Lean Six Sigma Green Belt Supplement

Transcription

1 Problem Solving and Process Improvement Tools and Techniques Guide Book Lean Six Sigma Green Belt Supplement Max Zornada, University of Adelaide Executive Education 7 th Floor, 10 Pultney Street, Adelaide, South Australia, Tel mzornada@hmg.com.au

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3 Contents Control Charts... 4 The X Chart... 9 The X-bar and R Chart The i and mr Chart The np Chart The p Chart The c Chart The u Chart Interpreting Control Charts Process Capability Box Plots Force Field Analysis The Seven Quality Management and Planning Tools...59 The Affinity Diagram Relations Diagram The Systematic Diagram (Tree Diagram) The Matrix Diagram The Decision Matrix The Prioritisation Matrix The Allocation Matrix The Process Decision Program Chart The Network Diagram (Critical Path) Page 3

4 Control Charts What are Control Charts? A Control Chart is a graphic comparison of how a process is performing with how the process should perform. It is basically a run chart with some additional information added, in the form of lines representing control limits for the process being monitored. Control limits are statistically determined upper and lower boundaries that define the range of measurements that would be considered to be normal for a given process. A Control Chart enables you to distinguish between variation that is a natural part of a process, and that caused by something outside the process that could indicate a problem. The Control Chart Principle The principles on which control charts are based are discussed in more detail in Appendix 1 - Understanding Variation and Data. To briefly summarise some of the key points: Control charts use the knowledge of the standard deviation of data around its mean, which has been collected about a process, to determine what the natural limits of performance for that process are; The standard deviation and mean can be calculated for any set of data; For data which follows the Normal distribution, the range defined between the mean minus three times the standard deviation (called the lower control limit) and the mean plus three times the standard deviation (called the upper control limit), will enclose 99.73% of all possible data points that can be obtained from such a process; The upper and lower control limits will therefore indicate the natuaral limits of variation of a process. ie % of the time, the process will yield a data point within these limits; Although not all data follows the Normal distribition, most data collected from real processes are close to normally distributed, and where they are not, the sampling plan used to collect the data can yield normally distributed points. Page 4

5 When to use Control Charts Control Charts can be used to address two broad categories of problems: Process analysis involves understanding the nature of variation in processes by looking at the effect of different process factors, so as to identify potential problems; Process control involves monitoring a process for which has been standardised, for which control limits have been set, to determine if the standardisation was correct and if it is being maintained. Process control and process analysis charts are made in the same way, except their purposes are different. Types of Data and Types of Charts The form of Control Chart used will depend on the type of data that is to be plotted. Data will be of two basic types: things that you measure, or; things that you count. For things that you measure, such as measurements of some physical dimension (eg. length in millimetres), time (eg. waiting time in minutes), weight (eg. weight in kilograms) or other unit of measurement, data can take any value depending on its measure. Such data can take on what are known to as "indiscrete" values or variables. ie. they can be anything and they can vary from item to item. Measured or variable data is referred to as Continuous data. For things that you count, for example, the number of defective items in a sample, or the number of errors per form, the data can only take on certain or "discrete" values. This type of data is called Attributes data. For attributes data, their can be two types - categorical data and occurrence data. The first type of attributes data is where we decide that we only want to know whether an "attribute" is present or not. For example, we may be monitoring the number of claim forms that have been incorrectly filled out. Any error present on the form qualifies the form as Page 5

6 "incorrectly filled out" irrespective of how many errors have been made on any particular form. It is either correct or not correct. Other examples of this type of data is typically of the form OK or Not OK, Pass or Fail, True or False etc. This type of data is called Categorical Data. The other type of "attribute" data is where we can have a count of more than one attribute per unit examined. For example, we may be interested in how many errors were made per claim form, not just whether the form was correct or incorrect. In this case we count the number of errors per form and can end up with a data count that is greater than the number of items examined. eg. if we examine 10 forms and find that there are an average of 3 errors per form, we can end up with 30 total errors for 10 forms. Such data is referred to as Occurrence Data. Different Control Charts are used for each of these different types of data, and in the case of attributes data depending on whether the sample size, also referred to as subgroup size, can be kept constant from measurement to measurement. This is summarised in Table 5.8. Table 5.8. Summary of Control Chart Types. Lots of data available Limited data available Continous Data X and R chart i and mr chart Sample size constant Sample size not constant Categorical Data np chart (pn chart) p chart Occurrence Data c chart u chart How to Construct a Control Chart The following is a general procedure for constructing a Control chart. The specific steps to be used in constructing the various charts given in Table 5.8. will be presented in subsequent sections. 1. Select the process is to be monitored and determine what data will be collected for it. 2. Determine which type of chart is suitable. Once the data to be collected, or characteristic to be monitored has been established, determine which chart is applicable to that type of data. Page 6

