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# Session statistical process control (spc)

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Session statistical process control (spc) in six sigma

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### Session statistical process control (spc)

1. 1. Variation Total Quality Management Statistical Process Control (SPC)  Variation is natural - it is inherent in the world around us.  Need  X bar and R charts  No two products or service experiences are exactly the same.  P chart  C chart  With a fine enough gauge, all things can be  Applications seen to differ.  One of the roles of management is work with all employees to reduce variation as much as possible. The Presence of Variation Types of Variation Common Cause Variation: The variation that naturally occurs and is expected in the system8’ -- normal -- random -- inherent Measuring -- stable 4’ 4’ 4’ 4’ Device 4’ 4’ 4’ 4’ Tape Measure Special Cause Variation: Variation which is abnormal - indicating something out of the ordinary has happened. 4.01’ 4.01’ 4.01’ 4.00’ Engineer Scale -- nonrandom 4.009’ 3.987’ 4.012’ 4.004’ -- unstable Caliper -- assignable cause variation 4.00913’ 3.98672’ 4.01204’ 4.00395’ Elec. Microscope Type of Variation Total Product or Process Travel Time to Work Example Variation Measurement of Interest: Time to get to work. Total variation = Common Cause + Special Cause Common Cause Variation Sources: -- traffic lights To reduce Total Variation -- traffic patterns -- weather First reduce or eliminate special cause variation -- departure time Reduce common cause variation Special Cause Variation Sources: -- accidents Identify the source and remove the causes -- road construction detours -- petrol refills 1
2. 2. Statistical Quality Types of Control Statistical Quality Control  Measures performance of a process  Uses mathematics (i.e., statistics) Statistical Quality Control  Involves collecting, organizing, & interpreting data Process Acceptance  Objective: provide statistical when Control Sampling assignable causes of variation are present Variables Attributes Variables Attributes Charts Charts  Used to – Control the process as products are produced – Inspect samples of finished products Quality Statistical Process Characteristics Control (SPC) Variables Attributes  Statistical technique used to ensure Measured values;  Has or Has not/Good process is making product to standard e.g., weight, length, or Bad/Pass or  All process are subject to variability volume,voltage, current etc. Fail/Accept or Reject – Natural causes: Random variations May be in whole or in  Characteristics for – Assignable causes: Correctable problems fractional numbers which you focus on Machine wear, unskilled workers, poor defects material Continuous random variables  Categorical or  Objective: Identify assignable causes discrete random  Uses process control charts variables Comparing Distributions Production Output Distributions Production Output Example What is the Difference? Units Produced Frequency Plant A Plant B Plant A 99 90 100 90 100 100 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 100 110 101 110 FrequencyX X  500  100 X X  500  100 Plant B n 5 n 5 No Differences!??? 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 2
3. 3. The Concept of StabilityMeasure of Variation (Sigma) 99.7% S = Standard Deviation 95% S (X  X ) 2 X - 3S X - 2S X - 1S 68% X +1S X +2S X + 3S Plant A n 1 Plant BX (X  X ) ( X  X )2 X (X  X ) ( X  X )299 99-100 = -1 12 =1 90 90 -100= -10 -102 =100100 100-100 = 0 02 = 0 90 90 -100= -10 -102 =100100 100-100 = 0 0 2=0 100 100 -100 = 0 02 = 0100 100-100 = 0 02 = 0 110 110 -100 = 10 102 =100101 101-100 = 1 12 = 1 110 110 -100 = 10 102 =100  0  2   0   400 2 400 X S  .707 S  10 4 4 X 2 400 Plant A S  .707 Plant B S  10 4 4 X  2S  98.586 X  2S  80 X  1S  100.707 X  3S  102.121 X  1S  110 X  3S  130 X X X  100 X  100 X  1S  99.293 X  1S  90 X  2S  101.414 X  2S  120 X  3S  97.879 X  3S  70 Under Normal Conditions: Under Normal Conditions: 68 percent of the time output will be between 99.293 and 68 percent of the time