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# IE-002 Control Chart For Variables

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• any one help me to sent identify at least 10 variables on telecommunication, hospital and how these company perform?
sent me email naseem_shan78@yahoo.com i m very thankful to his or her

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• You are absolutely, positively WRONG about the requirement of normality for the data to be used with a control chart. You obviously have NOT read or understood Shewhart's classic book. The control chart was developed empirically - not developed from theoretical calculations. Also, one should always used the +/- 3 sigma limits based on an average dispersion statistic (i.e., average mR for Individuals) and no other method!

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### IE-002 Control Chart For Variables

1. 1. Introduction • Variable - a single quality characteristic that can be measured on a numerical scale. • When working with variables, we should monitor both the mean value of the characteristic and the variability associated with the characteristic. Control Charts for Variables 1
2. 2. Control Charts for Variables 2
3. 3. Control Charts for x and R (1) Notation for variables control charts • n - size of the sample (sometimes called a subgroup) chosen at a point in time • m - number of samples selected • x i = average of the observations in the ith sample (where i = 1, 2, ..., m) • x = grand average or “average of the averages (this value is used as the center line of the control chart) Control Charts for Variables 3
4. 4. x and R(2) Control Charts for Notation and values • Ri = range of the values in the ith sample Ri = xmax - xmin • R = average range for all m samples •  is the true process mean •  is the true process standard deviation Control Charts for Variables 4
5. 5. x and R(3) Control Charts for Statistical Basis of the Charts • Assume the quality characteristic of interest is normally distributed with mean , and standard deviation, . • If x1, x2, …, xn is a sample of size n, then he average of x1  x 2    x n this sample is x n • x is normally distributed with mean, , and standard deviation, x   / n Control Charts for Variables 5
6. 6. Control Charts for x and R (4) Statistical Basis of the Charts • The probability is 1 -  that any sample mean will fall between    Z  / 2 x    Z  / 2 n and    Z  / 2 x    Z  / 2 n • The above can be used as upper and lower control limits on a control chart for sample means, if the process parameters are known. Control Charts for Variables 6
7. 7. x and R (5) Control Charts for x chart Control Limits for the UCL  x  A 2 R Center Line  x LCL  x  A 2 R • A2 is found in Appendix VI for various values of n. Control Charts for Variables 7
8. 8. Control Charts for x and R (6) Control Limits for the R chart UCL  D4 R Center Line  R LCL  D3 R • D3 and D4 are found in Appendix VI for various values of n. Control Charts for Variables 8
9. 9. Control Charts for x and R (7) Estimating the Process Standard Deviation • The process standard deviation can be estimated using a function of the sample average range. Relative range： W=R/σ R   E(W)=E(R/σ)=d2 d2 • This is an unbiased estimator of  Control Charts for Variables 9
10. 10. Control Charts for x and R (8) R=Wσ Var(W)=d32 σR=d3σ ^ σR=d3(R/d2) Control Charts for Variables 10
11. 11. Control Charts for x and R (9) Trial Control Limits • The control limits obtained from equations should be treated as trial control limits. • If this process is in control for the m samples collected, then the system was in control in the past. • If all points plot inside the control limits and no systematic behavior is identified, then the process was in control in the past, and the trial control limits are suitable for controlling current or future production. Control Charts for Variables 11
12. 12. Control Charts for x and R(10) Trial control limits and the out-of-control process • If points plot out of control, then the control limits must be revised. • Before revising, identify out of control points and look for assignable causes. – If assignable causes can be found, then discard the point(s) and recalculate the control limits. – If no assignable causes can be found then 1) either discard the point(s) as if an assignable cause had been found or 2) retain the point(s) considering the trial control limits as appropriate for current control. If future samples still indicate control, then the unexplained points can probably be safely dropped. Control Charts for Variables 12
