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Six Sigma Nitish Nagar
This document provides an overview of control charts and statistical process control. It discusses how control charts can show if a process is stable or experiencing special causes of variation. The document outlines different types of control charts for variable and attribute data, including how to generate and interpret them. It explains the role of control charts within the DMAIC improvement process for measuring process stability, establishing improved processes, and ongoing process monitoring and control.
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- 2. 2 Control Charts– Learning Objectives At the end of this section, delegates will: • Understand how control charts can show if a process is stable • Generate and interpret control charts for variable and attribute data • Understand the role of control charts within the DMAIC improvement process
- 3. 3 Control Charts– Agenda 1. Introduction to Statistical Process Control, SPC 2. Control Limits 3. Individual and Moving Range Chart 4. Workshop on Control Charts 5. Defective (Binomial) p-Chart 6. Defects (Poisson) u-Chart 7. Workshop on Attribute Control Charts 8. Uses of Control Charts 9. Summary
- 4. 4 What isStatistical Process Control? • Statistical Process Control is a method of monitoring and detecting changes in processes. • SPC uses an advanced form of Time Series plots. • SPC provides an easy method of deciding if a process has changed (in other words, is the process “in-control”?).
- 5. 5 We NeedWays of Interpreting Data • Everyday we are flooded by data and we are forced to make decisions: • Calls handled decreases by 4% • UK trade deficit rises by £5 billion • Company X’s earnings are $240Million less than the previous quarter • Should we take action ?
- 6. 6 How dowe manage data historically? Leave it alone - it ain’t broke Pain & suffering Pain & suffering Lower “Customer” Requirement Upper “Customer” Requirement This Method • Tells you where you are in relation to customer’s needs • It will NOT tell you how you got there or what to do next • Means that pressure to achieve customer requirements will cause you to: • Actually Fix The Process • Sabotage The Process • Sabotage The Data (Integrity)
- 7. 7 What doControl Charts detect? • Control Charts detect changes in a process. • All processes change slightly, but process control aims to detect ‘statistically significant’ changes that are not just random variation. • Processes can change in several different ways… • the process average can change • the process variation can change • the process may contain one-off events
- 8. 8 Process Control • Process control refers to the evaluation of process stability over time • Process Capability refers to the evaluation of how well a process meets specifications 0 5 10 15 20 25 Time LSL USL UCL LCL
- 9. 9 Why woulda Process be Incapable? There are a number of reasons why a process may not be capable of meeting specification: 1. The specification is incorrect! 2. Excessive variation 3. The process is not on target 4. A combination of the above 5. Errors are being made 6. The process is not stable
- 10. 10 The specificationis incorrect • This issue was discussed during the Customer Focus section of this course • If specifications are not clearly related to customer requirements, then it is always a good idea to challenge the specification before attempting to improve the process
- 11. 11 Excessive variation Upper Specification Limit Lower Specification Limit Target • Excessive variation means that we have a variation reduction issue • We will need to understand which process inputs are causing the variation in the process output
- 12. 12 The processis not on target Upper Specification Limit Lower Specification Limit Target • In this situation we have a process targeting issue • We will need to understand which process inputs are causing the process to be off-target • This situation is sometimes simple to solve!
