Quality Control. Ignacio Cascos Depto. Estadística, Universidad Carlos III 1

Transcription

1 Qualty Control 5 Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 1

2 Outlne 1. Introducton, control charts 2. Varables control charts (X-bar chart) 3. Attrbutes control charts (p and np charts) Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 2

3 Outlne 1. Introducton, control charts 2. Varables control charts (X-bar chart) 3. Attrbutes control charts (p and np charts) Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 3

4 Introducton An ndustral process s a sequence of transformatons addng value and varablty to the fnal product. A qualty product should satsfy the specfcatons mposed by the consumers. To ncrease the qualty of our producton, we should decrease ts varablty. Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 4

5 Introducton We wll dentfy a certan characterstc as the qualty of the product and analyse ts varablty. Sources of varablty of the fnal product: Raw materals Producton methods Manpower Adjustment of machnes Weather condtons Measurement erros Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 5

6 Varablty causes Assgnable causes. Important source of varablty (mproperly adjusted machnes, operator errors, or defectve raw materals). There are few. Each sngle one s an mportant source of varablty. Are occasonally present. Easy to detect snce they appear at an specfc stage of the process. Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 6

7 Varablty causes Chance causes. Natural varablty (background nose) due to causes that are an nherent part of the process. There are many, and some mght be unknown. Each of them causes only small varablty. Inherent part of the process. Dffcult (or mpossble) to get rd of. Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 7

8 Introducton Out of control. Some assgnable cause has occured and the qualty characterstc of the produced tems s substantally modfed. In-control state. Only chance causes have occured, the qualty characterstcs has a small varablty. Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 8

9 Introducton It s commonly assumed that the qualty characterstc follows a normal dstrbuton N(m,s). In the n-control state, the qualty characterstc of 99 73% of the produced tems les n an nterval of wdth 6s, (m ± 3s) Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 9 r

10 Control charts Basc elements of a control chart: Center lne. Mean level of the characterstc. Upper Control Lmt (UCL). Lower Control Lmt (LCL). Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 10

11 Meda Control charts A control chart montors some sample characterstc assocated wth the qualty of the producton Gráfco de control UCL LCS Tempo center lne lnea central LCI LCL Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 11

12 Outlne 1. Introducton, control charts 2. Varables control charts (X-bar chart) 3. Attrbutes control charts (p and np charts) Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 12

13 X-bar chart Montors sample means. If X ~ N( m, s n), the X-bar control chart s UCL m 3 center lne m LCL m 3 s n s n Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 13

14 X-bar chart Parameter estmaton, k samples of sze n 1 st sample: X 11,X 12,...,X 1n -th sample: X 1,X 2,...,X n k-th sample: X k1,x k2,...,x kn For each ndvdual sample X n j1 n X j ; S 2 n j1 ( X j n 1 X ) 2 Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 14

15 X-bar chart Grand mean. The grand mean s an unbased estmator of the process mean, but E[S ]=c 4 s. Therefore, we must transform S n order to obtan an unbased estmator of s, X 4 k 1 k Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 15 X k S S S 1 sˆ where c k

16 X-bar chart The X-bar chart s: UCL E[ S j ] m 3 c n center lne m E[ S j ] LCL m 3 c n 4 4 UCL x 3 c center lne x LCL 4 x 3 c 4 s s n n Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 16

17 X-bar chart Estmatng the control lmts: The upper and lower control lmts are estmated out of a set of tral samples. If the sample characterstc of any of the tral samples s out of the control lmts, the correspondng sample s deleted and the control lmts are estmated. The estmated control lmts are used to montor the sample characterstng n an on-gong process. Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 17

18 Meda X-bar chart X chart Gráfco de control para la meda 33 LCS UCL center lne Línea central Tempo LCI LCL Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 18

19 Meda X-bar chart X chart Gráfco de control para la meda 34 UCL LCS center lne Línea central Tempo LCL LCI Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 19

20 X-bar chart n c Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 20

21 Process capablty The process capablty of a varable control chart whose quanttatve characterstc X satsfes Var[X] = s 2 s gven by Capablty = 6s, where s s the standard devaton of X when the process s n the n-control state. Estmated capablty 6sˆ 6 Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 21 s c 4.