output will be between 90 and 110 units 100.707 units 95 percent of the time output will be between 80 and 120 units 95 percent of the time output will be between 98.586 and 99.7 percent of the time output will be between 70 units and 101.414 units 130 units 99.7 percent of the time output will be between 97.879 units and 102.121 unitsControl Limits Process Control Limits Control Limits are the statistical boundaries of a process Special Cause Variation which define the amount of variation that can be considered as normal or inherent variation Upper Control Limit UCL=X +3 Common Cause 3 sigma control limits are most common Average + 3S from the mean LCL =X - 3 Lower Control Limit If the process is in control, a value outside the control limit will occur only 3 time in 1000 ( 1 - .997 = .003) Special Cause Variation 3
4. 4. Relationship Between Sampling Distribution of Population and Sampling Means, and Process Distributions DistributionThree population distributions Sampling Distribution of sample means distribution of the Beta means Mean of sample means  x Normal Standard deviation of x Process  x  the sample means n distribution of the sample Uniform  3 x  2 x 1 x xσ 1 x  2 x  3 x (mean) xm 95.5% of all x fall within  2  x ( mean ) 99.7% of all x fall within  3 x Theoretical Basis Theoretical Basis of Control Charts of Control Charts Central Limit TheoremCentral Limit Theorem Mean Standard deviation As sample size sampling distribution x gets large becomes almost normal regardless of X  x  n enough, population distribution. X X  X X Process Control Limit Concepts Process Control Limit Concepts (continued)  Control Limits Define the limits of stability  Measures inside control limits are assumed to come  The ULC and LCL are calculated so that, if the from a stable process - Measures outside the control process is stable, almost all of the process limits are unexpected and considered the result of a output will be located within the control limits. special cause  3 sigma control limits  The most commonly used  The control limits are computed directly from the  UCL is 3 standard deviations above the sample data selected from the process -- The limits average and the average are not the choice of management or the operator - Formulas exist.  LCL is 3 standard deviations below the average  If the process is stable, only about 3 out of  The control limits define the range of inherent 1000 process outputs will fall outside the variation for the process as it currently exists, not control limits. how we would like it to be 4
5. 5. Control Chart Purposes Control Chart Types Categorical or Discrete  Show changes in data pattern Continuous Control Numerical Data Numerical Data – e.g., trends Charts Make corrections before process is out of control Variables Attributes Charts Charts  Show causes of changes in data – Assignable causes R X P C Data outside control limits or trend in data Chart Chart Chart Chart – Natural causes Random variations around average Statistical Process Control Steps Commonly Used Control Charts Produce Good NoStart Provide Service  Variables data Assign.  x-bar and R-charts Take Sample Causes?  x-bar and s-charts Yes Inspect Sample  Charts for individuals (x-charts) Stop Process  Attribute data Create Find Out Why Control Chart  For “defectives” (p-chart, np-chart)  For “defects” (c-chart, u-chart) X Chart X Chart Control Limits UCL  x  A R From Type of variables control chart x 2 Table Shows sample means over time LCL  x  A R Sample x 2 Range at Monitors process average Sample Time i n Mean at Example: Weigh samples of coffee &  xi Time i n  Ri compute means of samples; Plot x  i  R  i 1 n n # Samples 5
