S TATISTICAL PROCESS CONTROL CUSTOMER & COMPETITIVE INTELLIGENCE FOR PRODUCT, PROCESS, SYSTEMS & ENTERPRISE EXCELLENCE
DEPARTMENT
OF STATISTICS
REDGEMAN@UIDAHO.EDU
OFFICE: +1-208-885-4410
DR. RICK EDGEMAN, PROFESSOR & CHAIR – SIX SIGMA BLACK BELT
Quality Management:
Statistical Process Control
Statistical Process Control Statistical Process Control (SPC) can be thought of as the application of statistical methods for the purposes of quality control and improvement. Quality Improvement is perhaps foremost among all areas in business for application of statistical methods.
Data Driven Decision Making “In God we trust. ... all others must bring data.” --- The Statistician’s Creed SPC is one method that assists in enabling “data-driven decision making”. SPC is a key quantitative aid to quality improvement efforts.
Control Charts: Recognizing Sources of Variation • Why Use a Control Chart? – To monitor, control, and improve process performance over time by studying variation and its source.
• What Does a Control Chart Do? – Focuses attention on detecting and monitoring process variation over time; – Distinguishes special from common causes of variation, as a guide to local or management action; – Serves as a tool for ongoing control of a process; – Helps improve a process to perform consistently and predictably for higher quality, lower cost, and higher effective capacity; – Provides a common language for discussing process performance.
Control Charts: Recognizing Sources of Variation • How Do I Use Control Charts? – There are many types of control charts. The control charts that you or your team decides to use should be determined by the type of data that you have. – Use the following tree diagram to determine which chart will best fit your situation. Only the most common types of charts are addressed.
Control Chart Selection: Variable Data Measured & Plotted on a Continuous Scale such as Time, Temperature, Cost, Figures. n=1
2<n<9 median
n is ‘small’ 3<n<5
n is ‘large’ n > 10
X & Rm
X&R
X&R
X&S
Control Chart Selection: Attribute Data Counted or Plotted as Discrete Events Such as Shipping Errors, Waste or Absenteeism.
Defect or Nonconformity Data Constant sample size
Variable sample size
c chart
u chart
Defective Data Constant n > 50 p or np chart
Variable n > 50 p chart
Control Chart Construction • Select the process to be charted; • Determine sampling method and plan; – How large a sample needs to be selected? Balance the time and cost to collect a sample with the amount of information you will gather. – As much as possible, obtain the samples under the same technical conditions: the same machine, operator, lot, and so on. – Frequency of sampling will depend on whether you are able to discern patterns in the data. Consider hourly, daily, shifts, monthly, annually, lots, and so on. Once the process is “in control”, you might consider reducing the frequency with which you sample. – Generally, collect 20-25 groups of samples before calculating the statistics and control limits. – Consider using historical data to establish a performance baseline.
Control Chart Construction • Initiate data collection: – Run the process untouched, and gather sampled data. – Record data on an appropriate Control Chart sheet or other graph paper. Include any unusual events that occur. • Calculate the appropriate statistics and control limits: – Use the appropriate formulas. • Construct the control chart(s) and plot the data.
Control Chart Interpretation: Time, Production & Spatial Analysis: Still-Life Photography An event taken in isolation or a group of items each selected from a process during the same (brief) time span can generally provide information about process performance ONLY during that brief span. Unless process performance is static through time this will be true. Dynamic processes vary through time.
Control Chart Interpretation: Time, Production & Spatial Analysis: The Video Generation If a process varies through time, it is often useful to know how the process varies so that it can be “controlled” or “guided” in its behavior. This requires monitoring through time, similar to videotaping the process - in some sense, the process has a “life of its own” and we want to nurture that life.
Control Chart Interpretation: Persistence Through Time • A process can be characterized by: – Examining its behavior during a sufficiently brief interlude of time – Examining its behavior across a greater expanse of time.
