Department of Industrial Engineering Statistical Quality Control presented by Dr. Eng. Abed Schokry

Transcription

1 Department of Indstrial Engineering Statistical Qality Control presented by Dr. Eng. Abed Schokry

2 Department of Indstrial Engineering Statistical Qality Control C and U Chart presented by Dr. Eng. Abed Schokry

3 Control Charts for Nonconformities (Defects) There are many instances where an item will contain nonconformities bt the item itself is not classified as nonconforming. There is a single sitation in which any nmber of incidents may occr, each with a small probability. Typical of sch incidents are scratches in tables, fire alarms in a city, typesetting errors in a newspaper. It is often important to constrct control charts for the total nmber of nonconformities or the average nmber of nonconformities for a given area of opportnity. The inspection nit mst be the same for each nit.

4 Control Charts for Nonconformities (Defects)- C and U Charts Why need it? Control the nmber of nonconformities A nonconforming prodct does not satisfy one or more of the specifications nonconformity/defect: Each specific point at which a specification is not satisfied, e.g., stones on a bottle defective fse on a car body typos on a paper A nit may not be nonconforming, even thogh it has several nonconformities. So, nonconforming defects or nonconformities Assmption: The occrrence of nonconformities in samples of constant size is well modeled by the Poisson distribtion. The nmber of potential location for nonconformities is infinitely large, and the probability of occrrence of a nonconformity at any location is small and constant

5 Control Charts for Attribte Data: C Chart Nmber Nonconforming per Unit: c Chart Reasons for Using a c Chart 1. One or more types of nonconformities 2. Poisson distribtion

6 Control Charts for Nonconformities (Defects) Poisson Distribtion The nmber of nonconformities in a given area can be modeled by the Poisson distribtion. Let c be the parameter for a Poisson distribtion, then the mean and variance of the Poisson distribtion are eqal to the vale c. The probability of obtaining x nonconformities on a single inspection nit, when the average nmber of nonconformities is some constant, c, is fond sing: p(x) e c x! c x

7 The c Chart Constrction 1. Collect m samples of data, each of size n 2. Determine the nmber of nonconformities for the ith nit. Call this vale ci 3. Find the average nmber of nonconformities per nit, c 4. Compte the 3-sigma control limits 5. Constrct the chart

8 c Chart Eqations c = c i m UCL = c + 3 c Center Line = c LCL = c - 3 c

9 Door Panels Example c Chart for Door panels I I I I I I I I I I I I I I I I I I I I Sample Nmber I I I I I UCL LCL

10 c Chart Eqations c = c i m UCL = c + 3 c Center Line = c LCL = c - 3 c

11 Procedres with Constant Sample Size c-chart Standard Given: UCL c 3 c CL c No Standard Given: LCL c 3 c UCL c 3 c CL c LCL c 3 c

12 Control Chart for Temperatre

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14 A paper mill ses a control chart to monitor the imperfection in finished rolls of paper. Prodction otpt is inspected for 20 days, and the reslting data are shown below. Use these data to set p a control chart for nonconformities per roll of paper. Does the process appear to be in statistical control? What center line and control limits wold yo recommend for controlling crrent prodction? Day Rolls Prodced Nmber of Imperfections Day Rolls Prodced Nmber of Imperfections

15 Example (Cont d) n i [LCL i, UCL i ] 18 [0.1088, ] 20 [0.1392, ] 21 [0.1527, ] Control Chart for Paper Imperfections [0.1653, ] U [0.1881, ] Day

16 Procedres with Constant Sample Size Choice of Sample Size: The Chart If we find c total nonconformities in a sample of n inspection nits, then the average nmber of nonconformities per inspection nit is = c/n. The control limits for the average nmber of nonconformities is UCL 3 n CL LCL 3 n

17 Procedres with Variable Sample Size Three Approaches for Control Charts with Variable Sample Size 1. Variable Width Control Limits 2. Control Limits Based on Average Sample Size 3. Standardized Control Chart

