Statistical Process Control

Statistical Process Control

Overview Variation Control charts R charts X-bar charts P charts

Statistical Quality Control (SPC)
Measures performance of a process Primary tool - statistics Involves collecting, organizing, & interpreting data Used to: Control the process as products are produced Inspect samples of finished products Points which might be emphasized include: - Statistical process control measures the performance of a process, it does not help to identify a particular specimen produced as being “good” or “bad,” in or out of tolerance. - Statistical process control requires the collection and analysis of data - therefore it is not helpful when total production consists of a small number of units - While statistical process control can not help identify a “good” or “bad” unit, it can enable one to decide whether or not to accept an entire production lot. If a sample of a production lot contains more than a specified number of defective items, statistical process control can give us a basis for rejecting the entire lot. The issue of rejecting a lot which was actually good can be raised here, but is probably better left to later.

Bottling Company Machine automatically fills a 20 oz bottle.
Problem with filling too much? Problems with filling to little? So Monday the average is 20.2 ounces. Tuesday the average is 19.6 ounces. Is this normal? Do we need to be concerned? Wed is 19.4 ounces.

Natural Variation Machine can not fill every bottle exactly the same amount – close but not exactly.

Assignable variation A cause for part of the variation

SPC Objective: provide statistical signal when assignable causes of variation are present

Continuous Numerical Data Categorical or Discrete Numerical Data
Control Chart Types Continuous Numerical Data Categorical or Discrete Numerical Data Control Charts Variables Attributes Charts Charts This slide simply introduces the various types of control charts. R X P C Chart Chart Chart Chart

Measuring quality Attributes Variables
Characteristics that you measure, e.g., weight, length May be in whole or in fractional numbers Continuous random variables Characteristics for which you focus on defects Classify products as either ‘good’ or ‘bad’, or count # defects e.g., radio works or not Categorical or discrete random variables Once the categories are outlined, students may be asked to provide examples of items for which variable or attribute inspection might be appropriate. They might also be asked to provide examples of products for which both characteristics might be important at different stages of the production process.

Control Chart Purposes
Show changes in data pattern e.g., trends Make corrections before process is out of control Show causes of changes in data Assignable causes Data outside control limits or trend in data Natural causes Random variations around average

Figure S6.7

Steps to Follow When Using Control Charts
TO SET CONTROL CHART LIMITS Collect samples of n=4 or n=5 a stable process compute the mean of each sample. Calculate control limits Compute the overall means Calculate the upper and lower control limits.

Steps to Follow When Using Control Charts - continued
TO MONITOR PROCESS USING THE CONTROL CHARTS: Collect and graph data Graph the sample means and ranges on their respective control charts Determine whether they fall outside the acceptable limits. Investigate points or patterns that indicate the process is out of control. Assign causes for the variations. Collect additional samples and revalidate the control limits.

R Chart Monitors variability in process Variables control chart
Interval or ratio scaled numerical data Shows sample ranges over time Difference between smallest & largest values in inspection sample

R Chart Control Limits From Table S6.1 Sample Range at Time i
# Samples

West Allis Industries Control Charts for Variables
The management of West Allis Industries is concerned about the production of a special metal screw ordered by several of their largest customers. The diameter of the screw is critical. The next series of slides presents Example 7.1. The series builds in steps to the conclusion of the Example showing the development of key equations along the way. The first example is of variables charting using x-bar and R-charts. 1

Control Charts for Variables Special Metal Screw Sample Sample
Number 1 2 3 4 5 This slide and the next mimic the recording of the sample data for the five new observations of the screw diameter. 2

Control Charts for Variables Special Metal Screw
Sample Sample Number Should be at least 20 samples of size 4 to calculate the control limits. 3

Number R This slide and the next six show in a step-by-step fashion how the raw data is used to derive ranges and means for each observation. 4

Number R This slide adds the ‘post-it’ worksheet used for the actual calculations. This slide advances automatically. 5

Sample Sample Number R Here the range for the first observation is calculated. – = 6

Sample Sample Number R The range value is added to the original data list in the appropriate row. This slide advances automatically. – = 7

