Lucrare1 Weibull_3metode

8/12/2019 Lucrare1 Weibull_3metode

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ESTIMAREA PARAMETRILOR REPARTIIEI WEIBULL PRINMETODE PARAMETRICE

Se consider un eantion de 6 produse, ncercate pe un stand, pentru care valoriletimpilor la care s-a produs defectarea, sunt urmtoarele:

ti=16, 34, 53, 75, 3 i 1!" ore

1# S se determine parametrii reparti$iei %ei&ull prin:1#1# metoda 'rafic(

1#!# metoda celor mai mici ptrate(1#3# metoda verosimilit$ii ma)ime#

!# S se determine e)presia fia&ilit$ii i a nonfia&ilit$ii

3# S se determine cte produse se vor mai afla n func$iune dup !"" ore, dintr-un lotde !""" de produse#

1.1. DETERMINAREA PARAMETRILOR REPARTIIEI WEIBULL PRIN

METODA GRAFICt1, t!, ###, ti, ###, tn i- numarul produsului e) la 1 i=1,la ! i=!####

n=numarul de produse in acest e) avem 6

4#"

3#"*+

+=

n

itF in #

ti=16, 34, 53, 75, 3 i 1!" ore

r#crt#

ti ( ) 4#"3#"

+= nitF in ( ) 1""-in tF

1

!3

4

56

16

3453

75

31!"

"#1"1

"#!644"#4!14

"#57.6

"#7355"#.1"

1"#1

!6#444!#14

57#.6

73#55.#1"

[ ]1""-*+, ini tFt

( )

=t

etF 1

( )

=t

etR

( ) ( ) 1=+ tFtR


2/8

ReliaSoft Weibull++ 7 - www.ReliaSoft .com

Probabil ity - Weibull

=1.4429, =76.0821, =0.9956

Time, (t)

Unreliability,

F(t)

1. 1.1.1.

!.

".

"".Probabilit y-Weibull

#ata 1Weibull-$PRR%SR&&'#F&F*S

#ata PointProbability ine

onel &art inecuUnieritat yTranilania of /rao1$*7*$110!"0! &

,

"#

ore".#76

,44!#1

=

=

=

( )44#1

.1#76

==tt

eetR

t=!""


3/8

( ) ====

44#1

.1#76

!""

!"" eetR

t

"#45

( ) ( )( )

=="N

tNtR

1.2. DETERMINAREA PARAMETRILOR REPARTIIEI WEIBULL PRINMETODA CELOR MAI MICI PTRATE

,1*+

=t

etF

( )

=

ttF *+1ln #

( )[ ]

=

ttF ln*+1lnln

,

( )[ ] lnln*+1lnln = ttF #

( ) lnln

*+1

1lnln =

t

tF

( )

=

==

=

#

,ln

,ln

,*+1

1lnln

!

1

tx

tFy

xy!1

+= #

t1, t!, ###, ti, ###, tn

,lnsi*+1

1lnln ii

in

i txtF

y =

=

4#"

3#"*+

+

=n

itF in


4/8

ln,4#"

3#"*+,

*+1

1lnln

###

ln,4#"

3#"*+,

*+1

1lnln

###

,ln,4#"

3#"!*+,

*+1

1lnln

ln,4#"

3#"1*+,

*+1

1lnln

!!!

!

!

111

1

1

=+

=

=

=+

=

=

=+

=

=

=+=

=

.txn

ntF

tFy

,txn

itF

tFy

txn

tFtF

y

,txn

tFtF

y

nnn

n

n

iii

i

i

S y xi ii

n

= + == + * min# 1 !

!

1

[ ] [ ] [ ] [ ] #*+###*+###*+!*+ !!1!

!1

!

!!1!

!

1!11 nnii xyxyxyxyS +++++++++=

=

=

"

"

!

1

S

S

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

=++++

=++++

"*+!###*+!###*+!*+!

"*+!###*+!###*+!*+!

