1 3.500 1 1/2 1/120 x5 1

0 0 0

6 42.00012

48.000 -


Z = 0 - 4 - 3 0 0 0

0 x3 27.000 13 14 x1 3.500 1 1/2 1/120 x5 1

0 0 0

6 612

16 -


Z = 0 - 4 - 3 0 0 0

0 x3 27.000 13 1 -1/24 x1 3.500 1 1/2 1/120 x5 1

0 0 0

1 612

0 -


Z = 0 - 4 - 3 0 0 0

0 x3 27.000 13 1 -1/24 x1 3.500 1 1/2 1/120 x5 4.500 1

0 0 0

9 42.00012

36.000 -


Z = 0 - 4 - 3 0 0 0

0 x3 27.000 13 1 -1/24 x1 3.500 1 1/2 1/120 x5 4.500 9/2 1

0 0 0

6 912

9 -


Z = 0 - 4 - 3 0 0 0

0 x3 27.000 13 1 -1/24 x1 3.500 1 1/2 1/120 x5 4.500 9/2 -3/4 1

0 0 0

1 912

0 -


Z = 0 - 4 - 3 0 0 0

0 x3 27.000 13 1 -1/24 x1 3.500 1 1/2 1/120 x5 4.500 9/2 -3/4 1

Z = 14.000 0 0 0


Z = 0 - 4 - 3 0 0 0

0 x3 27.000 13 1 -1/24 x1 3.500 1 1/2 1/120 x5 4.500 9/2 -3/4 1

Z = 14.000 0 -1 0 1/3 0

x2

x1

1

2

3

1 32

x3 = 0

x4 = 0

x5 = 0

4

4


Z = 0 - 4 - 3 0 0 0

0 x3 27.000 13 1 -1/24 x1 3.500 1 1/2 1/120 x5 4.500 9/2 -3/4 1

Z = 14.000 0 -1 0 1/3 0

x2

x1

1

2

3

1 32

x3 = 0

x4 = 0

x5 = 0

4

4


Z = 0 - 4 - 3 0 0 0

0 x3 27.000 13 1 -1/2 2.0774 x1 3.500 1 1/2 1/12 7.0000 x5 4.500 9/2 -3/4 1 1.000

Z = 14.000 0 -1 0 1/3 0


Z = 0 - 4 - 3 0 0 0

0 x3 27.000 13 1 -1/2 2.0774 x1 3.500 1 1/2 1/12 7.0000 x5 4.500 9/2 -3/4 1 1.000

Z = 14.000 0 -1 0 1/3 0

x3 1x1 1x2 1.000 1

Z 0 0 0


Z = 0 - 4 - 3 0 0 0

0 x3 27.000 13 1 -1/2 2.0774 x1 3.500 1 1/2 1/12 7.0000 x5 4.500 9/2 -3/4 1 1.000

Z = 14.000 0 -1 0 1/3 0

0 x3 14 x1 13 x2 1.000 1 -1/6 2/9

Z 0 0 0


Z = 0 - 4 - 3 0 0 0

0 x3 27.000 13 1 -1/2 2.0774 x1 3.500 1 1/2 1/12 7.0000 x5 4.500 9/2 -3/4 1 1.000

Z = 14.000 0 -1 0 1/3 0

0 x3 14.000 1 5/3 -26/94 x1 3.000 1 1/6 -1/93 x2 1.000 1 -1/6 2/9

Z 0 0 0


Z = 0 - 4 - 3 0 0 0

0 x3 27.000 13 1 -1/2 2.0774 x1 3.500 1 1/2 1/12 7.0000 x5 4.500 9/2 -3/4 1 1.000

Z = 14.000 0 -1 0 1/3 0

0 x3 14.000 1 5/3 -26/94 x1 3.000 1 1/6 -1/93 x2 1.000 1 -1/6 2/9

Z = 15.000 0 0 0 1/6 2/9

x2

x1

1

2

3

1 32

x3 = 0

x4 = 0

x5 = 0

4

4

x2

x1

1

2

3

1 32

x3 = 0

x4 = 0

x5 = 0

4

4

0 x3 14.000 1 5/3 -26/94 x1 3.000 1 1/6 -1/93 x2 1.000 1 -1/6 2/9

Z = 15.000 0 0 0 1/6 2/9

PROCEDIMIENTO SIMPLEX1. PARTE DE UNA BASE FACTIBLE, Y

CALCULA EL Z2. SE PREGUNTA SI LA SOLUCIN ES

PTIMA. SI LO ES SE TERMINA EL PROCEDIMIENTO.

3. SI NO LO ES, SE PASA A LA BASE VECINA QUE MEJORE MS A Z, SELECCIONANDO UNA VARIABLE QUE INGRESE A LA BASE Y OTRA QUE SALGA.