Sample Sample Number R – = = 9

Number R This slide removes the ‘post-it’ note and fills in all the range and mean values for the five observations. 10

Number R R = To complete the exercise, the R-bar and grand mean values are calculated and added at the bottom of the appropriate columns. 11

Control Charts for Variables UCLR = D4R LCLR = D3R
Control Charts – Special Metal Screw R-Charts R = UCLR = D4R LCLR = D3R The data set is replaced by a worksheet showing the given values for R-bar and the appropriate equations for the R-chart. The following segment will show x-double bar. 13

Control Charts for Variables Control Chart Factors
Factor for UCL Factor for Factor Size of and LCL for LCL for UCL for Sample x-Charts R-Charts R-Charts (n) (A2) (D3) (D4) It is necessary to select the values for D3 and D4 from Table 7.1 in the text. Part of the Table is reproduced here to demonstrate the proper use of the Table. 14

Control Charts for Variables Control Charts - Special Metal Screw
Control Chart Factors Factor for UCL Factor for Factor Size of and LCL for LCL for UCL for Sample x-Charts R-Charts R-Charts (n) (A2) (D3) (D4) Control Charts - Special Metal Screw R - Charts R = D4 = Here the correct sample size (not the number of samples, a mistake many students make) is highlighted along with the appropriate values of D3 and D4. 15

Control Charts—Special Metal Screw R-Charts R = D4 = 2.282 D3 = 0 UCLR = D4R LCLR = D3R Removing the Table, the values are now shown on the worksheet. 16

Control Charts—Special Metal Screw R-Charts R = D4 = 2.282 D3 = 0 UCLR = D4R LCLR = D3R In this slide the values are substituted into the UCL equation and the resulting value is shown. UCLR = (0.0021) = in. 17

Control Charts—Special Metal Screw R-Charts R = D4 = 2.282 D3 = 0 UCLR = D4R LCLR = D3R In a similar fashion, this slide shows the substitution and calculation for the LCL. The arrows will disappear after a few seconds leaving only the calculations. UCLR = (0.0021) = in. LCLR = 0 (0.0021) = 0 in. 18

Control Charts—Special Metal Screw R-Charts R = D4 = 2.282 D3 = 0 UCLR = D4R LCLR = D3R This slide can be used to review and discuss the calculations. UCLR = (0.0021) = in. LCLR = 0 (0.0021) = 0 in. 19

Range Chart - Special Metal Screw
This slide shows the blank control chart for the R-chart with the mean and control limit values already plotted. This is a screen shot from the OM Explorer software. The process Range is comfortably in control. (The picky person might suggest there is not enough variability in the process, but with so few observations it is difficult to draw such conclusions.) 20

X Chart Monitors process average Variables control chart
Interval or ratio scaled numerical data Shows sample means over time

X Chart Control Limits
From Table S6.1 Sample Range at Time i Sample Mean at Time i The following slide provides much of the data from Table S4.1. # Samples

Number 3

Control Charts for Variables Special Metal Screw _ Sample Sample
Number R x _ This slide and the next six show in a step-by-step fashion how the raw data is used to derive ranges and means for each observation. 4

Number R x _ This slide adds the ‘post-it’ worksheet used for the actual calculations. This slide advances automatically. 5

Control Charts for Variables Special Metal Screw _
Sample Sample Number R x _ In a similar fashion the mean value is calculated and added to the data set. The arrow will disappear in a few seconds leaving just the calculations. This is a good opportunity to review the steps of the calculation. This slide advances automatically. ( )/4 = 8

Control Charts for Variables Special Metal Screw _
Sample Sample Number R x _ ( )/4 = 9

Number R x _ This slide removes the ‘post-it’ note and fills in all the range and mean values for the five observations. 10

Number R x R = x = _ To complete the exercise, the R-bar and grand mean values are calculated and added at the bottom of the appropriate columns. = 11

Control Charts for Variables = UCLx = x + A2R LCLx = x - A2R
Control Charts—Special Metal Screw X-Charts R = x = = UCLx = x + A2R LCLx = x - A2R = We return to the calculations, this time for the x-bar control limits. Example 7.1 22

