cap01 - programación dinámica

5/24/2018 Cap01 - Programaci n Din mica

1/70

P D

1


2/70

P D (PD) T

B D 1957 S

:

.(R).

PD: T , .

2


3/70

A P D

C P D

E B

P D D P D P

3


4/70

P D U EE.UU.

, .

A ,

. E .

E

. L

.

4


5/70

P D

E

C

C ( ) ?

1

2

2

3

3 3

3

3

3

4

4

4

4

4

4

1

6

7

6

5

610

9

8

3

2 5

4 7

1

5


6/70

P D

1

2

2

3

3 3

3

3

3

4

4

4

4

4

4

1

6

7

6

5

6

10

9

8

3

2 5

4 7

1

S 1: 126910 > C 13

.146 126

S 2: ,

(13321=18)

6


7/70

P D

S 3:PROGRAMACIN

DINMICA

P . E ,

, .

D A : E 4

7


8/70

P DM S

V ,

E, (=1,2,3,4) .

E

),( nn xsf C , S (

) X .

10;1 44321 = xxxxx

( S ), ( S ), ( S ), ( S )1234

* * * *

*n

( S ) = MinXn

( S, Xn) =( S, Xn)n }{ n *

n

Xn

( S0 ,Xn )n

( S1 ,Xn )n

C S X

8


9/70

P DM S

ETAPA 4 ; = 4

S X4

8 3 10 9 4 10

(S)4

**

ETAPA 3 ; = 3

S 8 9 X3

5 1 + 3 = 4 4 + 4 = 8 4 8

6 6 + 3 = 9 3 + 4 = 7 7 9

7 3 + 3 = 6 3 + 4 = 7 6 8

X3

(S,X3 )= CSX + (X3 )3*

*

3 4

(S)3*

9


10/70

P DM S

ETAPA 2 ; = 2

S 5 6 7 X2

2 7 + 4 = 11 4 + 7 = 11 6 + 6 = 12 11 5 6

3 4 + 6 = 10

4

X2

(S,X2 )= CSX + (X3 )2*

2 3

*(S)2

*

ETAPA 1 ; = 1

S 2 3 4 X1

1 2 + 11 = 13 4 + 7 = 11 3 + 8 = 11 11 3 4

X1

(S,X1 )= C1X + (X1 )1*

1 2

*(S)1*

Identifiquemos

la solucinoptima.

}1 3 5 8 10

1 4 5 8 10

1 4 6 9 10

E , PD.

10


11/70

C PD C : P

. P:

S .

E .

11


12/70

E PD E ,

. C . E ,

. P . E

.

E . U .

12


13/70

E PD L

. E :

D ,

(1, , +1). E

.

13


14/70

E PD S

, (+1). U ,

, .

( S ) = Mx Min ( S, Xn)*

n n{ }Xn

( S ) = Min CSXn + ( Xn)* *

Xn{ }

n +1n

..

E A ()

R

(+1)

=),( nn xsf

)(* sfn

*

nx2x1x 3xs

14


15/70

P D

DE 1:

U . L .

E .

P , . S , . L

:

N decargas

Tienda1 2 3

0 0 0 0

1 5 6 4

2 9 11 9

3 14 15 13

4 17 19 18

5 21 22 20

15


16/70

E 1:

N decargas

Tienda1 2 3

0 0 0 0

1 5 6 4

2 9 11 9

3 14 15 13

4 17 19 18

5 21 22 20

16

P D

D


17/70

E 1:

: (=1,2,3): , 0,1,2,3,4,5

: +1=

:

*

*

17

P D

D


18/70

E 1:

E 3, =3, 4=0

3 3

0 0 0

1 4 1

2 9 2

3 13 3

4 18 4

5 20 5

*

18

P D

D


19/70

E 1:E 2, =2

2/2 0 1 2 3 4 5 *

2() *2

0 0 0 0

1 0+4=4 6+0=6 6 1

2 0+9=9 6+4=10 11+0=11 11 2

3 0+13=13 6+9=15 11+4=15 15+0=15 15 123

4 0+18=18 6+13=19 11+9=20 15+4=19 19+0=19 20 25 0+20=20 6+18=24 11+13=24 15+9=24 19+4=23 22+0=22 24 123

19

P D

D


20/70

E 1:E 1, =1

1/1 0 1 2 3 4 5 1*(1)

*1

5 0+24=24 5+20=25 9+15=24 14+11=25 17+6=23 21+0=21 25 13

1 2 2

3 2 0

S 1 S 2 S 3

2

20

P D

D


21/70

E 2: I

U : 1, 1 ; 2, 3 ; 3, 2; 4, 4 .