- 13. 13 Excessive variationand not on target Upper Specification Limit Lower Specification Limit Target • In this situation we have both excessive variation and a process targeting issue • We will need to understand which process inputs are causing the excessive variation and which are causing the process to be off-target
- 14. 14 Errors arebeing made Upper Specification Limit Lower Specification Limit Target • A situation such as this might indicate that errors are being made which result in occasional excursions outside of the specification • This is often an indication of a mistake proofing issue
- 15. 15 The processis not stable Upper Specification Limit Lower Specification Limit Target Last week This week Next week? • A situation such as this is an indication that the process is unstable • Whenever this situation is encountered in a DMAIC activity, then the reason(s) for the instability must be found and removed before assessing process capability
- 16. 16 The processis not stable Upper Specification Limit Lower Specification Limit Target Last week This week Next week? • Causes of process instability are sometimes referred to as “special causes” • Removing these special causes may result in the process becoming capable of consistently meeting the target
- 17. 17 Unstable Process “Special” causes of variation are present Total Variation Time Target
- 18. 18 Stable Process Time Target Total Variation Only “Common” causes of variation are present
- 19. 19 Capable Process Time Spec Limits CAPABLE NOT CAPABLE Process is Stable but Process is not Capable Process is Stable and Process is Capable Management Action (DMAIC) to reduce common cause variation
- 20. 20 Control Chartstest for Stability (Control Chart of Average) 0 5 10 15 20 25 Time 1.3 1.29 1.28 1.27 1.26 Process Average Upper Control Limit Lower Control Limit
- 21. 21 Transactional ImprovementProcess Analyse Define Measure Improve Control Select Project Define Project Objective Form the Team Map the Process Identify Customer Requirements Identify Priorities Update Project File Control Critical x ’s Monitor y’s Validate Control Plan Identify further opportunities Close Project LSL USL 15 20 25 30 35 Phase Review 1 5 10 15 20 10.2 10.0 9.8 9.6 Upper Control Limit Lower Control Limit y Phase Review Develop Detailed Process Maps START PROCESS STEPS DECISION STOP Identify Critical Process Steps (x’s) by looking for: – Process Bottlenecks – Rework / Repetition – Non-value Added Steps – Sources of Error / Mistake Map the Ideal Process Identify gaps between current and ideal Phase Review Brainstorm Potential Improvement Strategies Select Improvement Strategy Criteria A B C D Time + s - + Cost + - + s Service - + - + Etc s s - + Plan and Implement Pilot Verify Improvement LSL USL 15 20 25 30 35 Implement Countermeasures Phase Review Phase Review Define Measures (y’s) Check Data Integrity Determine Process Stability Determine Process Capability Set Targets for Measures
- 22. 22 Role ofControl Charts Measure Phase: • used during capability studies to assess process stability Improve Phase: • used to establish if the modified, improved process is stable Control Phase • used to control critical process input variables (x’s) in order to reduce variability in process outputs (y’s) • used to monitor process outputs (y’s) on an ongoing basis to ensure that the process remains in control
- 23. 23 Control Charts Variable Data No Subgroups Subgroups n = 2-9 Subgroups n 9 Individuals Moving Range Chart X Bar R Chart X Bar s Chart Attribute Data Defect Data (Poisson) Defective Data (Binomial) Varying Subgroup Size Constant Subgroup Size Varying Subgroup Size Constant Subgroup Size u Chart c Chart p Chart np Chart
- 24. 24 What doControl Charts tell us? 0 5 10 15 20 25 Subgroup 1.3 1.29 1.28 1.27 1.26 X-bar • Is the process stable? • Should we be taking action? • Are there any special causes? • What is the average process output? • What is the variability?
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- 26. 26 Total Variation Total Variation Within Subgroup Variation Between Subgroup Variation
- 27. 27 Control LimitsUse Within Subgroup Variation • The total variation and Within Subgroup variation are the same only if the process is stable • The Within Subgroup variation is an estimate of what the total variation would be if the process were stable • The Within Subgroup variation is used to calculate the control limits since these limits represent the range of values expected for a stable process
- 28. 28 Controls Limits • Controls limits are always: Average ± 3 Standard Deviations Where the average and standard deviation are the average and standard deviation of whatever data is plotted: −4s −3s −2s −1s 0 +1s +2s +3s +4s 99.7%
- 29. 29 Control Limits Upper Control Limit Lower Control Limit • Control Limits are statistical boundaries which tell us whether or not the process is stable • Based on the normal distribution, 99.7% of the points plot within the control limits if the process is stable • The chance of a point outside the control limits, falsely indicating the process is unstable, is only 0.3% or 1 in 370
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- 31. 31 Individuals ControlChart • Used when only a single observation per time period (subgroup): Monthly reporting data: • On-time shipments, In-process Inventory, Complaints, etc. Rare events Sales Stock Price Inventory Levels Customer Response Time Lost Time Accidents Complaints Anything that can be measured and varies
- 32. 32 Within SubgroupVariation • The best estimate is obtained by taking the differences between consecutive samples i.e. the Moving Range (MR). • We can use the the average MR, R, or the median MR, R • When using R the Short Term standard deviation is estimated by: R 1.128 R d 2 Within = = ˜ • When using R˜ the standard deviation is estimated by: ~ R 0.954 ~ R d 4 Within = =
- 33. 33 Table ofConstants for ImR charts Sample size d 2 d 3 d 4 D 3 D 4 D 5 D 6 E 2 E 5 2 3 4 5 6 7 8 9 10 0.853 0.888 0.880 0.864 0.848 0.833 0.820 0.808 0.797 0.954 1.588 1.978 2.257 2.472 2.645 2.791 2.915 3.024 3.267 2.574 2.282 2.114 2.004 1.924 1.864 1.816 1.777 2.970 3.078 0 3.865 0 2.744 0 2.376 0 2.179 0.209 1.075 1.029 1.809 0.975 0.992 0 0.055 0.119 0.168 2.054 1.967 1.901 1.850 1.128 1.693 2.059 2.326 2.534 2.704 2.847 0 0 0 0 0 0.076 0.136 0.184 0.223 2.660 1.772 1.457 1.290 1.184 1.109 1.054 1.010 3.145 1.889 1.517 1.329 1.214 1.134 We would generally calculate the differences between consecutive samples, which corresponds to a “sample size” of 2 in this table. Minitab will calculate the control chart limits for us!