4. SE VUELVE AL PUNTO 2.

x2

x1

1

2

3

1 32

x3 = 0

x4 = 0

x5 = 0

4

4

0 x3 48.000 6 16 10 x4 42.000 12 6 10 x5 36.000 9 9 1

Z = 0 -4 -3 0 0 0

x2

x1

1

2

3

1 32

x3 = 0

x4 = 0

x5 = 0

4

4

0 x3 27.000 13 1 -1/24 x1 3.500 1 1/2 1/120 x5 4.500 9/2 -3/4 1

Z = 14.000 0 -1 0 1/3 0

x2

x1

1

2

3

1 32

x3 = 0

x4 = 0

x5 = 0

4

4

0 x3 14.000 1 5/3 -26/94 x1 3.000 1 1/6 -1/93 x2 1.000 1 -1/6 2/9

Z = 15.000 0 0 0 1/6 2/9

x2

x1

1

2

3

1 30 2

x3 = 0

x4 = 0

x5 = 0

4

4

SISTEMA DE INECUACIONES

6 x1 + 16 x2 4800012 x1 + 6 x2 420009 x1 + 9 x2 36000

Z = 4 x1 + 3 x2

SISTEMA DE ECUACIONES

6 x1 + 16 x2 + x3 = 4800012 x1 + 6 x2 + x4 = 420009 x1 + 9 x2 + x5 = 36000

Z = 4 x1 + 3 x2 + 0 x3 + 0 x4 + 0 x5

FORMULACIN MATRICIAL DE LAS CONDICIONES DE VNCULO

6 x1 + 16 x2 + x3 = 48.00012 x1 + 6 x2 + x4 = 42.000 9 x1 + 9 x2 + x5 = 36.000

Z = 4 x1 + 3 x2 + 0 x3 + 0 x4 + 0 x5

6 16 112 6 19 9 1

x1x2x3x4x5

=.

48.000

42.000

36.000

FORMULACIN MATRICIAL DEL FUNCIONAL

4 3 0 0 0

x1x2x3x4x5

= .Z

6 x1 + 16 x2 + x3 = 48.00012 x1 + 6 x2 + x4 = 42.000 9 x1 + 9 x2 + x5 = 36.000

Z = 4 x1 + 3 x2 + 0 x3 + 0 x4 + 0 x5

DEFINICIN DE VECTORES

6

129

A1=

6 x1 + 16 x2 + x3 = 48.00012 x1 + 6 x2 + x4 = 42.000 9 x1 + 9 x2 + x5 = 36.000

Z = 4 x1 + 3 x2 + 0 x3 + 0 x4 + 0 x5

A2= A3= A4= A5=16

69

1

00

0

10

0

01

FORMULACIN VECTORIAL

6

129

x1 +

6 x1 + 16 x2 + x3 = 48.00012 x1 + 6 x2 + x4 = 42.000 9 x1 + 9 x2 + x5 = 36.000

Z = 4 x1 + 3 x2 + 0 x3 + 0 x4 + 0 x5

16

69

1

00

0

10

0

01

x3 + x2 + x4 + x5 =

48.000

42.000

36.000


A1 x1 +

6 x1 + 16 x2 + x3 = 48.00012 x1 + 6 x2 + x4 = 42.000 9 x1 + 9 x2 + x5 = 36.000

Z = 4 x1 + 3 x2 + 0 x3 + 0 x4 + 0 x5

A2 A3 A4 A5 x3 + x2 + x4 + x5 =

48.000

42.000

36.000


6 x1 + 16 x2 + x3 = 48.00012 x1 + 6 x2 + x4 = 42.000 9 x1 + 9 x2 + x5 = 36.000

Z = 4 x1 + 3 x2 + 0 x3 + 0 x4 + 0 x5

BxA jj =

=

=

=

35

24

13

bxbxbx

=

+

+

=

=

3

2

1

543

5

4

3

bbb

x100

x010

x001

xxx

B

=m

1kkxAB

554433(1) xc xc xc Z ++=

=m

1kk

)1( xcZ

5544332211 xcxcxcxcxcZ ++++=

+

+

=

=

100

a010

a001

aaaa

A 31211131

21

11

1

+

+

=

=

100

a010

a001

aaaa

A 32221232

22

12

2

=m

1kijj AaA

=+

=+

=+

35131

24121

13111

bxxabxxabxxa

=

=

=

0xabx0xabx0xabx

13135

12124

11113

31

31

21

21

11

11

abx

abx

abx

==ij

i1 a

bminx

Cambio de base

=+

=+

=+

3531

2421

1311

bxabxabxa

Nueva base

=m

1kijj AaA

+=m

1kijj

m

1kk AaAxAB=

m

1kkxAB

)ax(AAB ijm

1kkj +=

)ax(ccZ ijm

1kkj

)2( +=

=m

1kijj AaA

)ax(ccZ ijm

1kkj

)2( +=

+=m

1ijk

m

1kkj

)2( acxccZ

+=m

1ijk

)1(j

)2( acZcZ

= j

m

1ijk

)1()2( cacZZ

( )jj)1()2( czZZ =

( )jj)P()1P( czZZ =+

Slide Number 1Slide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29Slide Number 30Slide Number 31Slide Number 32Slide Number 33Slide Number 34Slide Number 35Slide Number 36Slide Number 37Slide Number 38Slide Number 39Slide Number 40PROCEDIMIENTO SIMPLEXSlide Number 42Slide Number 43Slide Number 44Slide Number 45SISTEMA DE INECUACIONESSISTEMA DE ECUACIONESFORMULACIN MATRICIAL DE LAS CONDICIONES DE VNCULOFORMULACIN MATRICIAL DEL FUNCIONALDEFINICIN DE VECTORESFORMULACIN VECTORIALFORMULACIN VECTORIALFORMULACIN VECTORIALSlide Number 54Slide Number 55Slide Number 56Cambio de baseNueva baseSlide Number 59Slide Number 60Slide Number 61

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