D ,

3US$. A, 1 . A , 50 .

L 5 . L

4 , .S 0 .

L

. 21

P D

D


22/70

E 2

: , =1,2,3,4,5: , 0,1,2,3,4,5

: : , 0,1,2,3,4

*

: , =1,2,3,41=05=0+1=+

22

P D

D


23/70

E 2

E 4, =4, 4=4, 5=0

4\4 0 1 2 3 4 5

0 7

1 6

2 5

3 4

4 0

5=4+44

* *

4 4(4) 4

0 7 4

1 6 3

2 5 2

3 4 1

4 0 0

*

23

P D

D


24/70

E 2

E 3, =3, 3=2

3*(3) *3

3\3 0 1 2 3 4 5

0 5+7 =12 6.5+6=12.5 8+5=13 9.5+4=13.5 12 2

1 4+7=11 5.5+6=11.5 7+5=12 8.5+4=12.5 10+0=10 10 5

2 0+7=7 4.5+6=10.5 6+5=11 7.5+4=11.5 9+0=9 7 0

3 0.5+6=6.5 5+5=10 6.5+4=10.5 8+0=8 6.5 0

4 1+5=6 5.5+4=9.5 7+0=7 6 0

4= 3+330


25/70

E 2

E 2, =2, 2=3

*2(2) *2

2\2 0 1 2 3 4 5

0 6+12=18 7.5+10=17.5 9+7=16 16 5

1 5+12=17 6.5+10 =16.5 8+7=15 9.5+6.5=16 15 4

2 4+12=16 5.5+10=15.5 7+7 = 14 8.5+6.5=15 10+6=16 14 3

3 0+12=12 4.5+10=14.5 6+7=13 7.5+6.5=14 9+6=15 12 0

4 0.5+10=10.5 5+7=12 6.5+6.5=13 8+6=14 10.5 0

*

3=2+220


26/70

E 2

E 1, =1, 1=1, 1=0

2=1+110


27/70

E 2

1 5 0 4

1 3 2 4

0 2 0 0

M 1 M 2 M 3 M 4

27

P D

D


28/70

E 3:

U .

N , , .

C

, . E :

Das

Puntos deCalificacinEstimados

Cursos

1 2 3 4

1 3 5 2 6

2 5 5 4 7

3 6 6 7 94 7 9 8 9

28

P D

D


29/70

E 3:

: =(1,2,3,4): , (1,2,3,4)

: +1=

:

*

*

29

P D

D


30/70

E 3:

E 4, =4, 5=0

4 4

1 6 1

2 7 2

3 9 3

4 9 4

*

30

P D

D

Das

Puntos deCalificacinEstimados

Cursos

1 2 3 41 3 5 2 6

2 5 5 4 7

3 6 6 7 9

4 7 9 8 9


31/70

E 3:E 3, =3

3/3 1 2 3 4 3(3) 3

1

2 2+6=8 8 1

3 2+7=9 4+6=10 10 2

4 2+9=11 4+7=11 7+6=13 13 3

5 2+9=11 4+9=13 7+7=14 8+6=14 14 34

31

P D

D4 4

*(4) 4

1 6 1

2 7 2

3 9 3

4 9 4


32/70

E 3:E 2, =2

2/2 1 2 3 4 2(2) 2

1

2

3 5+8=13 13 1

4 5+10=15 5+8=13 15 1

5 5+13=18 5+10=15 6+8=14 18 16 5+14=19 5+13=18 6+10=16 9+8=17 19 1

32

P D

D


33/70

E 3:E 1, =1

1/1 1 2 3 4 1*() 1

7 3+19=22 5+18=23 6+15=21 7+13=20 23 2

2 1 3 1

C 1 C 2 C 3 C 4

33

P D

D


34/70

E 4: E

. A , .

P ( )

. L : . (P , , , ).