- 34. 34 Control Limits The controls limits for the average, based on R are: X 3 X 3 2 X E R X 2.66R R ± = ± = ± = ± = ± 1.128 X 3 R d 2 Within The controls limits for the range, based on R are: LCL = D R = 0 × R = 0 Range 3 UCL = D R = 3.267R Range 4
- 35. 35 Control Limits 3.145 X R ~ X E R ~ R ~ X 3 X 3 5 ± = ± = ± = ± = ± 0.953 X 3 R ~ d 4 Within ~ The controls limits for the average, based on R are: ~ The controls limits for the range, based on R are: ~ ~ LCL = D R = 0 × R = 0 Range 5 3.865 R ~ UCL D R ~ = = Range 6
- 36. 36 ¯ ˜ R Versus R • Some of the differences may be contaminated by shifts in the mean (special Causes). • R, the median MR, is more robust to this contamination so is generally preferred. • When many of the differences are zero, it might be necessary to use R instead. • A conversion factor can be developed: R ~ 182 . 1 R ~ = s = 1.128 0.954 R ~ R 0.954 R 1.128 within = × = × ~
- 37. 37 Call OutTime Sample Number Call Out Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 5.35 3.28 1.07 1.06 4.29 3.23 5.40 6.42 3.25 8.55 4.26 7.48 5.35 2.14 4.24 6.44 3.21 9.66 4.28 5.33
- 38. 38 Call OutTime Sample Number Call out Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Difference 5.35 3.28 1.07 1.06 4.29 3.23 5.40 6.42 3.25 8.55 4.26 7.48 5.35 2.14 4.24 6.44 3.21 9.66 4.28 5.33 2.07 2.21 0.01 3.23 1.06 2.07 1.02 5.30 4.29 3.22 2.13 3.21 2.10 2.20 3.23 6.45 5.38 1.05 2.21 4.71 2.82 Average Average Difference Ordered 0.01 1.02 1.05 1.06 2.07 2.07 2.10 2.13 2.20 2.21 2.21 3.21 3.22 3.23 3.23 4.29 5.30 5.38 6.45 Median
- 39. 39 Individuals chart- Minitab Open Worksheet: Call Out Time StatControl ChartsVariables Charts for IndividualsI-MR Select Variable: Call Out Time Click “I-MR” Options Click “Estimate” – select “Median moving range” Click “Tests” - select “1 point 3 standard deviations from center line”
- 40. 40 IMR Chart– Minitab (using R) Observation Individual Value 2 4 6 8 10 12 14 16 18 20 12 8 4 0 UC L=11.66 _ X=4.71 LC L=-2.24 Observation Moving Range 2 4 6 8 10 12 14 16 18 20 8 6 4 2 0 UC L=8.538 __ MR=2.613 LC L=0 I-MR Chart of Call out Time ˜ 21 . 2 R ~ Median = ~ R = R × 1 . 182 = 2 . 613
- 41. 41 Lognormal andother non – normal data • When dealing with lognormal and other non-normal data we need to be cautious. • X-bar and Range charts will be acceptable for most non-normal data with a sub-group size of 5 or larger. • I-MR charts may give false indications of instability.