18 If sample size varies, it is always to se a chart rather than a c chart Approaches Control limits varies with each sample size, bt the center line is constant LCL 3 UCL 3 CL Use a control limits based on an average sample size m ni n i1 m n Use a standardized control chart (this is preferred option), with UCL=3, LCL=-3, Center line=0. This chart can be sed for pattern recognition i Z i n i i ; n i ;

19 Procedres with Variable Sample Size Variable Width Control Limits Determine control limits for each individal sample that are based on the specific sample size. The pper and lower control limits are 3 n i

20 Procedres with Variable Sample Size Control Limits Based on an Average Sample Size Control charts based on the average sample size reslts in an approximate set of control limits. The average sample size is given by The pper and lower control limits are n m n i 1 m i 3 n

21 Procedres with Variable Sample Size The Standardized Control Chart The points plotted are in terms of standard deviation nits. The standardized control chart has the follow properties: Centerline at 0 UCL = 3 LCL = -3 The points plotted are given by: Z i i n i

22 Demerit Systems When several less severe or minor defects can occr, we may need some system for classifying nonconformities or defects according to severity; or to weigh varios types of defects in some reasonable manner.

23 Demerit Systems Demerit Schemes 1. Class A Defects - very serios 2. Class B Defects - serios 3. Class C Defects - Moderately serios 4. Class D Defects - Minor Let c ia, c ib, c ic, and c id represent the nmber of nits in each of the for classes.

24 Demerit Systems Demerit Schemes The following weights are fairly poplar in practice: Class A-100, Class B - 50, Class C 10, Class D - 1 d i = 100c ia + 50c ib + 10c ic + c id d i - the nmber of demerits in an inspection nit

25 Demerit Systems Control Chart Development Nmber of demerits per nit: D = where n = nmber of inspection nits n d i i1 i D n

26 Example The nmber of nonconformities fond on final inspection of a cassette deck is shown here. Can yo conclde that the process is in statistical control? What center line and control limits wold yo recommend for controlling ftre prodction? What are the center line and control limits for a control chart for monitoring ftre prodction based on the total nmber of defects in a sample of 4 cassette decks? Deck # # of Nonconformities Deck # # of Nonconformities

27 Demerit Systems Control Chart Development where and 100 UCL CL LCL A 50 B 3ˆ 3ˆ 10 C D ˆ A n B 2 C D 1/ 2

28 The Operating-Characteristic Fnction The OC crve (and ths the P(Type II Error)) can be obtained for the c- and -chart sing the Poisson distribtion. For the c-chart: P( x UCL c) P( X LCL c) where x follows a Poisson distribtion with parameter c (where c is the tre mean nmber of defects).

29 The Operating-Characteristic Fnction For the -chart: P(x UCL ) P(x LCL ) P(c nucl ) P(c nlcl )

30 Dealing with Low-Defect Levels When defect levels or cont rates in a process become very low, say nder 1000 occrrences per million, then there are long periods of time between the occrrence of a nonconforming nit. Zero defects occr Control charts ( and c) with statistic consistently plotting at zero are ninformative.

31 Dealing with Low-Defect Levels Alternative Chart the time between sccessive occrrences of the conts or time between events control charts. If defects or conts occr according to a Poisson distribtion, then the time between conts occr according to an exponential distribtion.

32 Dealing with Low-Defect Levels Consideration Exponential distribtion is skewed. Corresponding control chart very asymmetric. One possible soltion is to transform the exponential random variable to a Weibll random variable sing x = y 1/3.6 (where y is an exponential random variable) this Weibll distribtion is wellapproximated by a normal. Constrct a control chart on x assming that x follows a normal distribtion.

33 Gidelines for Implementing Control Charts 1. Determine which process characteristics to control. 2. Determine where the charts shold be implemented in the process. 3. Choose the proper type of control chart. 4. Take action to improve processes as the reslt of SPC/control chart analysis. 5. Select data-collection systems and compter software.