6. 6. Factors for Computing Control Chart Limits R Chart Sample Mean Upper Lower  Type of variables control chart Size, n Factor, A2 Range, D4 Range, D3 2 1.880 3.268 0 – Interval or ratio scaled numerical data 3 1.023 2.574 0 4 0.729 2.282 0  Shows sample ranges over time 5 0.577 2.115 0 – Difference between smallest & largest 6 0.483 2.004 0 values in inspection sample 7 0.419 1.924 0.076 8 0.373 1.864 0.136  Monitorsvariability in process 9 0.337 1.816 0.184 10 0.308 1.777 0.223  Example: Weigh samples of coffee & compute ranges of samples; Plot 0.184 R Chart Out-of-control…when? Control Limits UCL R  D 4 R From Table LCL R  D 3R n Sample Range at  Ri Time i R  i 1 n # SamplesProcess is Out of Control Process is Out of Control Trend: 8 or more points moving in the same direction - up or down Process Control Chart Process Control Chart 200 200 180 160 Shift in Process Average 180 140 160 Process Average Trend Up 140 UCL Measure 120 UCL 120 Measure 100 Average 100 80 80 Average 60 LCL 60 40 40 LCL 20 20 0 0 200 201 202 203 204 205 206 207 208 209 200 201 202 203 204 205 206 207 208 209 210 Sam ple Num ber Sample Number 6
7. 7. Process is Out of Control Process is Out of ControlNonrandom Patterns Present in the Data Nonrandom Patterns Present in the Data Process Control Chart Process Control Chart 200 150 180 140 160 130 UCL 140 UCL Measure Measure 120 120 100 Average 110 80 100 Average 60 LCL 90 40 80 20 70 LCL 0 200 201 202 203 204 205 206 207 208 209 210 60 Sample Number 200 201 202 203 204 205 206 207 208 209 210 Sample Number Using X and R Process Control Signals of Control Problems Charts  A point outside the control limits Situation: Boise Cascade is interesting in monitoring the  7 or more points in a row above or length of logs that arrive at a mill yard. In the long run, they below the average (center-line) Shift want the average to be 18 feet and the variation should continue to decline  8 or more points in a row moving in the same direction, up or down. Trend The process output measure is length of the logs.  Nonrandom patterns in the data An X and R chart will be developed to monitor the log lengths. Use Common sense and Good Judgment Log length Example: Data Developing X and R Charts 30 days (subgroups) -- subgroup size =4  Define Process Measurement of Interest Day Log Length (feet)  Determine Subgroup (sample) size (3-6) 1 2 3 4  Determine data gathering methods 1 16 18 21 23  where, how, who 2 26 20 19 19 3 20 22 18 18  Determine number of subgroups (20-30) 4 24 16 22 20 5 17 19 24 17  Collect Data 6 17 17 15 18  Compute X and R for each subgroup 7 22 12 20 22 8 24 19 19 17  Plot X and R on separate charts 9 18 18 20 14 10 17 23 19 15  Compute Control Limits 11 20 20 17 21  Draw Control Limits and Centerline on Charts 12 21 17 21 23 13 22 17 22 17 14 16 19 18 19 15 17 18 15 23 7
8. 8. Log Length Data Compute X for Each(continued) Subgroup X Where: Day Log Length (feet) X = the values in X = the subgroups 1 2 3 4 16 19 17 21 17 n = subgroup size n 17 19 19 13 16 18 21 14 17 16 19 18 17 25 18 20 20 18 20 19 21 23 21 23 21 First Subgroup: 22 20 20 20 14 23 18 18 26 15 24 20 22 23 21 25 23 22 21 24 16 + 18 + 21 + 23 26 22 14 21 19 X1 = = 19.5 27 28 18 19 20 20 18 16 22 14 4 29 21 19 16 20 30 22 22 19 21 Compute R for Each Log Length Example: Data Subgroup 30 days (subgroups) -- subgroup size = 4 Day Log Length (feet)R = Subgroup High - Subgroup Low 1 2 3 4 Average = X Range = R 1 16 18 21 23 19.5 7 2 26 20 19 19 21 7 3 20 22 18 18 19.5 4 First Subgroup: 4 24 16 22 20 20.5 8 5 17 19 24 17 19.25 7 6 17 17 15 18 16.75 3 7 22 12 20 22 19 10 R1 = 23 - 16 8 9 24 18 19 18 19 20 17 14 19.75 17.5 7 6 10 17 23 19 15 18.5 8 11 20 20 17 21 19.5 4 12 21 17 21 23 20.5 6 = 7 13 14 22 16 17 19 22 18 17 19 19.5 18 5 3 15 17 18 15 23 18.25 8 Log Length Data Plot the X Values (continued) P lot of S ubgroup Ave ra ge s Day Log length (feet) 50 45 1 2 3 4 Average = X Range = R 40 16 19 17 21 17 18.5 4 35 Su b g ro u p A verag e 17 19 19 13 16 16.75 6 18 21 14 17 16 17 7 30 19 18 17 25 18 19.5 8 25 20 20 18 20 19 19.25 2 21 23 21 23 21 22 2 20 22 20 20 20 14 18.5 6 15 23 18 18 26 15 19.25 11 24 20 22 23 21 21.5 3 10 25 23 22 21 24 22.5 3 5 26 22 14 21 19 19 8 27 18 20 18 22 19.5 4 0 11 13 15 17 19 21 23 25 27 29 1 3 5 7 9 28 19 20 16 14 17.25 6 29 21 19 16 20 19 5 Su b g r o u p 30 22 22 19 21 21 3 8