22 Process capablty Estmatng the process capablty: Take k samples of sze n (tral samples) Buld an X-bar control chart Delete the samples that have been taken when the process was out of control (f any) and recalculate the control lmts (test data s normaly dstrbuted) Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 22

23 Process Capablty Rato The Process Capablty Rato compares the process capablty wth the product specfcatons PCR Specfcat ons Capablty USL-LSL 6s where USL (Upper Specfcaton Lmt) and LSL (Lower Specfcaton Lmt) consttute lmt bounds for an admssble product. PCR > 1 few nonconformng unts produced; PCR < 1 large number of nonconformng unts produced; PCR % nonconformng unts. Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 23

24 Average Run Length If the process suffers a shft n ts mean (or standard devaton) such that the probablty of detectng t wth the X-bar chart s p, the mean number of samples untl the shft s detected s Average Run Length. ARL = 1/p It s the mean of a Geometrc random varable. Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 24

25 Analyss of Patterns of Control Charts In a control chart t s possble to detect nonrandom patterns. Example: long sequence of subsequent ponts above or below the center lne. Such arrangements of ponts are called volatng runs. Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 25

26 Outlne 1. Introducton, control charts 2. Varables control charts (X-bar chart) 3. Attrbutes control charts (p and np charts) Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 26

27 Attrbutes charts Attrbutes control charts are desgned to montor qualty characterstcs that cannot be measured on a quanttatve scale. In a p chart, the proporton of tems possessng a certan attrbute p s montored Proporton of nonconformng (defectve) unts Proporton of clents that clam Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 27

28 p chart Sample sze of the -th sample: n The number of nonconformng unts n the -th sample s d and follows a Bnomal dstrbuton wth parameters n and p. n r r n r P( d r) p (1 p) f r 0,1,2,, n Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 28

29 p chart for thesample fracton defectve We know that: E[ ˆ After the CLT, t holds: p Var[ pˆ ] p ] p(1 n p) pˆ d n pˆ N p, p(1 n Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 29 p)

30 p chart The p chart s gven by: UCL E[ pˆ center lne E[ pˆ LCL E[ pˆ where p ] 3 ] ] 3 1 k Var[ pˆ Var[ pˆ k 1 d n ] ] UCL center lne LCL p 3 p p 3 p(1 p) n p(1 p) n, average fracton defectve Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 30

31 p p chart p chart Gráfco P 0,024 UCL 0,02 0,016 0,012 0,008 center lne línea central = 0,01 0, Muestra LCL Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 31

32 p chart Estmatng the control lmts: We take k samples (at least 20) of sze n (commonly at least 30) The sample proporton of nonconformng unts n each sample s computed p s estmated by the average fracton defectve The control lmts are computed If any sample fracton defectve les out of the control lmts, the sample s deleted and lmts recomputed Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 32

33 p chart Capablty. Proporton of conformng unts produced n the n-control state Capablty 1 p Estmated capablty 1 p Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 33

34 np chart The np chart s equvalent to the p chart. It montors the number of nonconformng unts n a sample d (nstead of ts proporton). It s useful when: The number of nonconformng unts s more mportant than the proporton. The sample sze s constant. Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 34

35 np chart Let n = n constant. After the CLT d npˆ N np, np(1 p) UCL E[ d ] 3 Var[ d ] UCL np 3 np(1 p) center lne E[ d ] center lne np LCL E[ d ] 3 Var[ d ] LCL np 3 np(1 p) Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 35

36 np chart Capablty. As for the p chart, the process capablty s the proporton of conformng unts produced n the n-control state Capablty 1 p Estmated capablty 1 p Ignaco Cascos Depto. Estadístca, Unversdad Carlos III 36