N deEquiposMdicos

Miles de aos

de vidaadicional del

hombre

Pas

1 2 3

0 0 0 0

1 45 20 50

2 70 45 703 90 75 80

4 105 110 100

5 120 150 130

Q ,

? C

? Q

? 34

P D

D


35/70

E 4:

E = P; = 1,2,3V: X = E

E: S = E

N deEquiposMdicos

Miles de aos

de vidaadicional delhombre

Pas

1 2 30 0 0 0

1 45 20 50

2 70 45 70

3 90 75 80

4 105 110 100

5 120 150 130

)( nn xP A () X .

=

=

=

=

=

ii

i

ni

i

ni

ii

xx

Sxas

xPZMax

;0

.

)(

3

3

N ;

35

P D

D


36/70

E 4:

36

P D

D


37/70

E 4:

E = P; = 1,2,3V: X = E

E: S = E

N deEquiposMdicos

Miles de aos

de vidaadicional delhombre

Pas

1 2 30 0 0 0

1 45 20 50

2 70 45 70

3 90 75 80

4 105 110 100

5 120 150 130

)( nn xP A

() X .

=

=

=

=

=

ii

i

ni

i

ni

ii

xx

Sxas

xPZMax

;0

.

)(

3

3

N ;

37

P D

D


38/70

E 4:

=

=

+=

=

+=

+=

ii

nn

i

nii

ni

ii

nnnnn

xx

xSxas

xPZMax

xPxsf

;0

.

)(

)(),(

3

1

3

1

:),( nn xsfA , , S X.

)(* 1 nnn xsf +

( S, Xn

) = Pn

( Xn

)+ ( S - Xn

)n+1n

*

L

38

P D

D


39/70

E 4:

s *3x

0 0 0

1 1 50

2 2 70

3 3 80

4 4 100

5 5 130

)(*3 sf

( S, Xn) = Pn ( Xn)+ ( S - Xn)n+1n

* N deEquiposMdicos

Miles de aosde vida

adicional delhombre

Pas

1 2 30 0 0 0

1 45 20 50

2 70 45 70

3 90 75 80

4 105 110 100

5 120 150 130

39

P D

D


40/70

E 4:

( S, Xn) = Pn ( Xn)+ ( S - Xn)n+1n

*

Miles de aos de vida adicionales

del hombre

N de Equipos Pas

Mdicos 1 2 3

0 0 0 0

1 45 20 50

2 70 45 70

3 90 75 80

4 105 110 100

5 120 150 130

0 1 2 3 4 5

0 0+0=0 0 0

1 0+50=50 20+0=20 50 0

2 0+70=70 20+50=70 45+0=45 70 01

3 0+80=80 20+70=90 45+50=95 75+0=75 95 2

4 0+100=100 20+80=100 45+70=115 75+50=125 110+0=110 125 3

5 0+130=130 20+100=120 45+80=125 75+70=145 110+50=160 150+0=150 160 4

)()(),( 2*

32222 xsfxPxsf += )(*2 sf*

2x

40

P D

D

P D


41/70

E 4:

( S, Xn) = Pn ( Xn)+ ( S - Xn)n+1n

*

Miles de aos de vida adicionales

del hombre

N de Equipos Pas

Mdicos 1 2 3

0 0 0 0

1 45 20 50

2 70 45 70

3 90 75 80

4 105 110 100

5 120 150 130

s1x

0 1 2 3 4 5

5 0+160=160 45+125=170 70+95=165 90+70=160 105+50=155 120+0=120 170 1

)()(),( 1*

21111 xsfxPxsf +=)(*1 sf

*

1x

1 3 1

P 1 P 2 P 3

41

P D

D

P D


42/70

E 5: V MU

M. T .

S ,

1, 2 3 0.40, 0.60 0.80. P , (0.40)(0.60)(0.80)=0.192 (19.2%).

D ,

.

N de

Cientficos

Nuevos

Probabilidad de Falla

Equipo de Investigacin

1 2 3

0 0,40 0,60 0,80

1 0,20 0,40 0,50

2 0,15 0,20 0,30

)( nn xP P X

=++

=

;0

2..

)(*)(*)(

321

332211

ix

xxxas

xPxPxPZMin

42

P D

D

P D


43/70

E 5: V M

=

=

= +=

+=

;0

..