7 Refer to Table 5.8. or refer to the decision tree given in Figure 5.6 to decide which chart is appropriate. 3. Decide how often data is to be collected. The more frequently the event you wish to monitor occurs, the more frequently you will need to take measurements. For example: Measure queue outpatient waiting times at 10:00am and 2:00pm, measure manufactured pin diameters every hour. etc. The time interval between successive measurements or data collection is referred to as the sampling interval. 4. Decide how many data points to record each time. That is, how many measurements will you take or how much data will you collect each time measurements are take or data collected. This is the sample or subgroup size. eg. measure waiting times for first five customers, measure first 10 pins manufactured each hour. 5. Collect the data. Implement you sampling/data collection plan. During each sampling interval or at the predetermined time, collect one sample of data values. 6. Calculate the statistics of interest. Use the prescribed statistical formulas for the particular chart you have chosen to calculate the position of the: Centre Line or Mean (CL) Upper Control Limit (UCL) Lower Control Limit (LCL) Page 7

8 Measuring Are you counting or measuring? Counting Continuous Data No Can you have more than one count per unit Yes Not much data available Lots of data available Categorical Data Occurence Data i and mr Charts X-bar ( X ) and R Charts Constant Sample Size np Chart Sample size not constant p Chart Constant Sample Size c Chart Sample size not constant u Chart Figure 5.6 Control chart selection logic tree 7. Draw the control chart This involves laying out the chart format, and then plotting the data points. To lay out the chart format: Draw and scale the vertical and horizontal axes; Draw the Centre Line, Upper and Lower Control Limits. Then plot the data values collected on the Control Chart 8. Analyse the Control Chart. In general, if all the points plotted fall within the control limits, the process can be said to be in control and operating normally. The existence of points outside the control limits or the existed or certain characteristic trends would suggest that the process is not in control and that special causes are present. Page 8

9 The X Chart The X-chart is the most basic of control charts. It provides a plot of all of the individual points measured. The X-chart provides a direct respresentation of the process. X Chart Example Consider a company which manufacturers and delivers product to meet specific customer orders. The days taken to fill the past 25 orders have been recorded and are given in the following table. Table 5.1 Number of days to fill customer orders Order Number Days to Deliver Order Number Days to Deliver Total Calculate the mean value _ Calculate the mean value, where the mean ( X ) is calculated as: _ X = Σ X n n Where: Σ X n = represents the sum of all the data points or values n = the number of values recorded Page 9

10 In our example: X = = = = Calculate the standard deviation The standard deviation, usually referred to by the symbol σ or the letters SD or S, is calculated using the equation: _ σ = Σ (X n - X) 2 n - 1 These symbols have been previously defined. In our example, this is calculated as follows: σ = (15-22) 2 + (18-22) 2 + (23-22) (18-22) 2 24 = = = 6.94 = Calculate the Upper and Lower Control Limits. The general formulas for the calculation of Upper and Lower Control Limits are: _ Upper Control Limit (UCL) = X + 3 σ _ Lower Control Limit (LCL) = X - 3 σ For example, substituting our value of standard deviation, we get: UCL = X 6.9 = = 42.7 LCL = = 1.3 Page 10

11 4. Draw the chart This is shown in Figure UCL = Mean X = LCL = Figure 5.7 X-Chart of Delivery Performance 5. Interpret the chart. The control chart suggests that the process is in statistical control, centred on a mean performance level of 22 days. The natural limits of variation for this process are between 1.3 and 42.7 days. Page 11