• Stable process: one which performs with a high degree of consistency at an essentially constant level for an extended period of time – “In-control”
• A process that is not stable is referred to as being in an outof-control state
Data Plot with PAT Zones 36.10 (A)
36
34.36 (B) 34 32.62 (C) 32 30.88 30 29.14 (C) 28
27.40 (B)
26
25.66 (A) 0
5
10
15
Item
20
25
30
Control Chart Interpretation: Pattern Analysis Tests (PATs) • PAT 1: One point plots beyond zone A on either side of the mean • PAT 2: Nine points in a row plot on the same side of the mean • PAT 3: Six consecutive points are strictly increasing or strictly decreasing • PAT 4: Fourteen consecutive points which alternate up and down
Control Chart Interpretation: Pattern Analysis Tests â&#x20AC;˘ PAT 5: Two out of three consecutive points plot in zone A or beyond, and all three points plot on the same side of the mean â&#x20AC;˘ PAT 6: Four out of five consecutive points plot in zone B or beyond, and all five points plot on the same side of the mean
Control Chart Interpretation: Pattern Analysis Tests â&#x20AC;˘ PAT 7: Fifteen consecutive points plot in zones C, spanning both sides of the mean â&#x20AC;˘ PAT 8: Eight consecutive points plot at more than one standard deviation away from the mean with some smaller than the mean and some larger than the mean
Control Chart Interpretation: Monitoring & Improving Processes • The performance of every process will be composed of two primary components: – Controlled or guided performance which is predictable in both an instantaneous and long-term sense – Uncontrolled variation • Special or assignable causes • Common causes
Control Chart Interpretation: Monitoring & Improving Processes • True process improvement is typically a result of either: – Breakthrough thinking – Efforts to identify and reduce or eliminate common causes of variation; methodical quantitatively oriented tools which monitor a process over time --- the approach taken generally by “control charts”.
Control Chart Interpretation â&#x20AC;˘ The vertical axis coordinate of a point plotted on the chart corresponding to the value of an appropriate PPM and the horizontal axis coordinate of a point plotted on the chart corresponding to the time in sequence at which the observation was made with the time between observations divided into equal increments.
Control Charts: Colors Used UCL A
*
**
C
*
*C B A
U2SWL
*
B
* * * *
*
* *
*
U1SL CL L1SL
*
L2SWL LCL
P Charts for the Process Proportion Based on m preliminary samples from the process. While the number of items, n, may vary from sample to sample, it is customary for each of the samples in a given application to include the same number of items, n. For the ith of these m samples, let
Yi =
number of defective units in the sample
Then the proportion defective for the ith sample is:
pi = Yi / ni
Control Chart Interpretation • Center line (CL) positioned at the estimated mean • Upper and lower one standard deviation lines (U1SL and L1SL) positioned one standard deviation above and below the mean. • Upper and lower two standard deviation warning lines (U2SWL and L2SWL) positioned at two standard deviations above and below the mean. • Upper and lower control lines (UCL and LCL) positioned at three standard deviations above and below the mean.
P Charts for the Proportion An estimate of the overall process proportion defective is
p = (Y1+Y2+...+ Ym) / (n1+n2+...+ nm) = (total defectives) / (total items) When all samples have n items each then p = (p1 + p2 + ... + pm)/m
The estimated standard deviation of the process proportion defective is
Sp =
â&#x2C6;&#x161;
p (1-p)/ ni
P Chart Control Lines & Limits The coordinates for the seven lines on the P chart are positioned at: CL = U1SL = U2SWL = UCL =
p p + Sp p + 2Sp p + 3Sp
L1SL = p - Sp L2SWL = p - 2Sp LCL = p - 3Sp
South of the Borders, Inc. Custom Wallpapers & Borders
Free Estimates (013) 555-9944
South of the Borders, Inc. South of the Borders, Inc. is a custom wallpapers and borders manufacturer. While their products vary in visual design, the manufacturing process for each of the products is similar. Each day a sample of 100 rolls of wallpaper border is sampled and the number of defective rolls in the sample is noted. The number of defective rolls in samples from 25 consecutive production days follows. Determine all coordinates; construct & interpret the p chart. PATs 1, 2, 3 and 4 apply to p charts.
