Graficas Control

INGENIERIA Y CONTROL DE LA CALIDAD

CONTROL ESTADISTICO DEL PROCESO

GRAFICAS DE CONTROL PARA DEFECTOS

Csar A. Acosta-Meja

GRAFICOS DE CONTROL PARA ATRIBUTOSCLASIFICACIONGrficos de control para unidades defectuosas

La grfica p fraccin defectuosaLa grfica np nmero de unidades defectuosas

Grficos de control para defectos

La grfica c nmero de defectos. (tamao constante) La grfica u nmero de defectos por unidad (tamao variable)

de inspeccin

GRAFICA DE CONTROL cEsta grfica controla si la media del nmero de defectos en una unidad inspeccionada permanece constante

Todas las muestras son iguales a una unidad inspeccionada.

Puede ser una pieza, una caja de 12 piezas, un tramo de 100 mts. de tela o 1000 litros de pintura

Se usa comunmente en industrias de proceso contnuo, como

Industrial textilProductos qumicos (lquidos)Vidrio

Se asume que el nmero de defectos en una unidad inspeccionada es una v.a. de Poisson que puede aproximarse por una Normal

GRAFICA DE CONTROL cRecurdese que en la grfica p

X : # de defectuosos en una muestra de tamao n Binomial (n,p)

En la fabricacin contnua no existen piezas producidas, por lo que

n y p 0 mantenindose = np constante.

Bajo estas condiciones

Binomial (n,p)Poisson ( = np )

Se acostumbra a llamar al parmetro como c.

GRAFICA DE CONTROL cBinomial (n,p)Poisson ( = np )

Sea p = / n

GRAFICA DE CONTROL cSea X el nmero defectos observados en una unidad inspeccionada

X se puede modelar como una v.a. Poisson ( = c) si

El nmero n de lugares potenciales para la ocurrencia de los defectos es infinito

La probabilidad p de ocurrencia de un defecto en cada uno de los lugares potenciales es pequea y constante


Supongamos que X se puede modelar como una v.a. Poisson ( = c)Entonces

X Poisson (c)E(X) = cVar(X) = c

y los lmites de control son:E [X] 3 DS [X]

c 3 c


Supongamos que X se puede modelar como una v.a. Poisson ( = c)Entonces

X Poisson (c)E(X) = cVar(X) = c

y los lmites de control son:E [X] 3 DS [X]

c 3 c

La aproximacin normal a la Poisson es aceptable si c = > 5 y es mejor cuanto mayor es

GRAFICA DE CONTROL cClculo de los lmites de control

Los Lmites de control son: c 3 c (Poisson normal)

Si c no se conoce, se le estimaa partir de m muestras previas, con

As los lmites de control son: 3

GRAFICA DE CONTROL cEjemplo

La tabla mostrada presenta el nmero de defectos encontrados en un rollo de tela. Todos los rollos de tela son de igual tamao.Construya una grfica c y determine si el proceso est en control.

MuestraDefectos encontrados1921131349515613788169101017111012101391451512166171518101972052192212


La tabla mostrada presenta el nmero de defectos encontrados en un rollo de tela. Todos los rollos de tela son de igual tamao. Construya una grfica c y determine si el proceso est en control.SOLUCION

Estimamos c a partir del total de defectosc = 231 / 22 = 10.5 defectos por rollo

MuestraDefectos encontrados192113134951561378816910101711101210139145151216617151810197205219m = 2212TOTAL231

Los lmites de control resultanc 3 c 10.5 3 10.5LSC = 20.22LIC = 0.7789


Los lmites de control resultanc 3 c 10.5 3 10.5LSC = 20.22LIC = 0.7789


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GRAFICA DE CONTROL cSupongamos que

- la unidad de inspeccin es un lote de 150 unids.- = 5 defectos por lote- el proceso esta en control estadstico- el proceso produce en promedio 5 defectos por loteLa grafica c permite probar la hiptesis Ho: = 5 defectos por lote

Supongamos que mejoramos el proceso a = 1 defecto por lotePara poder usar la aprox normal debemos aumentar la unidad de inspeccin. Por ejemplo

- = 5 defectos por cada 750 unids.

GRAFICOS DE CONTROL PARA ATRIBUTOSCLASIFICACIONGrficos de control para unidades defectuosas

La grfica p fraccin defectuosaLa grfica np nmero de unidades defectuosas

Grficos de control para defectos

La grfica c nmero de defectos. (tamao constante) La grfica u nmero de defectos por unidad (tamao variable)

de inspeccin

GRAFICA DE CONTROL uSe usa cuando el nmero de unidades inspeccinadas vara de muestra a muestra

Sea X el nmero defectos en n unidades de inspeccin y

U = X / n el nmero de defectos por unidad de inspeccin

Esta grfica controla si la media de U permanece constante

GRAFICA DE CONTROL uSea X el nmero defectos en n unidades de inspeccin, y U = X / n el nmero defectos en una unidad de inspeccin, entoncesX Poisson (c)E(X) = cVar(X) = c

E(U) = c/nVar(U) = c/n2

y los lmites de control son:E [U] 3 DS [U]

GRAFICA DE CONTROL uSea entonces los lmites se expresan como

Si no se conoce, se le estima

a partir de m muestras previas

GRAFICA DE CONTROL uEjemplo En una planta textil se inspecciona el producto controlando el nmero de defectos por cada 50 m2 de tela (sta es la unidad de inspeccin).En la tabla se muestran los datos de 10 rollos de tela de distinto tamao.Construya una grfica u y determine si el proceso est en control.

GRAFICA DE CONTROL uEjemplo SolucinEn una planta textil se inspecciona el producto controlando el nmero de defectos por cada 50 m2 de tela (sta es la unidad de inspeccin).En la tabla se muestran los datos de 10 rollos de tela de distinto tamao. Construya una grfica u y determine si el proceso est en control.

GRAFICA DE CONTROL uEjemplo SolucinEn una planta textil se inspecciona el producto controlando el nmero de defectos por cada 50 m2 de tela (sta es la unidad de inspeccin).En la tabla se muestran los datos de 10 rollos de tela de distinto tamao. Construya una grfica u y determine si el proceso est en control.

u = 153 / 107.5 = 1.423 ui

GRAFICA DE CONTROL uEjemplo SolucinLa lnea central de la grfica de control es igual al nmero promedio de disconformidades por unidad de inspeccin (50 m2 de tela),

u = 153 / 107.5 = 1.423

es decir, en promedio, 1.423 defectos por cada 50 m2 de tela.Este es el parmetro que deseamos controlar.

Los lmites de control resultan entonces

los cuales varan segun el nmero de unidades de inspeccin ni

GRAFICA DE CONTROL uEjemplo SolucinPor ejemplo, para el ltimo rollo (ui = 23 / 12.5 = 1.84), los lmites de control son

= 1.42 3 (0.337)

LSC10= 2.43

LIC10= 0.41

GRAFICA DE CONTROL uEjemplo - MINITABVariable: defectos Subgroups in: unidades

metrosunidadesdefectos5001014400812650132050010114759.575001010600122152510.516600121962512.523

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