12 The X-bar and R Chart _ An X-bar and R chart is one that shows both the mean value X (called X bar) and the range, R. It is the most common type of control chart for continuous data. The X-bar portion of the chart mainly shows changes in the mean value of the process (the location), while the R portion shown any changes in the dispersion or spread of the process. X-bar and R Chart Example An industrial equipment manufacturer and retailer is contemplating launching a new marketing initiative using their superior delivery performance as a source of differentiation from their competitors. Among several options, they are considering giving customers a rock solid guarantee that their order will be delivered within a predetermined number of days and offering a substantial cash rebate if an order is delivered late. In order to determine what they can and cannot guarantee, they have decided to monitor the performance of their existing order fulfilment process. When collecting data for X-bar and R charts, the data needs to be collected in the form of subgroups or samples. Ideally, each sample should include at least 2 data items but no more than 10. The optimum is approximately 4 or 5. In this example, we shall be tracking the delivery performance of 5 randomly selected orders each day, for a period of 10 days. The results are shown in Table 5.9. Table 5.9 Delivery performance in days from receipt of order Day Order A Order B Order C Order D Order E Average (Sample) Page 12

13 With reference to Table 5.9, we have sampled 5 orders each time we have taken a sample ie. 5/day. The subgroup size, usually referred to as n, is therefore 5. (ie. 5 orders). Calculate the statistics of interest. _ 1. Determine the location of the centre-line for the X Chart. Calculate the average waiting time for the period being monitored. For the above example this is the sum of all the daily averages divided by 10 (10 days). = Average Mean X = = 10 Average Mean X = 4.4 Days _ The daily averages are referred to as X. The average of the averages which defines the centre-line is called X double bar. 2. Determine the centre-line for the Ranges chart. To determine the centreline of the ranges chart, we must first determine the range of each group of data collected. The range is the difference between the largest and smallest value in each sample. eg. for the orders sampled during day one, the fastest was filled in 2 days, the slowest in 11 days. The range is therefore 11 days - 2 days = 9 days. The ranges for each of the samples are given in Table Table 5.10 Ranges of order fulfillment samples. Day Range Page 13

14 The range is referred to as R. Calculate the average range for the period: _ Average Range R = _ 10 Average Range R = 5.3 Days This defines the centre-line for the range chart. _ 3. Calculate the Upper and Lower Control Limits for the X Chart. The formulas for the upper and lower control limits are as follows: = _ Upper Control Limit (UCL) = X + A 2 R = _ Lower Control Limit (LCL) = X - A 2 R Where the values of the coefficient A 2 may be obtained from Table 5.11 Table 5.11 Coefficients for X and R Charts _ X chart R chart LCL R chart UCL Sub-group size (n) A 2 D 3 D Page 14

15 For the example used here, we sampled 5 orders each day, therefore the subgroup size n = 5. From Table A 2 = Substituting into the above equations: UCL = x 5.3 = 7.5 LCL = x 5.3 = Calculate the Upper and Lower Control Limits for the R Chart. The equations for the upper and lower control limits for the range chart are: UCL = D 4 R and LCL = D 3 R From Table 5.11, D 4 = and D 3 = 0. The Upper and Lower Control Limits can therefore be calculated as: UCL = x 5.3 = 11.2 LCL = 0 5. Lay out the X-bar and R chart format. Draw the centre-lines, upper and lower control limits for the X chart on one graph and the centre-line, upper and lower control limits for the R chart on another. 6. Plot the values. Plot the daily averages (means) on the X chart and the values of the daily ranges on the R chart. See Figure 5.8.a. (X-bar chart) and 5.8.b. (R-chart). X Chart 10 8 UCL = 7.5 Days to Fill Order Mean X = 4.4 LCL = Sample Days Figure 5.8.a The X-bar Chart Page 15

16 14 R Chart 12 Days to Fill Order Sample Days Figure 5.8.b. The R Chart 7. Analyse the charts. In general, a normal control chart should have points evenly distributed between the control limits. This indicates that variation is occurring, but that it is within the normal limits for the process. Changes in the mean of the process will bring about changes in the X Chart. The R Chart will remain unchanged. Changes in the spread of a process will effect both the R Chart and the X Chart. Increases in spread will cause points on the R Chart to increase and points on the X Chart to show a greater spread and possibly go beyond the control limits. When interpreting X and R charts, the R chart should always be read first. The R chart shows process capability, it shows the degree of variation in the process and the degree of variation due to common causes. If the R chart is out of control, it is pointless to attempt to make any process adjustments on the basis of the X chart. If the R chart looks stable and the X chart is not, this suggests that the process is probably inherently stable, and that incorrect or unnecessary adjustment or some other factor is causing the X chart instability. Page 16