South of the Borders, Inc. Day Defective Rolls 1 13 2 4 3 7 4 11 5 8 6 10 7 2 8 9 9 12 10 6 11 4 12 7 13 9
Day 14 15 16 17 18 19 20 21 22 23 24 25
Defective Rolls 8 9 3 5 14 10 11 6 6 9 3 10
South of the Borders, Inc. Total # of items sampled = 2500 Total # of defective items = 196 p = 196/2500 = .0784 Sp = â&#x2C6;&#x161; .0784(.9216)/100 = .02688
South of the Borders, Inc. CL =
.0784
UCL =
.0784 +
3(.0269) =
.1590
LCL =
.0784 -
.0806 =
-.0022 (na)
U2SWL = .0784 +
2(.0269) =
.1322
L2SWL = .0784 -
.0538 =
.0246
U1SL =
.0784 +
.0269 =
.1053
L1SL =
.0784 -
.0269 =
.0515
P Chart for Defective Wallpaper Rolls 3.0SL=0.1590
Proportion
0.15
2.0SL=0.1322 1.0SL=0.1053
0.10
P=0.07840 -1.0SL=0.05152
0.05
-2.0SL=0.02464 0.00 Subgroup Rolls
-3.0SL=0.000 0
5
10
15
20
25
8
6
9
11
10
Proportion of Defective Rolls Received
South of the Borders, Inc. P Chart Interpretation • No violations of PATs one through four are apparent. This implies that the process is “in a state of statistical control”. • It does not indicate that we are satisfied with the performance of the process. • It does, however, indicate that the process is stable enough in its performance that we may seriously engage in PDCA for the purpose of long-term process improvement.
C and U Charts for Nonconformities • When data originates from a Poisson process, it is customary to monitor output from the process with a defects or C chart • Recall the Poisson Distribution with mean = c and standard deviation = √c • P(y) = cye-c/y!
C & U Charts for Nonconformities • C represents the average number of defects (nonconformities) per measured unit with all units assumed to be of the same “size” and all samples are assumed to have the same number of units • m = 20 to 40 initial samples • C = (number of defects in the m samples) / m • Estimated standard deviation=
√C
C Control Chart Coordinates • • • •
CL = C UCL = C+3 C and U2SWL= C+2 C and U1SL = C+ C and
LCL = C-3 C L2SWL = C- 2 C L1SL = C- C
Scientific & Technical Materials, Inc.
Scientific & Technical Materials, Inc. â&#x20AC;˘ Scientific & Technical Materials, Inc. produces material for use as gaskets in scientific, medical, and engineering equipment. Scarred material can adversely affect the ability of the material to fulfill its intended use. â&#x20AC;˘ A sample of 40 pieces of material, taken at a rate of 1 per each 25 pieces of material produced gave the results on the following slide. Use this information to construct and interpret a C chart.
Scientific & Technical Materials, Inc. Piece Scars
1 4
Piece Scars
11 12 13 14 15 16 17 18 19 20 1 1 2 3 0 4 3 2 2 1
Piece Scars
21 22 23 24 25 26 27 28 29 30 2 1 0 3 5 4 2 1 4 2
Piece Scars
31 32 33 34 35 36 37 38 39 40 2 1 1 3 2 0 1 5 9 1
2 4
3 2
4 3
5 1
6 2
7 0
8 2
9 10 3 1
Scientific & Technical Materials, Inc. • • • • • • •
C= 90/40= 2.25= CL, Sc= 2.25 = 1.5 UCL= 2.25+ 3(1.5) = 6.75 LCL= 2.25- 4.5 = -2.25 (NA) U2SWL= 2.25+ 2(1.5)= 5.25 L2SWL= 2.25- 3 = -0.75 (NA) U1SL= 2.25+ 1.5 = 3.75 L1SL= 2.25- 1.5 = 0.75
Scientific & Technical Materials, Inc. C Chart for Gasket Material Data 10 9 8 7
UCL
6
U2SWL
5 4
U1SL
3
CL
2 1
L1SL
37 39
33 35
29 31
25 27
21 23
17 19
13 15
9 11
7
5
3
1
0
Scientific & Technical Materials, Inc. C Chart Interpretation • Application of PATs one through four indicates a violation of PAT 1 at sample number 39 where 9 scars appear on the surface of the sampled material. • Corrective measures would be identified and implemented. • After process stability was (re) assured, we would move into PDCA mode.