- 42. 42 Workshop –Individuals Control Chart Open Minitab worksheet: PAYMENT TIMES.MTW Use Minitab to create: • Individuals and Moving Range chart • Use the Median with a moving range of 2 • Assess the stability of the process • What would you want to do next? • Prepare a short report of your findings
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- 44. 44 Control Chartsfor Defective Items (Binomial) • A p-Chart is used to track the proportion defective • The p-Chart is constructed using data on the number of defectives from varying (or fixed) subgroup sizes • The data opposite shows the number of defective orders from random samples taken over 10 working days • The subgroups should be large enough to contain 5 or more defective items Sample Number Defectives (np) Subgroup Size 1 8 96 2 12 104 3 13 99 4 8 100 5 7 103 6 13 110 7 6 97 8 7 88 9 10 111 10 8 105
- 45. 45 Control Chartsfor Defective Items (Binomial) • The p-chart must satisfy the requirements for the Binomial Distribution. The particular requirements affecting the p-chart are: 1. Each unit (e.g., transaction, invoice, …) can only be classified as pass or fail 2. If one unit (e.g., transaction, invoice, …) fails, then the chance of the next unit failing is not affected • If the Binomial distribution is not appropriate then it may be possible to use the Individuals control chart already discussed • Since we are charting defective items this chart should not be used when the number of defectives is zero or there are a large number of zeros (80-90%)
- 46. 46 P ChartConstruction Sample Number Defectives (np) Subgroup Size (n) Proportion Defective (p) Average Proportion Defective (pbar) 3 Sigma UCL(p) LCL(p) 1 8 96 0.083 0.091 0.088062 0.179062 0.002938 2 12 104 0.115 0.091 0.084608 0.175608 0.006392 3 13 99 0.131 0.091 0.086718 0.177718 0.004282 4 8 100 0.08 0.091 0.086283 0.177283 0.004717 5 7 103 0.068 0.091 0.085017 0.176017 0.005983 6 13 110 0.118 0.091 0.082268 0.173268 0.008732 7 6 97 0.062 0.091 0.087607 0.178607 0.003393 8 7 88 0.08 0.091 0.091978 0.182978 -0.00098 9 10 111 0.091 0.091 0.081896 0.172896 0.009104 10 8 105 0.076 0.091 0.084204 0.175204 0.006796 Total 92 1013 p(1-p) n p(1-p) , LCL p 3 p-3 = = = = p(1-p) n 92 np UCL p 3 p 3 n 0.091, 1013 n p p = + = + p = − =
- 47. 47 P Chart- Minitab P Chart of Defectives 1 2 3 4 5 6 7 8 9 10 Sample Proportion 0.20 0.15 0.10 0.05 0.00 UCL=0.1749 _ P=0.0908 LCL=0.0067 Tests performed with unequal sample sizes Open Worksheet P Chart StatControl ChartsAttributes ChartsP Variable: Defectives Subgroups in: “Subgroup Size” Click “P Chart – Options” Click “Tests” – select “1 point 3 standard deviations from center line”
- 48. 48 Control Chartsfor Defects (Poisson) • A u Chart is used to track the number of defects per unit (e.g., transaction, invoice, …). • The u chart is constructed using data on the number of defects from varying subgroup sizes (number of units). • The data opposite shows the number of defects in the given number of invoices sampled randomly from 10 weeks of invoicing. Sample Number Defects c Invoices n 1 7 40 2 4 45 3 8 33 4 5 40 5 3 39 6 8 46 7 5 27 8 7 45 9 9 38 10 4 39
- 49. 49 Control Chartsfor Defects (Poisson) • Since the u-chart is based on the Poisson Distribution, the data should be tested to see if it fits the Poisson distribution (e.g., some “count data” such as complaints and late shipments may not fit the Poisson distribution) • If the Poisson distribution does not fit then it may be possible to use the Individuals control chart already discussed • Since we are charting defects, this chart should not be used when the number of defects is zero or there are a large number of zeros (80-90%)
- 50. 50 U Chart- Construction Sample Number Defects c Invoices n DPU u u bar LCL(u) UCL(u) 1 7 40 0.175 0.153 0 0.34 2 4 45 0.089 0.153 0 0.328 3 8 33 0.242 0.153 0 0.307 4 5 40 0.125 0.153 0 0.339 5 3 39 0.077 0.153 0 0.341 6 8 46 0.174 0.153 0 0.326 7 5 27 0.185 0.153 0 0.378 8 7 45 0.156 0.153 0 0.327 9 9 38 0.237 0.153 0 0.343 10 4 39 0.103 0.153 0 0.341 Total 60 392 0.153 u n 0.153, u = = = = u , LCL u 3 u 3 n 60 392 c n u UCL u 3 u 3 = + s = + = − s = − u u