17 The i and mr Chart When there are few data points and/or data is only available at irregular intervals, we may not be able to collect enough points to construct X-bar and R-charts. For example: a process that runs in single batches with a long cycle time; situations where it is uneconomical to take multiple samples; where destructive testing is used to take measurements; etc., etc.; In such situations the i - individuals, and mr - moving range charts are used. These charts are similar to the X-bar and R-charts except that they use individual points. An i and mr Chart Example A mining company operates a process for extracting small amounts of previously unrecoverable precious metals for waste slurry. The slurry is stored in tanks as it is generated, every few days when enough slurry has been accumulated, the batch is processed. The number of grams recovered from each batch is recorded every time the recovery process is run. The results of the past 25 runs is shown in Table Page 17

18 Table 5.12 Precious Metals recoveries during last 25 runs Run Grams recovered Moving Range Total Page 18

19 1. Calculate the moving ranges. This has already been done, and is given in Table To calculate the moving range values, subtract the previous value of grams recovered from the current value. If the result is a negative number, record it as a positive number ie. the magnitude of the difference, ignoring the sign. The moving range value for the first number is recorded as a zero, because there is no previous number to subtract from it. 2. Calculate the mean of the moving ranges. The mean of the moving ranges is calculated using the formula: _ R = R 1 + R R n-1 n-1 Where, n is the number of values recorded. Note: the sum of the range values is divided by n-1 because the first one will always be zero and therefore be ignored. In our example, the average moving range is: _ R = = = Calculate the control limits for the mr-chart. As for the R-chart in the X-bar and R-chart pairing, the Upper and Lower Control Limits for the mr-chart are calculated using the same equations. Namely: _ LCL = D 3 R _ UCL = D 4 R Because our subgroup size will always be 2 for an mr chart, D 3 and our LCL will always be equal to zero. Page 19

20 Similarly, D 4 will always equal (ie. D 4 for n=2 is 3.267). Therefore: _ UCL = x R Which in our case is: UCL = x 5.7 = Draw the mr-chart 20 UCL = Mean R = LCL = Figure 5.9.a. The moving range chart (mr-chart) _ If the moving range chart is in control, the mean moving range R may be used to calculate the UCL and LCL for the i chart. If the chart is not in control, remove the values contributing to the outliers from the source data. 5. Calculate the value of the centre line for the i - chart The average of all the values defines the centre line for the i chart. This is calculated as : _ X = Σ X n n _ X = X 1 + X 2 + X X n n Page 20

21 Which in our example is: _ X = = = = Calculate the Upper and Lower Control Limits These are calculated using the equations used for the X-bar chart. That is: UCL = X + A 2 R LCL = X - A 2 R The value of A 2 is Therefore: UCL = X R, and LCL = X R In our example, X = 22 and R = 5.7. Substituting gives: UCL = x 5.7 = = = 37.2 LCL = = 6.84 = 6.9 Page 21

22 7. Draw the i - chart UCL = Mean X = LCL = Figure 5.9b. The completed individuals chart (i-chart) 8. Interpret the chart One point appears to be due to a special cause. However, given that this is above the control limit, indicating a higher than expected recovery rate, it warrants investigation to see if the process can be improved so that the mean is shifted to this higher level. Page 22

23 The np Chart. The np chart or Number Defective chart, also referred to as the pn chart is a chart that shows the number of defective items produced by a process. np charts are used with Categorical data, ie. attribute data where values are measured in terms of OK or Not OK. The np Chart is based on the assumption that count data, where we are counting OK/Not OK tend to follow the Binomial distribution, which approximates the Normal distribution. As subgroup size increases, the accuracy of this approximation increases. One restriction applying to the use of the np chart is that it requires sample or subgroup size to be constant. ie. each sample has the same number of items in it. np Chart Example An automatic packaging line which fills and seals bulk wine containers for export, traditionally has damaged a certain percentage of containers during the process. These are scrapped and the wine is recycled. The extent of the problem is going to be investigated using an np Chart. The first 200 containers filled each hour were examined and the results recorded for 25 consecutive production hours. The collected data is given in Table Page 23