13. 13. Control Charts for x and R(11) • Before revising, identify out of control points and look for assignable causes. – When many of the initial samples plot out of control against the trial limits, (as few data will remain which we can recompute reliable control limits.) it is better to concentrate on the pattern formed by these points. Control Charts for Variables 13
14. 14. Control Charts for x and R(12) • When setting up x and R control charts, it is best to begin with the R chart. Because the control limits on the x chart depend on the process variability, unless process variability is in control, these limits will not have much meaning. Control Charts for Variables 14
15. 15. Example 5-1 Control Charts for Variables 15
16. 16. Estimating Process Capability • The x-bar and R charts give information about the capability of the process relative to its specification limits. Control Charts for Variables 16
17. 17. The fraction of nonconforming of Example 5-1 • Assumes a stable process. • Assume the process is normally distributed, and x is normally distributed, the fraction nonconforming can be found by solving: P(x < LSL) + P(x > USL) Control Charts for Variables 17
18. 18. 製程能力(process capability) • 意義 – 對於穩定之製程所持有之特定成果，能夠合 理達成之能力界限。 • 製程能力的表示法 – 製程準確度(Ca值) – 製程精密度(Cp值)製程能力比(process capability ratio，PCR) – 製程能力指數(Cpk值) Control Charts for Variables 18
19. 19. 製程準確度Ca (capability of accuracy) 製程平均值-規格中心值 Ca= 規格公差/2 = X-μ = T/2 T：規格公差=規格上限-規格下限 Control Charts for Variables 19
20. 20. 製程準確度之評價方法 • A級 – 小於等於12.5%理想的狀態，故維持現狀。 • B級 – 大於12.5%，小於等於25%儘可能調整改進至A級。 • C級 – 大於25%，小於等於50%應立即檢討並加以改善 • D級 – 大於50%，小於100%採取緊急措施，並全面檢討， 必要時應考停止生產。 Control Charts for Variables 20
21. 21. 製程精密度Cp (process capability ratio，PCR) 雙邊規格： 規格公差 T Cp= 6個估計標準差 = 6σ 單邊規格： = 規格上限-製程平均值 SU-X = Cp= 3σ 3個估計標準差 或 = 製程平均值-規格下限 X-SL = Cp= 3σ 3個估計標準差 Control Charts for Variables 21
22. 22. 製程精密度評價方式 • A+級 – 大於等於1.67可考慮放寛產品變異，尋求管理的簡單化或成 本降低方法。 • A級 – 小於1.67，大於等於1.33理想的狀態，故維持現狀。 • B級 – 小於1.33，大於等於1.00確實進行製程管制，儘可能改善為 A級。 • C級 – 小於1.00，大於等0.67產生不良品，產品須全數選別，並管 理改善製程。 • D級 – 小於0.67採取緊急對策，進行品質改善，探求原因並重新檢 討規格。 Control Charts for Variables 22
23. 23. 製程能力指數Cpk Cpk=(1-|Ca|)Cp = = X-SL SU-X Cpk=min{ } 3σ ， 3σ 當Ca=0時Cpk=Cp Control Charts for Variables 23
24. 24. 製程能力指數評價方式 • A級 – 大於等於1.33製程能力足夠 • B級 – 小於1.33，大於等於1.00能力尚可應再努 力 • C級 – 小於1.00應加以改善 Control Charts for Variables 24
25. 25. Process Capability Ratios (Cp) (assume the process is centered at the midpoint of the specification band) If Cp > 1, then a low number of nonconforming items will be produced. • If Cp = 1, (assume norm. dist) then we are producing about 0.27% nonconforming. • If Cp < 1, then a large number of nonconforming items are being produced. Control Charts for Variables 25
26. 26. Control Charts for Variables 26
27. 27. The percentage of the specification band  ˆ   1 100% P C   p **The Cp statistic assumes that the process mean is centered at the midpoint of the specification band – it measures potential capability. Control Charts for Variables 27
28. 28. Revision of Control Limits and Center Line • The effective use of control chart will require periodic revision of the control limits and center lines. – Every week, every month, or every 25, 50, or 100 samples. – When revising control limits, it is highly desirable to use at least 25 samples or subgroups in computing control limits. • If the R chart exhibits control, the center line of the X Chart will be replaced with a target value. This can be helpful in shifting the process average to the desired value. Control Charts for Variables 28
29. 29. Phase II Operation of Charts • Use of control chart for monitoring future production, once a set of reliable limits are established, is called phase II of control chart usage • A run chart showing individuals observations in each sample, called a tolerance chart or tier diagram, may reveal patterns or unusual observations in the data Control Charts for Variables 29
30. 30. Control Charts for Variables 30