Control Charts for Variables UCLx = x + A2R LCLx = x - A2R
Control Chart Factors Factor for UCL Factor for Factor Size of and LCL for LCL for UCL for Sample x-Charts R-Charts R-Charts (n) (A2) (D3) (D4) Control Charts - Special Metal Screw x - Charts R = x = UCLx = x + A2R LCLx = x - A2R Here we select the proper value of A2 from Table 7.1 as before. Example 7.1 23

Control Charts—Special Metal Screw x- Charts R = A2 = 0.729 x = = UCLx = x + A2R LCLx = x - A2R = The A2 value is now shown with the other required values for the calculations. 24

Control Charts—Special Metal Screw x-Charts R = A2 = 0.729 x = = UCLx = x + A2R LCLx = x - A2R = Here the required values are substituted in the UCL equation and the value is calculated. The arrows on this slide will disappear after a few seconds and the second calculation will appear. UCLx = (0.0021) = in. Example 7.1 25

Control Charts—Special Metal Screw x-Charts R = A2 = 0.729 x = = UCLx = x + A2R LCLx = x - A2R = This slide completes the substitution and calculation for the LCL. UCLx = (0.0021) = in. LCLx = – (0.0021) = in. 26

x-Chart Special Metal Screw
Here the control chart for the x-bar chart is displayed with the mean and control limits in place and a plot of the same means. This is a screen shot from the OM Explorer software. This slide advances automatically. 27

And an observation outside the upper control limit is discovered. This indicates the probable presence of assignable causes of error in the process. 27

Investigate Cause The four steps of process improvement are shown on the note. 27

p Chart Shows % of nonconforming items Attributes control chart
Nominally scaled categorical data e.g., good-bad

p Chart Control Limits z = 2 for 95.5% limits; z = 3 for 99.7% limits
# Defective Items in Sample i Size of sample i

Hometown Bank HOMETOWN BANK The operations manager of the booking services department of Hometown Bank is concerned about the number of wrong customer account numbers recorded by Hometown personnel. Each week a random sample of 2,500 deposits is taken, and the number of incorrect account numbers is recorded. The records for the past 12 weeks are shown in the following table. Is the process out of control? Use 3-sigma control limits. Changing banks, the next set of slides goes though attribute sampling and charting at the Hometown Bank. This slide introduces the problem. This slide advances automatically. 1

UCLp = p + zp LCLp = p - zp Control Charts for Attributes
Sample Wrong Number Account Number 1 15 2 12 3 19 4 2 5 19 6 4 7 24 8 7 9 10 10 17 11 15 12 3 Total Hometown Bank n = 2500 UCLp = p + zp LCLp = p - zp p = p(1 - p)/n Total defectives Total observations p = This slide overlays the original data set for the account number errors and shows the sample size and the equation for the mean value. The format of the data set is slightly different to allow space for calculations. 3

Control Charts for Attributes UCLp = p + zp LCLp = p - zp
Sample Wrong Number Account Number 1 15 2 12 3 19 4 2 5 19 6 4 7 24 8 7 9 10 10 17 11 15 12 3 Total Hometown Bank n = 2500 UCLp = p + zp LCLp = p - zp p = p(1 - p)/n 147 12(2500) p = This slide substitutes in the appropriate values in the equation 4

Control Charts for Attributes UCLp = p + zp LCLp = p - zp
Sample Wrong Number Account Number 1 15 2 12 3 19 4 2 5 19 6 4 7 24 8 7 9 10 10 17 11 15 12 3 Total Hometown Bank n = 2500 UCLp = p + zp LCLp = p - zp p = p(1 - p)/n p = And this slide solves the equation for the mean. 5

Control Charts for Attributes UCLp = p + zp LCLp = p – zp
Hometown Bank n = p = UCLp = p + zp LCLp = p – zp p = p(1 – p)/n The mean value is now shown and the next slides will use these values to calculate the control limits. 7

Hometown Bank n = p = UCLp = p + zp LCLp = p – zp First the values are substituted in the equation for p. p = (1 – )/2500 8

Hometown Bank n = p = UCLp = p + zp LCLp = p – zp And the value for p is calculated. p = 9