- 51. 51 U Chart- Minitab U Chart of Defects 1 2 3 4 5 6 7 8 9 10 Sample Sample Count Per Unit 0.4 0.3 0.2 0.1 0.0 UCL=0.3410 _ U=0.1531 LCL=0 Tests performed with unequal sample sizes Open Worksheet: U Chart StatControl ChartsAttributes ChartsU Variable: Defects Subgroups in: “Units” Click “U Chart Options” Click “Tests” – select “1 point 3 standard deviations from center line”
- 52. 52 Workshop -Attributes Control Chart • Using the packets of sweets provided (assume that each packet has been taken from a different batch of production over the last few days): Randomly select 20 sweets from each packet Inspect the sweets for two types of defect- • Badly mis-shaped/damaged sweet • Missing or poorly printed logo • Using Minitab, assess the stability of the process • Prepare a short report of your findings
- 53. Uses of ControlCharts
- 54. 54 Control Charts Variable Data No Subgroups Subgroups n = 2-9 Subgroups n 9 Individuals Moving Range Chart X Bar R Chart X Bar s Chart Attribute Data Defect Data (Poisson) Defective Data (Binomial) Varying Subgroup Size Constant Subgroup Size Varying Subgroup Size Constant Subgroup Size u Chart c Chart p Chart np Chart
- 55. 55 Improvement •Control charts are one of many variation reduction tools • Controls charts detect change of the output variable (y) • The output changes because a critical input variable (x) has changed • Control charts provide clues that can help to identify these critical inputs (x’s)
- 56. 56 Clues toDiscovering Critical x’s • When did the change occur? • What patterns are emerging? Shifts • Gradual or Sudden? Trends • Increasing or Decreasing? Unusual patterns or cycles?
- 57. 57 Identification ofCritical x’s • To determine the critical x, i.e., the input causing the shift, we need to consider: Delayed detection Multiple inputs causing shifts Lack of information on inputs • We can also use screening experiments, scatter diagrams, … to determine critical x’s
- 58. 58 Transmission ofVariation, y = f(x) • Control charts can help to discover critical x’s that are causing the process to shift • Tighter control of these critical x’s will make the process more stable O U T P U T Relationship Between Input and Output Variation of Input INPUT Transmitted Variation
- 59. 59 Transactional ImprovementProcess Analyse Define Measure Improve Control Select Project Define Project Objective Form the Team Map the Process Identify Customer Requirements Identify Priorities Update Project File Control Critical x ’s Monitor y’s Validate Control Plan Identify further opportunities Close Project LSL USL 15 20 25 30 35 Phase Review 1 5 10 15 20 10.2 10.0 9.8 9.6 Upper Control Limit Lower Control Limit y Phase Review Develop Detailed Process Maps START PROCESS STEPS DECISION STOP Identify Critical Process Steps (x’s) by looking for: – Process Bottlenecks – Rework / Repetition – Non-value Added Steps – Sources of Error / Mistake Map the Ideal Process Identify gaps between current and ideal Phase Review Brainstorm Potential Improvement Strategies Select Improvement Strategy Criteria A B C D Time + s - + Cost + - + s Service - + - + Etc s s - + Plan and Implement Pilot Verify Improvement LSL USL 15 20 25 30 35 Implement Countermeasures Phase Review Phase Review Define Measures (y’s) Check Data Integrity Determine Process Stability Determine Process Capability Set Targets for Measures
- 60. 60 Control Charts- Summary • Charts can be constructed for variable or attribute data • I-MR Charts should always be considered for attribute data • Control Charts are used during capability studies to determine process stability • Real-Time control charts are used to detect shifts so that causes of shifts can be identified and eliminated • Should be used to control critical x’s (process input variables) in order to reduce variability in process outputs (y’s). • Used to monitor y’s on an ongoing basis to ensure that the process remains in control.