Variation of the C chart where Sample size may vary U = (u1+u2+...+um) / (n1+n2+...+nm) = (total # of defects) / (total # of units in the m samples)
• • • •
CL = U UCL = U+ 3 U/ni, LCL= U-3 U/ni U2SWL= U+ 2 U/ni, L2SWL= U- 2 U/ni U1Sl= U+ U/ni, L1SL= U- U/ni
U Chart
Control Charts for the Process Mean and Dispersion ‘X bar’ Chart Typically used to monitor process centrality (or location) Limits depend on the measure is used to monitor process dispersion (R or S may be used).
‘S’ or ‘Standard Deviation’ Chart: Used to monitor process dispersion
‘R’ or ‘Range’ Chart: Also used to monitor process dispersion
Sample Summary Information • • • •
m = 20 to 40 initial samples of n observations each. Xi = mean of ith sample Si = standard deviation of ith sample Ri = range of ith sample X = (X1 + X2 +... + Xm) / m R = (R1 + R2 + ... +Rm)/m S = (S1 + S2 + ... + Sm)/m σ = R/d2 where d2 depends only on n
Coordinates for the X-bar Control Chart: “R” • • • • • • •
CL= X, UCL= X+ A2R, UCL= X- A2R U2SWL= X+ 2A2R/3 L2SWL= X- 2A2R/3 U1SL= X+ A2R/3 L1SL= X- A2R/3 A2 is a constant that depends only on n.
Coordinates for an R Control Chart • • • • • • • •
CL= R UCL= D4R LCL= D3R U2SWL= R+ 2(D4-1)R/3 L2SWL= R- 2(D4-1)R/3 U1SL= R+ (D4-1)R/3 L1SL= R- (D4-1)R/3 where D3 and D4 depend only on n
Championship Card Company
Championship
Championship Card Company Championship Card Company (CCC) produces collectible sports cards of college and professional athletes. CCCs card-front design uses a picture of the athlete, bordered all-the-way-around with one-eighth inch gold foil. However, the process used to center an athlete’s picture does not function perfectly. Five cards are randomly selected from each 1000 cards produced and measured to determine the degree of off-centeredness of each card’s picture. The measurement taken represents percentage of total margin (.25”) that is on the left edge of a card. Data from 30 consecutive samples is included with your materials, and summarized on the following slides.
Championship Card Company Sample X-bar R 1 55.6 22 2 61.0 23 3 45.2 20 4 46.2 11 5 46.8 18 6 7 8 9 10
49.8 46.8 44.2 50.8 48.4
23 18 20 32 16
Sample X-bar R 11 51.2 15 12 49.4 14 13 44.0 32 14 51.6 14 15 53.2 12 16 17 18 19 20
52.4 50.6 56.0 50.2 44.0
23 8 18 19 23
Sample 21 22 23 24 25
X-bar 50.0 47.0 50.6 48.8 44.6
R 11 14 15 16 22
26 27 28 29 30
46.8 49.2 45.6 57.6 51.4
16 8 19 40 17
Championship Card Company Summary Information n=5
A3 = 1.427
X = 49.63
B3 = NA
S = 7.42
B4 = 2.089
R = 18.63
D3 = NA
d2 = 2.326
D4 = 2.115
A2 = 0.577
Ď&#x192; = R/d2 = 8.01
Championship Card Company X-bar and R Control Chart Limits UCL U2SWL U1SL CL L1SL L2SWL LCL
X based on R
R
60.38 56.80 53.22 49.63 46.05 42.47 38.89
39.40 32.48 25.55 18.63 11.71 4.79 ------
Championship Card Company X Bar Chart for Sports Cards Centering Values Limits Based on R 1 3.0SL=60.38
60
SampleMean
2.0SL=56.80 1.0SL=53.22 50
X=49.63 - 1.0SL=46.05 - 2.0SL=42.47
40
- 3.0SL=38.89 0
10
20
30
Sample Number
Samples of 5 from each 1000 Cards Printed
Championship Card Company R Chart for Sports Card Centering 40
3.0SL=39.40
Sample Range
2.0SL=32.48 30 1.0SL=25.55 20
R=18.63 - 1.0SL=11.71
10
- 2.0SL=4.791 - 3.0SL=0.000
0 0
10
20
30
Sample Number
Samples of 5 Cards from each 1000 Produced
Championship Card Company X-bar & R Chart Interpretation • Application of all eight PATs to the X-bar chart indicated a violation of PAT 1 (one point plotting above the UCL) at sample 2. Apparently, a successful process adjustment was made, as suggested by examination of the remainder of the chart. • Application of PATs one through four to the R chart indicated a violation of PAT 1 at sample 29. Measures would be investigated to reduce process variation at that point. The violation was a “close call” and was out of character with the remainder of the data. • We are close to being able to apply PDCA to the process for the purpose of achieving lasting process improvements.