Control Charts for Attributes UCLp = 0.0049 + 3(0.0014)
Hometown Bank n = p = UCLp = (0.0014) LCLp = – 3(0.0014) This value, along with the value for p-bar, is substituted into the control limit equations. p = 10

Control Charts for Attributes UCLp = 0.0049 + 3(0.0014)
Hometown Bank n = p = UCLp = (0.0014) LCLp = – 3(0.0014) This value, along with the value for p-bar, is substituted into the control limit equations. Why 3? 3-sigma limits Also to within 99.7% p = 10

Control Charts for Attributes UCLp = 0.0091 LCLp = 0.0007
Hometown Bank n = p = UCLp = LCLp = Finally the values for the control limits are calculated. p = 11

p-Chart Wrong Account Numbers
This slide shows the control chart for the example with the control limits and mean values in place. The sample proportions are added to the chart in the standard fashion. Note the suspiciously large variability, not to mention the out-of-control point for sample 7. This slide advances automatically. 12

p-Chart Wrong Account Numbers
Here the point for sample 7 is highlighted. 12

p-Chart Wrong Account Numbers Investigate Cause
This slide once again overlays the summary note for the problem solving steps when the situation is an undesirable condition. 12

Figure S6.7

Which control chart is appropriate?
Webster Chemical Company produces mastics and caulking for the construction industry. The product is blended in large mixers and then pumped into tubes and capped. Webster is concerned whether the filling process for tubes of caulking is in statistical control. The process should be centered on 8 ounces per tube. Several samples of eight tubes are taken and each tube is weighed in ounces.

Webster Chemical Company produces mastics and caulking for the construction industry. The product is blended in large mixers and then pumped into tubes and capped. Webster is concerned whether the filling process for tubes of caulking is in statistical control. The process should be centered on 8 ounces per tube. Several samples of eight tubes are taken and each tube is weighed in ounces. X-bar and R charts

A sticky scale brings Webster’s attention to whether caulking tubes are being properly capped. If a significant proportion of the tubes aren’t being sealed, Webster is placing their customers in a messy situation. Tubes are packaged in large boxes of 144. Several boxes are inspected. The number of leaking tubes in each box is recorded.

A sticky scale brings Webster’s attention to whether caulking tubes are being properly capped. If a significant proportion of the tubes aren’t being sealed, Webster is placing their customers in a messy situation. Tubes are packaged in large boxes of 144. Several boxes are inspected. The number of leaking tubes in each box is recorded. P charts

c Chart Type of attributes control chart
Discrete quantitative data Shows number of nonconformities (defects) in a unit Unit may be chair, steel sheet, car etc. Size of unit must be constant

c Chart Control Limits Use 2 for 95.5% limits Use 3 for 99.7% limits
# Defects in Unit i # Units Sampled

Woodland Paper Company
The Woodland Paper Company produces paper for the newspaper industry. As a final step in the process, the paper passes through a machine that measures various quality characteristics. This slide introduces the c-chart (Example 7.4) which is valuable since it illustrates an out-of-control condition that is a desirable situation. This slide advances automatically. 16

When the process is in control, it averages 20 defects per roll. Setup the control chart for number of defects. Five rolls have 16, 21, 17, 22, and 24 defects. A sixth roll from a different supplier has 5 defects. Are the two suppliers equivalent? (use 2 sigma limits) This slide introduces the c-chart (Example 7.4) which is valuable since it illustrates an out-of-control condition that is a desirable situation. This slide advances automatically. 16

Control Charts for Attributes Woodland Paper Company c = z = 2 UCLc = c + z c LCLc = c – z c This slide shows the equations for the c-chart and the necessary data from historical records. 16

Control Charts for Attributes Woodland Paper Company c = z = 2 UCLc = LCLc = 20 – The data are substituted in the equations. 16

Control Charts for Attributes Woodland Paper Company c = z = 2 UCLc = 28.94 LCLc = 11.06 And the values for the UCL and LCL are calculated. 16

Control Charts for Attributes Woodland Paper Company This slide presents the complete control chart with the control limits and the six new observations plotted. Notice that Sample 6, the one using pulp from the different supplier, is below the lower control limit. This slide advances automatically. 16

Control Charts for Attributes Woodland Paper Company This slide presents the complete control chart with the six new observations plotted. Notice that Sample 6, the one using pulp from the different supplier, is below the lower control limit. 16

Control Charts for Attributes Investigate Cause When the assignable causes result in an improvement to the process, the objective should be to exploit the improvement and incorporate it into the process. 16

At Webster Chemical, lumps in the caulking compound could cause difficulties in dispensing a smooth bead from the tube. Even when the process is in control, there will still be an average of four lumps per tube of caulk. Testing for the presence of lumps destroys the product, so Webster takes random samples.