)(

*)(),(3

1

31

i

n

ni

i

iini

nnnn

x

xsxas

xPZMin

xPxsf

1)0(*4 =f

),( nn xsf P , +1,..3, S, X .

)(* 1 nn xsf +

)(*)(),( * 1 nnnnnn xsfxPxsf = +43

P D

D

P D


44/70

E 5: V M N deCientficos

Nuevos



1 2 3

0 0,40 0,60 0,80

1 0,20 0,40 0,50

2 0,15 0,20 0,30

0 1 2

0 0.6*0.8=0.48 0.48 0

1 0.6*0.5=0.3 0.4*0.8=0.32 0.3 0

2 0.6*0.3=0.18 0.4*0.5=0.2 0.2*0.8=0.16 0.16 2

)(*)(),( 2*

32222 xsfxPxsf =

)(*

2 sf

*

2xs2x

0 0.8 0

1 0.5 1

2 0.3 2

)(*

3 sf*

3xs

44

P D

D

P D


45/70

E 5: V MN de

Cientficos

Nuevos



1 2 3

0 0,40 0,60 0,80

1 0,20 0,40 0,50

2 0,15 0,20 0,30

0 1 2

2 0.4*0.16=0.064 0.2*0.3=0.06 0.15*0.48=0.072 0.06 1

)(*)(),( 2*

21111 xsfxPxsf =)(*1 sf

1xs

*

1x

1 0 1

E 1 E 2 E 3

45

P D

D

P D


46/70

E 6: P V ( )L S O A. L

, .L .

A, , .

1. P .2. R , =3,

=5, =US$ 1200, 1=$ 200, 2=$ 300, 3=$100 :

46

P D

D


47/70

1. F: N (=1,2K).

: C .

: D .

: D .

: C .

: B .

n

nX

nS

nH

nC

( )n ng X

47


48/70

F O:

1n n n nS S C X + = A :

C :

*

S1 = P

H1

= M

fn+1 = 0

Hn+1 > 0

Sn+1 > 0

48


49/70

2.1 2 3

200, 300, 100, 3, 5

3

C US C US C US K M

n

= = = = =

=

0 1 2 3 4 5

(0-99; 0-5)

(100-1200; 0)0 - - - - - 0 0

(100-199; 1-5)

(200-1200; 1)0 1 - - - - 1 1

(200-299; 2-5)

(300-1200; 2)0 1 3 - - - 3 2

(300-399; 3-5)

(400-1200; 3)0 1 3 3 - - 3 23

(400-499; 4-5)

(500-1200; 4)0 1 3 3 2 - 3 23

(500-1200; 5) 0 1 3 3 2 1 3 23

*3f *3X(S3;H3)

*

49


50/70

O

f3*(S3,H3) =

0

1

3

Si S3 < 100 o H3 = 0

Si (100 < S3 < 200 y H3 > 1)

o (S3 >100 y H3 = 1)

De otra manera

50


51/70

2n =

0 1 2 3 4 5

200 0 0 - - - - - 0 0

400 1 0+1=1 1+0=1 - - - - 1 01

600 2 0+3=3 1+1=2 4+0=4 - - - 4 2

800 3 0+3=3 1+3=4 4+1=5 - - - 5 2

1000 4 0+3=3 1+3=4 4+3=7 6+1=7 - - 7 23

1200 5 0+3=3 1+3=4 4+3=7 6+3=9 8+0=8 - 9 3

2H2S *2f *2X*

51


52/70

1n=

0 1 2 3 4 5

1200 5 0+9=9 1+7=8 2+5=7 3+4=7 4+1=5 5+0=5 9 0

1H1S *1f *1X

S ( )

L : 0*US 200 + 3*US 300 + 2*US 100 =

*

52

P D


53/70

P D

PE R I

E . E

.

A , , .

53

P D


54/70

P D

P E I

Sn XnDecisin

P1

P2

PN

1

C1

C2

CN

2

N

(1)*n+1

(1)*

n+1

(1)*

n+1

Etapa n ProbabilidadContribucin

de la etapa n Etapa n+1

Estado:

(Sn, Xn)

n

54

P D


55/70

P D

P E I

E 1: P

L C. I H A M . S .

E ( ), .

A, L ; , ()L.

55

P D


56/70

P D

P E I

E 1: P ()

S CM ( ) $ 100 ( ) .