9. 9. Compute Centerlines for Plot of R Values (Ranges) Each Chart Plot of R Values 18 X Chart: 16 14 12 X = X i = 577.5 = 19.25 k 30Range (R) 10 8 6 4 R Chart: 2 R 0 171 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 i Subgroup R = = = 5.7 k 30 Plot the the Centerline on X Chart Plot of Centerline on R Chart P lot of S ubgroup Ave ra ge s Plot of R Values 50 45 18 40 16 35 14 Su b g ro u p A verag e 30 12 Range (R) 25 10 8 20 X = 19.25 6 15 R= 5.7 4 10 2 5 0 0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 11 13 15 17 19 21 23 25 27 29 1 3 5 7 9 Su b g r o u p Subgroup Compute the Control Limits on Compute X Control Limits the X Chart Table n A2 D3 D4 1 2.66 n A2 D3 D4 2 1.88 0.0 3.27 3 1.02 0.0 2.57 1 2.66 4 0.73 0.0 2.28 2 1.88 0.0 3.27 5 6 0.58 0.48 0.0 0.0 2.11 2.00 3 1.02 0.0 2.57 4 0.73 0.0 2.28 UCL = X + A 2 R = 19.25 + .73(5.7) = 23.41 5 0.58 0.0 2.11 6 0.48 0.0 2.00 LCL = X - A 2 R = 19.25 - .73(5.7) = 15.09 Now Plot the Control Limits on the X Chart 9
10. 10. Compute Control Limits for R Plot Control Limits on X Chart Chart Plot of Subgroup Ave ra ge s 30 n A2 D3 D4 25 1 2.66 UCL 2 1.88 0.0 3.27 23.41 3 1.02 0.0 2.57 Subgroup Averag e 20 X = 19.25 4 0.73 0.0 2.28 5 0.58 0.0 2.11 15 15.09 6 0.48 0.0 2.00 LCL 10 5 UCL = D 4 R = 2.28(5.7) = 13.00 0 DR 11 13 15 17 19 21 23 25 27 29 1 3 5 7 9 Subgroup LCL = 3 = 0.00(5.7) = 0.00 Plot the Control Limits on R Chart R Chart with Control Limits Utilizing the Control Charts Plot of R Values  Continue to Collect Subgroup data 18  Plot Values to X and R charts 16  Examine the R Chart First - Then the X Chart 14 12 UCL 13.0  Look for Signals  A point outside the control limitsRange (R) 10  7 points in a row above or below the centerline 8 6 4 5.7  8 points in a row moving in the same direction 2  any nonrandom patterns LCL 0 0.0  Take action when signal indicates 11 13 15 17 19 21 23 25 27 29 1 3 5 7 9 Subgroup  Update Control limits when appropriate Special control charts for variable Special Variables Control Charts data X bar and s-Chart  x-bar and s charts S (X  X ) 2 For the associated x-chart, the control limits are derived from n 1  x-chart for individuals the overall standard deviation are: UCLS  B 4 S UCL  x  A s x 3 From Table LCL S  B3 S LCL  x  A s x 3 n Sample S.D.  Si at Time i S  i 1 59 n # Samples 10
11. 11. Set of observations measuring the percentage of cobalt in a chemical process X chart for individuals UCLx  x  3R / d 2 UCLx  x 3R / d 2Samples of size 1, however, do not furnish enough information for process variabilitymeasurement. Process variability can be determined by using a moving average ofranges, or a moving range, of n successive observations. For example, a movingrange of n=2 is computed by finding the absolute difference between two successiveobservations. The number of observations used in the moving range determines theconstant d2; hence, for n=2, from appendix b, d2=1.128. UCL R  D 4 R LCL R  D 3 R Charts for Attributes  Fraction nonconforming (p-chart)  Fixed sample size  Variable sample size  np-chart for number nonconforming  Charts for defects  c-chart  u-chart 11
12. 12. P Charts P Chart Example Plywood Veneer is graded when it comes out of the Used When the Variable of Interest is an Attribute and dryer. Sheets that graded incorrectly cause We are Interested in Monitoring the Proportion of Items problems later in the process. Management is in Sample that have this Attribute - interested in monitoring the rate of incorrectly graded veneer. Can accommodate unequal sample sizes. Sample sizes are usually 50 or greater. The variable of interest is the proportion ofExamples:Need 20-30 samples to construct the P-chart. incorrectly graded veneer. Each shift, n=100 sheets are selected and evaluated Proportion of Invoices with errors for grade. The number of mis-grades are Proportion of Incorrectly Sorted Logs recorded. Proportion of Items Requiring Rework P-Chart Data P Charts  Step 1:  Collect appropriate data.  Attribute data of the “yes/no” type A Sheet is inspected. Is it incorrectly graded - Yes or No? Record the number of “Yes” occurrences Fraction Nonconformance P Charts - p- Values  Step 2:  Calculate the fraction defective for each subgroup.  The fraction defective is known as the p value: number of nonconform ances in the subgroup p= size of the subgroup Key Point: The fraction defective is always expressed as a decimal value. Using the percentage value (i.e. 4.7% rather than .047) will cause later computations to be inaccurate. 12