Coordinates for the X bar Control Chart: “S” • • • • • • • •
CL= X UCL= X= A3S LCL= X- A3S U2SWL= X+ 2A3S/3 L2SWL= X- 2A3S/3 U1SL= X+ A3S/3 L1SL= X- A3S/3 where A3 depends only on n
Coordinates on an S Control Chart • • • • • • • •
CL= S UCL= B4S LCL= B3S U2SWL= S+ 2(B4-1)S/3 L2SWL= S- 2(B4-1)S/3 U1SL= S+ (B4-1)S/3 L1SL= S- (B4-1)S/3 where B3 and B4 depend only on n
Championship Card Company Sample X-bar S Sample X-bar S Sample X-bar S 1 55.6 9.63 11 51.2 6.83 21 50.0 5.15 2 61.0 8.63 12 49.4 5.46 22 47.0 5.15 3 45.2 7.40 13 44.0 14.35 23 50.6 5.55 4 46.2 4.09 14 51.6 5.18 24 48.8 6.50 5 46.8 7.22 15 53.2 5.36 25 44.6 8.96 6 7 8 9 10
49.8 46.8 44.2 50.8 48.4
8.76 6.72 8.53 11.95 6.19
16 17 18 19 20
52.4 50.6 56.0 50.2 44.0
9.48 3.44 7.00 7.60 8.46
26 27 28 29 30
46.8 6.50 49.2 3.19 45.6 7.96 57.6 14.38 51.4 6.80
Championship Card Company X-bar and S Chart Limits UCL U2SWL U1SL CL L1SL L2SWL LCL
X based on S
S
60.22 56.69 53.16 49.63 46.11 42.58 39.05
15.49 12.80 10.11 7.42 4.72 2.03 ------
Championship Card Company X Bar Chart for Sports Cards Centering Values 1
Limits Based on S
3.0SL=60.22
60
Sample Mean
2.0SL=56.69 1.0SL=53.16 50
X=49.63 - 1.0SL=46.11 - 2.0SL=42.58
40
- 3.0SL=39.05 0
10
20
30
Sample Number
Samples of 5 from each 1000 Cards Printed
Championship Card Company S Chart for Sports Card Centering Values 3.0SL=15.49
15
SampleStdev
2.0SL=12.80 1.0SL=10.11
10
S=7.416 5
- 1.0SL=4.724 - 2.0SL=2.032
0
- 3.0SL=0.000 0
10
20
30
Sample Number
5 Cards Sampled from each 1000 Cards Produced
Championship Card Company X-bar & S Chart Interpretation • Application of all eight PATs to the X-bar chart indicates a violation of PAT 1 (one pt. above the UCL) at sample 2. Judging from the remainder of the chart, the process was successfully adjusted. • Application of the first four PATs to the S chart indicates no violations. • In summary, the process appears to have been temporarily “out-of-control” w.r.t. its mean at sample 2. The process was successfully adjusted and may now be subjected to PDCA for permanent improvement purposes.
Common Questions for Investigating an Out-of-Control Process • Are there differences in the measurement accuracy of instruments / methods used? • Are there differences in the methods used by different personnel? • Is the process affected by the environment, e.g. temperature/humidity? • Has there been a significant change in the environment? • Is the process affected by predictable conditons such as tool wear? • Were any untrained personnel involved in the process at the time? • Has there been a change in the source for input to the process such as a new supplier or information? • Is the process affected by employee fatigue?