At Webster Chemical, lumps in the caulking compound could cause difficulties in dispensing a smooth bead from the tube. Even when the process is in control, there will still be an average of four lumps per tube of caulk. Testing for the presence of lumps destroys the product, so Webster takes random samples. C charts

What Is Acceptance Sampling?
Form of quality testing used for incoming materials or finished goods e.g., purchased material & components Procedure Take one or more samples at random from a lot (shipment) of items Inspect each of the items in the sample Decide whether to reject the whole lot based on the inspection results Here again it is useful to stress that acceptance sampling relates to the aggregate, not the individual unit. You might also discuss the decision as to whether one should take only a single sample, or whether multiple samples are required.

What Is an Acceptance Plan?
Set of procedures for inspecting incoming materials or finished goods Identifies Type of sample Sample size (n) Criteria (c) used to reject or accept a lot Producer (supplier) & consumer (buyer) must negotiate

Operating Characteristics Curve
Shows how well a sampling plan discriminates between good & bad lots (shipments) Shows the relationship between the probability of accepting a lot & its quality You can use this and the next several slides to begin a discussion of the “quality” of the acceptance sampling plans. You will find additional slides on “consumer’s” and “producer’s” risk to pursue the issue in a more formal manner in subsequent slides.

OC Curve 100% Inspection P(Accept Whole Shipment) 100% 0% 1 2 3 4 5 6
Keep whole shipment Return whole shipment 0% 1 2 3 4 5 6 7 8 9 10 Cut-Off % Defective in Lot

OC Curve with Less than 100% Sampling
P(Accept Whole Shipment) 100% 0% % Defective in Lot Cut-Off 1 2 3 4 5 6 7 8 9 10 Return whole shipment Keep whole shipment Probability is not 100%: Risk of keeping bad shipment or returning good one.

AQL & LTPD Acceptable quality level (AQL)
Quality level of a good lot Producer (supplier) does not want lots with fewer defects than AQL rejected Lot tolerance percent defective (LTPD) Quality level of a bad lot Consumer (buyer) does not want lots with more defects than LTPD accepted Once the students understand the definition of these terms, have them consider how one would go about choosing values for AQL and LTPD.

Producer’s & Consumer’s Risk
Producer's risk () Probability of rejecting a good lot Probability of rejecting a lot when fraction defective is AQL Consumer's risk (ß) Probability of accepting a bad lot Probability of accepting a lot when fraction defective is LTPD This slide introduces the concept of “producer’s” risk and “consumer’s” risk. The following slide explores these concepts graphically.

An Operating Characteristic (OC) Curve Showing Risks
 = 0.05 producer’s risk for AQL = 0.10 Consumer’s risk for LTPD Probability of Acceptance Percent Defective Bad lots Indifference zone Good lots LTPD AQL 100 95 75 50 25 10

OC Curves for Different Sampling Plans
1 2 3 4 5 6 7 8 9 10 % Defective in Lot P(Accept Whole Shipment) 100% 0% LTPD AQL n = 50, c = 1 n = 100, c = 2 This slide presents the OC curve for two possible acceptance sampling plans.

Average Outgoing Quality
Where: Pd = true percent defective of the lot Pa = probability of accepting the lot N = number of items in the lot n = number of items in the sample It is probably important to stress that AOQ is the average percent defective, not the average percent acceptable.

Developing a Sample Plan
Negotiate between producer (supplier) and consumer (buyer) Both parties attempt to minimize risk Affects sample size & cut-off criterion Methods MIL-STD-105D Tables Dodge-Romig Tables Statistical Formulas

Statistical Process Control

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Statistical Process Control

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