!1!1!!1!!1!111

!1!1!!1!1!11

nnniii

nnii

xyxxyxxyxxyx

xyxyxyxy

=+

=+

===

==n

i

ii

n

i

i

n

i

i

n

i

i

n

i

i

yxxx

yxn

11

!

!

1

1

11

!1

,

,

1

!

1

!

1 1 1!

= =

= = =

=

n

i

n

i

ii

n

i

n

i

n

i

iiii

xxn

yxyxn

(

1

!

1

!

1 111

!

1

= =

= ===

=

n

i

n

i

ii

n

i

n

i

ii

n

i

i

n

i

ii

xxn

yxxxy


5/8

#sauln=si

/

1

///

1!

/

== e

ti=16, 34, 53, 75, 3 i 1!" ore

r#crt#

ti ii tx ln= ( )!! ln ii tx = ( )4#"

3#"

+

=n

itFin

*+1

1lnln

in

itF

y

= !iy ii yx

1

!

34

56

16

34

5375

31!"

!#77!6

3#5!64

3#7"34#3175

4#53!64#7.75

7#6.73

1!#435!

15#763!1.#64"7

!"#5445!!#!"1

"#1"1

"#!644

"#4!14"#57.6

"#7355"#.1"

-!#15.3

-1#1."!

-"#6"3"-"#146"

"#!.51"#755

4#65.!

1#33"

"#3637"#"!13

"#".13"#63!.

-5#.4"

-4#16!"

-!#343-#"63"3

1#!!33#.".3

6

1

!3#"6. 7#" -3#""7" 7#15"! -.#"6

( ) ( )

=

=

==

#135#6

,43"1#1"6.#!3"#76

""7"#3"6.#!3"6#.6

1

!!

( )

===

=

ore76#31.

43"1#1

43"1#1

135#6

!

1

ee

1.3. DETERMINAREA PARAMETRILOR REPARTIIEI WEIBULL PRINMETODA VEROSIMILITII MAXIME

t1, t!, ###, ti, ###, tn


6/8

( )kitf ,###,( !1 #

( ) ( ) ( ) ( )

( )#,###,(

,###,(###,###,(###,###,(,###,(

1

!1

!1!1!1!!11

=

=

=n

i

ki

knkikk

tf

tftftftfL

( ) ( ) ( ) ( )[ ]( ) ( ) ( ) (

( )#,###,(ln

,###,(ln###,###,(ln###,###,(ln,###,(ln

,###,(###,###,(###,###,(,###,(ln

ln

1

!1

!1!1!1!!11

!1!1!1!!11

=

=

+++++=

==

n

i

ki

nkikk

knkikk

tf

tftftftf

tftftftf

L

=

=

=

#"ln

####

,"ln

,"ln

!

1

n

L

L

L

+="*

( )

=

t

e

t

tf

1

#

t1, t!, ###, ti, ###, tn

( )

#

######,

1

1

111

!

1

1

!1

=

=

=

n

i

t

i

t

n

t

i

tt

i

ni

et

et

et

et

et

L

( )

#lnlnln

ln###ln###lnln

ln

,ln

11

1

1

111

!

1

1

1

1

!1

=

=

=

=

+

+=

++

++

+

=

=

=

n

i

tn

i

i

n

i

t

n

t

i

tt

n

i

t

i

i

ni

i

et

et

et

et

et

et

L


7/8

( )

( )

=

+=

=

=+=

=

=

=

#"lnln1,ln

,",ln

1

11

n

i

iii

n

i

i

tttL

tnL

=

+

=

==

=+

#"lnln

"

11

11

n

i

ii

n

i

i

n

i

i

tttn

tn

n

tn

i

i== 1

#

ti=16, 34, 53, 75, 3 i 1!" ore#

=

+

=

==

=+

#"lnln6

"

6

6

1

6

1

6

11

i

ii

i

i

ii

ttt

t

6

6

1

== i

it

=

=

ore#5!6#73

,33#1


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