24 Table 5.12 Defective wine container data Sample (Subgroup) Number Sample (Subgroup) size Number defective Total 5, The Subgroup size indicates how many containers were sampled each time a sample was taken. In our case, this was 200 and remained constant through out the data collection period. The number of defectives indicates how many of these were defective. Note: Subgroup size n should be greater than 50, and the expected mean value of defectives for each subgroup should ideally range from 3 to 4. To calculate the required statistics for an np chart we proceed as follows. Page 24

25 1. First, calculate the average number defective. This calculation is to work out the average number defectives found per sample, overall. In our case, we took 25 samples and found 610 defective items. The average number of defectives is calculated as: _ np = total number of defectives = 610 number of samples 25 _ np = Calculate the average fraction or proportion defective for all the data collected. Average fraction or proportion defective is calculated as follows: _ p = Total defective = Σnp Total inspected Σn for the data in this example, _ p = 610 = = 12.2% 5,000 This defines the centre-line of the control chart. Given that the sample size is constant, this could also have been calculated as: p = average number defective = np sample size n = = = 12.2% 3. Calculate the Upper and Lower Control Limits. The equations for calculating the Upper and Lower Control Limits for an np chart are: _ UCL = np + 3 x np(1-p) _ LCL = np - 3 x np(1-p) UCL = = 38.3 LCL = = 10.5 Page 25

26 4. Layout the control chart. Draw in the centre line (np) and the upper and lower control limits on your control chart proforma. Plot your data points on the control chart. The completed control chart is shown in Figure np chart Number defective UCL = 38.3 Mean np = LCL = Sample Hours Figure 5.10.a. The completed np chart Note, the np chart shows some points lying outside the control limits. These have been highlighted in Figure 5.10.b. Page 26

27 50 np chart Number defective UCL = 38.3 Mean np = LCL = Sample Hours Figure 5.10.b. np chart with special causes highlighted Data points outside the control limits indicate that there are factors present and impacting on the process that are not a normal part of the process. These factors are special causes. A process with numerous points outside the control limits - known as outliers, is said to be out of control, or not in statistical control. Investigating and resolving the specific causes of the outliers become priority projects for quality and process improvement teams. Page 27

28 The p Chart The p Chart or Proportion Defective Chart, is a chart that shows the fraction of defective items produced by a process. As for np charts, p charts are used with Categorical data, ie. attribute data where values are measured in terms of OK or Not OK. The p Chart is based on the assumption that count data, where we are counting OK/Not OK tend to follow the Binomial distribution, which approximates the Normal distribution. As subgroup size increases, the accuracy of this approximation increases. The key difference between the np chart and the p chart is that it does not require sample or subgroup size to be constant, although it can still be used in cases where sample size is constant. This gives it several advantages over the np chart for monitoring categorical data. In some cases it may not be possible to establish a constant sample size eg. 100% sampling of market driven process - the sample size will probably differ how we structure our sampling plan. In other cases, we may be using a sampling plan based on taking a statistically significant smaller sample of a high volume process. As is common in high volume production environments. In such cases, if production volumes increase substantially over time, we may find ourselves having to increase our sample size in order to still have a statistically valid subgroup size. If sample size changes, before and after np charts will not be directly comparable. p charts will remain directly comparable regardless of what changes are made to sample size over time. p Chart Example With reference to the automatic packaging line which was used as the basis of the np chart example, suppose we change our sampling so that we record all the containers filled per hour, for 25 consecutive production hours. What we will find, is the number of containers filled each hour varies depending a number of factors, such as equipment problems, material availability problems, operator availability etc., giving us a sample size that is no longer constant, but which reflects what was actually made. The new data is shown in Table Page 28

29 Page 29 Table 5.12 Defective wine container data Sample (Subgroup) Number Sample (Subgroup) size Number defective % Defect. p (%) UCL (%) LCL (%) Total 5,