31. 31. Control Limits, Specification Limits, and Natural Tolerance Limits(1) • Control limits are functions of the natural variability of the process • Natural tolerance limits represent the natural variability of the process (usually set at 3-sigma from the mean) • Specification limits are determined by developers/designers. Control Charts for Variables 31
32. 32. Control Limits, Specification Limits, and Natural Tolerance Limits(2) • There is no mathematical relationship between control limits and specification limits. • Do not plot specification limits on the charts – Causes confusion between control and capability – If individual observations are plotted, then specification limits may be plotted on the chart. Control Charts for Variables 32
33. 33. Control Limits, Specification Limits, and Natural Tolerance Limits(3) Control Charts for Variables 33
34. 34. Rational Subgroups • X bar chart monitors the between sample variability (variability in the process over time). – Sample should be selected in such a way that maximizes the chances for shifts in the process average to occur between samples. • R chart monitors the within sample variability (the instantaneous process variability at a given time). – Samples should be selected so that variability within samples measures only chance or random causes. Control Charts for Variables 34
35. 35. Guidelines for the Design of the Control Chart(1) • Specify sample size, control limit width, and frequency of sampling • if the main purpose of the x-bar chart is to detect moderate to large process shifts, then small sample sizes are sufficient (n = 4, 5, or 6) • if the main purpose of the x-bar chart is to detect small process shifts, larger sample sizes are needed (as much as 15 to 25)…which is often impractical…alternative types of control charts are available for this situation…see Chapter 8 Control Charts for Variables 35
36. 36. Guidelines for the Design of the Control Chart(2) • If increasing the sample size is not an option, then sensitizing procedures (such as warning limits) can be used to detect small shifts…but this can result in increased false alarms. • R chart is insensitive to shifts in process standard deviation for small samples. The range method becomes less effective as the sample size increases, for large n, may want to use S or S2 chart. • The OC curve can be helpful in determining an appropriate sample size. Control Charts for Variables 36
37. 37. Guidelines for the Design of the Control Chart(3) Allocating Sampling Effort • Choose a larger sample size and sample less frequently? or, Choose a smaller sample size and sample more frequently? • The method to use will depend on the situation. In general, small frequent samples are more desirable. • For economic considerations, if the cost associated with producing defective items is high, smaller, more frequent samples are better than larger, less frequent ones. Control Charts for Variables 37
38. 38. Guidelines for the Design of the Control Chart(4) • If the rate of production is high, then more frequent sampling is better. • If false alarms or type I errors are very expensive to investigate, then it may be best to use wider control limits than three-sigma. • If the process is such that out-of-control signals are quickly and easily investigated with a minimum of lost time and cost, then narrower control limits are appropriate. Control Charts for Variables 38
39. 39. x Changing Sample Size on the and R Charts1 • In some situations, it may be of interest to know the effect of changing the sample size on the x-bar and R charts. Needed information: – Variable sample size. – A permanent change in the sample size because of cost or because the process has exhibited good stability and fewer resources are being allocated for process monitoring. Control Charts for Variables 39
40. 40. x Changing Sample Size on the and R Charts2 • = average range for the old sample size R old • R new = average range for the new sample size • nold = old sample size • nnew = new sample size • d2(old) = factor d2 for the old sample size • d2(new) = factor d2 for the new sample size Control Charts for Variables 40
41. 41. x Changing Sample Size on the and R Charts3 Control Limits R  chart x  chart  d ( new )  UCL  D 4  2  d 2 (new )   R old UCL  x  A 2   d 2 ( old )   R old  d 2 (old)   d ( new )  CL  R new   2  R old  d 2 (new )   d 2 ( old )  LCL  x  A 2   R old    d 2 ( new )   d 2 (old)  UCL  max  0 , D 3   R old   d 2 ( old )    A2、D3、D4 are selected for the new sample size Control Charts for Variables 41