13. 13. P Charts Plot of the p-Values  Step 3:  Plot the data on a graph.  Plot each p value P-Values and Centerline p Charts  Step 4:  Compute the center line for the p chart CL = .215 and plot on the chart  The center line of the p chart is p total number of nonconform ances in all subgroups p= total number of items examined in all subgroups 429 p=  .215 2,000p Charts P Control Chart Step 5  If the sample sizes are equal, compute the 3 sigma control limits using the following formulas - plot on control chart: UCL = .338 Upper Control Limit p (1 - p ) UCL = p + 3 n CL=.215 .215(1  .215) UCL  .215  3  .338 100 Lower Control Limit LCL = .092 p (1 - p ) LCL = p - 3 n .215(1  .215) LCL  .215  3  .092 100 13
14. 14. P Charts P Charts Step 5: Alternative - When sample sizes are Analyzing p Charts not equal  p charts are analyzed using the standard tests  Compute the 3-sigma upper and lower control for special cause variation: limits for the p chart.  If the size of the subgroup size varies, the control limit  A Point located outside the control limits calculations can be accomplished by two methods:  7 or more points above or below the centerline  Compute multiple control limits based on the largest and  8 or more points moving in the same direction smallest subgroup sizes  Other evidence of nonrandom patterns  The two sets of control limits are plotted on the p chart. By calculating control limits based on the largest and smallest subgroups, both the narrowest limits (largest subgroup size) and the widest limits (smallest subgroup size) are plotted.  Compute separate control limits for each fraction nonconformance. p Charts Using Multiple Control Limits:  In analyzing a control chart with multiple limits, it must be clear that:  Any value plotting outside the widest control limits is considered out of control  Any value plotting inside the narrowest control limits is considered in control  Only those values, if any, which plot between the two upper or two lower control limits raise questions needing further evaluation (calculate their individual limits) np-Charts for number Nonconforming  The np-chart is a useful alternative to the p-chart because it is often easier to understand for production personnel-the number of nonconforming items is more meaningful than a fraction.  To use the np-chart, the size of each sample must be constant. 14
15. 15. y1  y 2  .......  y n np  k Estimate of the standard deviation snp = np (1 - p ) wh ere p  ( n p ) / n Upper Control Limit UCL n p = np + 3 np (1 - p ) Lower Control Limit LCL n p = np - 3 np (1 - p ) Chart for defects  A defect is a single nonconforming characteristics of an item, while a defective refers to an item that has one or more defects.  In some situation, quality assurance personnel mat be interested not only in whether an item is defective but also in how many defects it has. For example, in complex assemblies such as electronics, the number of defects is just as important as whether the product is defective.  The c-chart is used to control the total number of defects per unit when subgroup size is constant. If subgroup sizes are variable, a u-chart is used to control the average 87 number of defects per unit.c Charts c Charts  Necessary Characteristics  A c chart is a process control tool for  Subgroups must be the same size (in practical use, if they vary less than + 15% from the average it is charting and monitoring the number acceptable to use the average subgroup size to compute the chart) of attributes per unit. Each unit must  Subgroup size must be large enough to provide an be like all other units with respect to average of at least 5 nonconformities per subgroup size, volume, height, or other  The attribute of interest is the number of nonconformities per unit measurement.  Each unit may have one or more nonconformities  The actual number of nonconformities is small compared with the number of opportunities for nonconformities 15