30 To calculate the required statistics for a p chart we proceed as follows. 1. Calculate the average fraction or proportion defective for all the data collected. Average fraction or proportion defective is calculated as follows: _ p = Total defective = Σnp Total inspected Σn for the data in this example, _ p = 610 = = 10.3% 5,925 This defines the centre-line of the control chart. 2. Calculate the Upper and Lower Control Limits. The equations used to calculate the upper and lower control limits for a p chart are as follows. _ UCL = p + 3 x p(1-p) n _ LCL = p - 3 x p(1-p) n Using our previously calculated value for p = gives: UCL = x n LCL = x n A note on calculating the control limits for p charts. The control limit formulas use the sample size n, as part of the calculation. Because our sample size is different for each sample, we will need to calculate the UCL and LCL separately, for each individual point that is plotted. This has been done and is displayed in the right-most two columns on Table If the subgroup size is constant, then the UCL and LCL will also be constant. Figure 5.11 shows the p chart for the data in Table Page 30

31 25 Percent Defective Upper Control Limit Mean 5 0 Lower Control Limit Sample Number Figure 5.11 Completed p chart Page 31

32 The c Chart The c - chart or non-conformities chart, is a chart that shows the number of occurrences of some event ie. a non-conformity per sampling period or interval. The c-chart is used to monitor Occurrence Data. For the p and np charts (Categorical data) we know both the number of defectives as well as the number of remaining (nondefectives) units in the sample. This is not always possible. In many situations we may be able to measure the number of non-conformities or occurrences of some event only. For example, if a retailer were monitoring customer complaints (the non-conformities) it may be impractical to try to measure the total number of customers which come into the store, so as to determine the total sample size. ie. they can easily measure the number of complaints per day, but the total number of customers per day is not known. In such cases, a c-chart is used. A c-chart is used when the sampling period, interval or unit is constant. The sampling unit may be a fixed length, area, quantity, time etc. Examples of fixed sampling units are: Complaints per day; Scratches per car; Errors per form; etc. A key difference between Occurrence data and Categorical data, is that for categorical data, we cannot obtain a count of the parameter we are monitoring which is greater than the number of items in the sample. ie. if we sample 100 forms to monitor the number or percent defective (ie. incorrectly filled out) the most we could conceivably measure is 100 assuming they were all defective. For Occurrence data, we may obtain a measure which is greater than the number sampled. For example, if we a measuring the number of errors per form (Occurrence data), and there is an average of 3 errors per form, we will obtain a measurement of 300 which is significantly greater than the sample size. Page 32

33 c - Chart Example The method for constructing a c-chart will now be illustrated by way of an example. Consider a textile firm manufacturing plain white fabric which is later dyed to meet specific customer orders. Weaving errors and stains (usually oil) which often find there way into the fabric during the process cannot be allowed in the finished product which goes to dying. The dye highlights flaws and stains, causing such fabric to be rejected. To minimise the amount of finished product which is scrapped, the plain white fabric is inspected. Any oil or other stains are manually cleaned during inspection, any sections with weaving faults are highlighted to be cut out the main roll before dying. The rolls of plan white fabric are all prepared in 3,000 metre rolls. Information is collected by the fabric inspectors on the number of weaving faults found. This information is used by the weaving operators and mechanics to try to improve the weaving process. Data collected from 25 rolls is shown in Table Table 5.14 Number of weaving faults found per roll Roll Number Number of Weaving Faults Roll Number Number of Weaving Faults Total 610 Page 33

34 1. Calculate the average number of non-conformities Calculate the average number of weaving faults for the 25 rolls for which data has been collected. _ c = X 1 + X 2 + X X 25 number of samples = = = 24.4 This defines the centre line of the control chart. 2. Calculate the Upper and Lower Control Limits. For occurrence data, the standard deviation is calculated as the square root of the average. That is: _ Standard Deviation = c The Upper Control Limit is calculated as: UCL = c + 3 c = = x 4.94 = = 39.2 The Lower Control Limit is calculated as: LCL = c - 3 c = = = 9.6 Page 34

35 3. Plot the chart UCL = Centre line c = LCL = Figure The completed c-chart 4. Interpret the chart The above chart suggests that the upstream process which contributes to weaving faults in the fabric is not stable or in statistical control. There is evidence of several special causes - points outside the control limits. Improvement efforts should focus on identifying and eliminating the causes of instability in the weaving process. Page 35