42. 42. Example 5-2 Control Charts for Variables 42
43. 43. Charts Based on Standard Values • If the process mean and variance are known or can be specified, then control limits can be developed using these values: X  chart R  chart UCL  D 2  UCL    A  CL  d 2  CL   LCL  D 1  LCL    A  • Constants are tabulated in Appendix VI Control Charts for Variables 43
44. 44. x and R Charts Interpretation of • In interpreting patterns on the X bar chart, we must first determine whether or not the R chart is in control. • Patterns of the plotted points will provide useful diagnostic information on the process, and this information can be used to make process modifications that reduce variability. – Cyclic Patterns – Mixture – Shift in process level – Trend – Stratification Control Charts for Variables 44
45. 45. Control Charts for Variables 45
46. 46. x The Effects of Nonnormality • In general, the X bar chart is insensitive (robust) to small departures from normality. • In most cases, samples of size four or five are sufficient to ensure reasonable robustness to the normality assumption. Control Charts for Variables 46
47. 47. The Effects of Nonnormality R1 • The R chart is more sensitive to nonnormality than the x chart • For 3-sigma limits, the probability of committing a type I error is 0.00461on the R-chart. (Recall that for x , the probability is only 0.0027). Control Charts for Variables 47
48. 48. The Operating Characteristic Function(1) • How well the x and R charts can detect process shifts is described by operating characteristic (OC) curves. • Consider a process whose mean has shifted from an in-control value by k standard deviations. If the next sample after the shift plots in-control, then you will not detect the shift in the mean. The probability of this occurring is called the -risk. Control Charts for Variables 48
49. 49. The Operating Characteristic Function(2) • The probability of not detecting a shift in the process mean on the first sample is    ( L  k n )   ( L  k n ) L= multiple of standard error in the control limits k = shift in process mean (#of standard deviations). Control Charts for Variables 49
50. 50. The Operating Characteristic Function(3) • The operating characteristic curves are plots of the value  against k for various sample sizes. Control Charts for Variables 50
51. 51. The Operating Characteristic Function(4) • If  is the probability of not detecting the shift on the next sample, then 1 -  is the probability of correctly detecting the shift on the next sample. Control Charts for Variables 51
52. 52. Average run length，ARL In-control ARL0=1/α Out-of-control ARL1=1/(1-) Control Charts for Variables 52
53. 53. Average time to signal，ATS and Expected number of individual units sampled，I ATS=ARLh I=nARL Control Charts for Variables 53
54. 54. Control Charts for Variables 54
55. 55. Construction and Operation of x and S Charts(1) • First, S2 is an “unbiased” estimator of 2 • Second, S is NOT an unbiased estimator of  • S is an unbiased estimator of c4  where c4 is a constant • The standard deviation of S is  1  c 2 4 Control Charts for Variables 55
56. 56. Construction and Operation of x and S Charts(2) • If a standard  is given the control limits for the S chart are: UCL  B6 CL  c4 LCL  B5 • B5, B6, and c4 are found in the Appendix for various values of n. Control Charts for Variables 56
57. 57. Construction and Operation of x and S Charts(3) No Standard σGiven • If  is unknown, we can use an average 1m sample standard deviation, S   Si m i1 UCL B4S CL S LCL B3S Control Charts for Variables 57
58. 58. Construction and Operation of x and S Charts(4) x Chart when Using S The upper and lower control limits for the x chart are given as UCL  x  A 3 S CL  x LCL  x  A 3 S where A3 is found in the Appendix Control Charts for Variables 58
59. 59. Construction and Operation of x and S Charts(5) Estimating Process Standard Deviation • The process standard deviation,  can be estimated by S   c4 Control Charts for Variables 59
60. 60. Example 5-3 Control Charts for Variables 60
61. 61. The x and S Control Charts with Variable Sample Size(1) • The x and S charts can be adjusted to account for samples of various sizes. • A “weighted” average is used in the calculations of the statistics. m = the number of samples selected. ni = size of the ith sample Control Charts for Variables 61