Common Questions for Investigating an Out-of-Control Process • Has there been a change in policies or procedures such as maintenance procedures? • Is the process frequently adjusted? • Did the samples come from different parts of the process? Shifts? Individuals? • Are employees afraid to report “bad news”?
Process Capability: The Control Chart Method for Variables Data 1.
Construct the control chart and remove all special causes. NOTE: special causes are “special” only in that they come and go, not because their impact is either “good” or “bad”. 2. Estimate the standard deviation. The approach used depends on whether a R or S chart is used to monitor process variability.
^
_ σ = R / d2
^ _
σ = S / c4
Several capability indices are provided on the following slide.
Process Capability Indices: Variables Data ^
^
CP = (engineering tolerance)/6σ = (USL – LSL) / 6σ This index is generally used to evaluate machine capability. tolerance to the engineering requirements. Assuming that the process is (approximately) normally distributed and that the process average is centered between the specifications, an index value of “1” is considered to represent a “minimally capable” process. HOWEVER … allowing for a drift, a minimum value of 1.33 is ordinarily sought … bigger is better. A true “Six Sigma” process that allows for a 1.5 σ shift will have Cp = 2.
Process Capability Indices: Variables Data ^
CR = 100*6σ / (Engineering Tolerance) =
^
100* 6σ /(USL –LSL)
This is called the “capability ration”. Effectively this is the reciprocal of Cp so that a value of less than 75% is generally needed and a Six Sigma process (with a 1.5σ shift) will lead to a CR of 50%.
Process Capability Indices: Variables Data ^
^
CM = (engineering tolerance)/8σ = (USL – LSL) / 8σ This index is generally used to evaluate machine capability. Note … this is only MACHINE capability and NOT the capability of the full process. Given that there will be additional sources of variation (tooling, fixtures, materials, etc.) CM uses an 8σ spread, rather than 6σ. For a machine to be used on a Six Sigma process, a 10σ spread would be used.
Process Capability Indices: Variables Data =
^
ZU = (USL – X) / σ
=
^
ZL = (X – LSL) / σ
Zmin = Minimum (ZL , ZU) Cpk = Zmin / 3 This index DOES take into account how well or how poorly centered a process is. A value of at least +1 is required with a value of at least +1.33 being preferred. Cp and Cpk are closely related. In some sense Cpk represents the current capability of the process whereas Cp represents the potential gain to be had from perfectly centering the process between specifications.
Process Capability: Example Assume that we have conducted a capability analysis using X-bar and R charts with subgroups of size n = 5. Also assume the process is in statistical control with an average of 0.99832 and an average range of 0.02205. A table of d2 values gives d2 = 2.326 (for n = 5). Suppose LSL = 0.9800 and USL = 1.0200
^
_ σ = R / d2 = 0.02205/2.326 = 0.00948 Cp = (1.0200 – 0.9800) / 6(.00948) = 0.703
CR = 100*(6*0.00948) / (1.0200 – 0.9800) = 142.2% CM = (1.0200 – 0.9800) / (8*(0.00948)) = 0.527 ZL = (.99832 - .98000)/(.00948) = 1.9 ZU = (1.02000 – .99832)/(.00948) = 2.3 so that Zmin = 1.9 Cpk = Zmin / 3 = 1.9 / 3 = 0.63
Process Capability: Interpretation Cp = 0.703 … since this is less than 1, the process is not regarded as being capable. CR = 142.2% implies that the “natural tolerance” consumes 142% of the specifications (not a good situation at all). CM = 0.527 = Being less than 1.33, this implies that – if we were dealing with a machine, that it would be incapable of meeting requirements. ZL = 1.9 … This should be at least +3 and this value indicates that approximately 2.9% of product will be undersized. ZU = 2.3 should be at least +3 and this value indicates that approximately 1.1% of product will be oversized. Cpk = 0.63 … since this is only slightly less that the value of Cp the indication is that there is little to be gained by centering and that the need is to reduce process variation.
S TATISTICAL PROCESS CONTROL End of Session DEPARTMENT
OF STATISTICS
REDGEMAN@UIDAHO.EDU
OFFICE: +1-208-885-4410
DR. RICK EDGEMAN, PROFESSOR & CHAIR – SIX SIGMA BLACK BELT