A, $ 300, . E . S , $ 1600.

E , .

56


57/70

O , K(xn) = 3*min(1,xn)

57


58/70

* *

1 1

*

1

*

4

Por lo tanto para s 1,

1 1(1, ) ( ) (1) 1 (0)

2 2

1( ) (1)

2

Donde (1) es igual a 16, el costo terminal por no haber obtenido art culos ace

n n

n

n

x x

n n n n n n

x

n n n

f x K x x f f

K x x f

f

+ +

+

=

= + + +

= + +

* *

10,1,...

ptables

De esta manera, la relaci n recursiva para los clculos de programaci n din mica es

1(1) ( ) (1)

2

1,2,3.

n

n

x

n n n nx

f mn K x x f

para n

+=

= + +

=

58


59/70

3

3

x

s

3

3 3 3 3

1(1, ) ( ) 16

2

x

f x K x x

= + +

*

3 3( )f s *

3x 0 1 2 3 4 5

0 0 0 0

1 16 12 9 8 8 8,5 8 3 4

P =3

2

2

x

s

2

*

2 2 2 2 3

1(1, ) ( ) (1)

2

x

f x K x x f

= + +

*

2 2( )f s *

2x 0 1 2 3 4

0 0 0 0

1 8 8 7 7 7,5 7 2 3

P =2

59


60/70

P =1

1

1

x

s

1

*

1 1 1 1 2

1(1, ) ( ) (1)

2

x

f x K x x f

= + +

*

1 1( )f s *1x 0 1 2 3 4

1 7 7,5 6,75 6,875 7,4375 6,75 2

E, ; , ; , . E

$675.

60

P D


61/70

P E I

E 2: J V

U . S , , , . C .

E 2/3 .

S , ( ) . L .

E .

61


62/70

F:E: (=1,2,3)

V.E : F .

V.D : C .

P

.

nS

nX

( , )n n n

f X S =

* *

1 1

1 2( , ) ( ) ( )

3 3n n n n n n n n nf X S f S X f S X+ += + +

62


63/70

P :=3

P .

0 01 0

2 0

3 2

4 1

>=5 1 0

2 / 3

*

3X*

3f

2 / 3

3S

63


64/70

=2

0 1 2 3 4

0 0 0

1 0 0 0

2 0 1 23 0, 2 3

4 1

>=5 1 1 0

2S

2X 2f *

2

f*

2

X

4 / 9 4 / 94 / 92 / 3 2 / 3 2 / 3

2 / 3 2 / 3 2 / 3 2 / 38 / 9

4 / 92 / 3

8 / 9

64


65/70

=1

R :

E 20/27

.

0 1 2 33 1

1S1X 1f *

1f*

1X

2 / 3 2 0 / 2 7 2 / 3 2 / 3 2 0 / 2 7

*3*

2 *

3

* *1 2*

3* *

2 2

, 0, 1

, 2 3

1 2 3 ( 1),

, 1 2 1,2,3 4 ( 2),

si gana X si gana X

si pierde X o

X o para X si gana X

si pierde X o o para X si pierde la apuesta esta perdida

== =

= =

=

= =

65

P D


66/70

P C ,

E 3: S

L S , 1/, 6 .

C 2 /. L

50 / .

D S, . L

1.S 6 ( ) . M S 6 .

Demanda diaria

Tiendas Galones Probabilidad

Tienda

1

1 0,60

2 0

3 0,40

Tienda2

1 0,50

2 0,10

3 0,40

Tienda3

1 0,40

2 0,30

3 0,30

66

F


67/70

F

67

F


68/70

F

$4.2

68


69/70

P E 3:

*

*

0 0 0

1 1 2

2 2 3.4

>=3 3 4.35

*

*

0 0 0 0

1 2 2 01 2

2 3.4 4 3.25 1 4

3 4.35 5.4 5.25 4.35 1 5.4

4 4.35 6.35 6.65 6.35 2 6.65

5 4.35 6.35 7.6 7.75 3 7.75

6 4.35 6.35 7.6 8.7 3 8.7

E 2:

69


70/70

P E 1:

*

*

6 8.7 9.75 9.75 9.45 1 2 9.75

P , 2 , ,

T 1 T 3T 2

1 3 2C 1

C 2 2 2 2

70

cap01 - programación dinámica

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