62. 62. The x and S Control Charts with Variable Sample Size(2) • The grand average can be estimated as: m n x i i x i 1 m n i i 1 • The average sample standard deviation is: 1/ 2  2 m   ni  1Si  S   i1m   ni  m     i1  Control Charts for Variables 62
63. 63. The x and S Control Charts with Variable Sample Size • Control Limits UCL  x  A3S UCL  B 4 S CL  x CL  S LCL  x  A3S LCL  B 3 S • If the sample sizes are not equivalent for each sample, then – there can be control limits for each point (control limits may differ for each point plotted) Control Charts for Variables 63
64. 64. The S2 Control Chart • There may be situations where the process variance itself is monitored. An S2 chart is S2 2 UCL    / 2,n 1 n 1 CL  S 2 S2 2 LCL  1(  / 2),n 1 n 1   / 2 , n  1 and 1 (  / 2 ),n 1 are points found 2 2 where from the chi-square distribution. Control Charts for Variables 64
65. 65. The Shewhart Control Chart for Individual Measurements(1) • What if you could not get a sample size greater than 1 (n =1)? Examples include – Automated inspection and measurement technology is used, and every unit manufactured is analyzed. There is no basis for rational subgrouping. – The production rate is very slow, and it is inconvenient to allow samples sizes of N > 1 to accumulate before analysis – Repeat measurements on the process differ only because of laboratory or analysis error, as in many chemical processes. • The X and MR charts are useful for samples of sizes n = 1. Control Charts for Variables 65
66. 66. 個別值與移動全距管制圖(2) • 使用理用 產品需經很久的時間才能製造完成 – 分析或測定一件產品的品質較為麻煩且費時 – 產品係非常貴重的物品 – 產品品質頗為均勻(自動化生產) – 屬破壞性試驗 – 在於管制製造條件如溫度、壓力、濕度等 – Control Charts for Variables 66
67. 67. The Shewhart Control Chart for Individual Measurements(3) Moving Range Chart • The moving range (MR) is defined as the absolute difference between two successive observations: MRi = |xi - xi-1| which will indicate possible shifts or changes in the process from one observation to the next. Control Charts for Variables 67
68. 68. The Shewhart Control Chart for Individual Measurements(4) X and Moving Range Charts • The X chart is the plot of the individual observations. The control limits are MR UCL  x  3 d2 CL  x MR LCL  x  3 d2 m  MRi where MR  i1 m Control Charts for Variables 68
69. 69. The Shewhart Control Chart for Individual Measurements(5) X and Moving Range Charts • The control limits on the moving range chart are: UCL D4 MR CL  MR LCL 0 Control Charts for Variables 69
70. 70. Example 5-5 Control Charts for Variables 70
71. 71. The Shewhart Control Chart for Individual Measurements(5) Example Ten successive heats of a steel alloy are tested for hardness. The resulting data are Heat Hardness Heat Hardness 1 52 6 52 2 51 7 50 3 54 8 51 4 55 9 58 5 50 10 51 Control Charts for Variables 71
72. 72. The Shewhart Control Chart for Individual Measurements(6) Example I and MR Chart for hardness 62 3.0SL=60.97 Individuals X=52.40 52 -3.0SL=43.83 42 Observation 0 1 2 3 4 5 6 7 8 9 10 3.0SL=10.53 10 Moving Range 5 R=3.222 0 -3.0SL=0.000 Control Charts for Variables 72
73. 73. The Shewhart Control Chart for Individual Measurements(7) Interpretation of the Charts • X Charts can be interpreted similar to x charts. MR charts cannot be interpreted the same as x or R charts. • Since the MR chart plots data that are “correlated” with one another, then looking for patterns on the chart does not make sense. • MR chart cannot really supply useful information about process variability. • More emphasis should be placed on interpretation of the X chart. Control Charts for Variables 73
74. 74. The Shewhart Control Chart for Individual Measurements(8) Average Run Lengths β Size of Shift ARL1 1σ 0.9772 43.96 2σ 0.8413 6.03 3σ 0.5000 2.00 The ability of the individuals control chart to detect small shifts is very poor. Control Charts for Variables 74
75. 75. The Shewhart Control Chart for Individual Measurements(9) Normal ProbabilityPlot • The normality assumption is often .999 .99 taken for granted. .95 lity .80 Probabil • When using the .50 .20 individuals chart, the .05 .01 normality .001 assumption is very 50 51 52 53 54 55 56 57 58 hardness important to chart Average: 52.4 Anderson-Darling Normality Test StDev: 2.54733 A-Squared: 0.648 N: 10 P-Value: 0.063 performance. Control Charts for Variables 75