36 The u Chart If we are monitoring occurrence data but the sampling interval/unit varies we can no longer use the c - chart, as this requires a constant sampling period/unit. For example, if a department store is monitoring the number of customer complaints per day and they have late night shopping on Friday and morning shopping on Saturday, the length of the day will not be the same for every day. A c-chart would not be used in such a situation. Where the data is occurrence data and the sampling unit/period is not constant a u-chart is used. The u-chart is to the c-chart what the p-chart is to the npchart. u - Chart Example The method for constructing a u-chart will now be illustrated with reference to the textiles example used for the c-chart. As part of the dying preparation process, the sections of fabric where significant weaving faults were found are cut out, and the remaining acceptable fabric sewn back into the roll. Rolls are also batched together or split into smaller rolls depending on the size of the customer order for a particular colour or pattern. This means that the rolls processed in the dying section will be of varying lengths depending on the customer order and the amount of flawed fabric that had to be cut out. Post-dying, the rolls are again inspected to identify any dying errors or faults. Sections with flaws or shade inconsistencies are cut out of the main roll and only fabric that meets the customer s quality specifications are batched into the finished order, ready to be fabricated into finished items. eg. garments, bedlinen, curtains etc. Because our sampling interval or unit (the roll) is no longer constant, we can no longer use a c-chart to monitor the dying process. A u-chart is now the appropriate chart to use. Data collected for 25 consecutive customer orders is given in the following table. Page 36

37 Table 5.15 Dying faults per customer order Order Number Roll length in Kilometres (n) Number of dying faults (c) Total Dying Faults per km (u=c/n) Page 37

38 1. Calculate u for each subgroup of data collected. Calculate u for each unit or subgroup of data using the relationship: u = c/n where n = size of the sampling unit or subgroup. For example, the first roll is 1.15 kilometres long and the number of faults is 15. Therefore n = 1.15 and c = 15. u is calculated as: u = c/n = 15/1.15 = 13 per kilometre. 2. Calculate the average nonconformities/unit (u). This can be calculated as follows: _ u = Σ c = 542 Σ n 66.2 = Calculate the Upper and Lower Control Limits Calculate the UCL and the LCL using the formula: UCL = u + 3 u n i LCL = u - 3 u n i Note, the UCL and LCL will need to be calculated for each roll, to reflect the changing sample size. For example, for the first roll, the roll length is 1.15 kilometres. Therefore, UCL and LCL will be: UCL = = = x 2.67 = = 16.2 Page 38

39 LCL = = = 0.19 = 0.2 The completed UCL and LCL calculations for all the rolls (data points) are given in Table Table 5.16 Dying faults per customer order, completed table Order Number Roll length in Kilometres (n) Number of dying faults (c) Dying Faults per km (u=c/n) UCL LCL Total Page 39

40 In the case of roll 18, the calculation for the LCL has given us a negative number. In practice we know that it is not possible to record a negative number of occurrences, therefore this would be recorded as a LCL of zero. 4. Draw the chart The completed chart is shown in Figure u = Figure The completed u chart 5. Interpret the chart. The chart suggests that the dying process is in control and centred around a mean of 8.2 occurrences of dying faults per kilometre of fabric dyed. Page 40

41 Interpreting Control Charts Several tests can be applied to Control Charts to assist with their interpretation. These are usually referred to as the "Shewhart tests". The foundation test is the presence of points beyond the control limits. The remaining tests are based on the fact that they represent events having the equivalent probability of occurring, in the absence of special causes, as a point beyond the control limits. The relevant tests are as follows: Points beyond the control limits Runs of points Trends Periodicity Hugging the centre-line Hugging the control limits Points beyond the control limits. According to the laws of probability, a point has a 3 in 1,000 chance of falling outside the control limits purely by chance, for a stable process. The presence of any point falling outside the control limits, should initially be taken as indicating the presence of a special cause, ie. an indication that the process is unstable, and investigated as such. If the subsequent investigation fails to find a special cause, it can be assumed that what was seen was one of the 3 in 1,000. Runs of Points. The following runs of points are an indication of an unstable process. Eight consecutive points fall on one side of the centre-line; Two consecutive points fall more than 2 standard deviations from the centre-line on the same side of the centre-line; Two out of three consecutive points fall more than 2 standard deviations from the centre-line on the same side of the centre-line; Four out of five consecutive points fall more than 1 standard deviation from the centre-line on the same side of the centre-line. Page 41

42 Trends. Any tendency of points to drift so as to give rise to a trend, is an indication that the process may be unstable. Periodicity. When points in a chart show a regular size and fall, the resulting pattern is referred to as periodicity. Periodicity is an indication that something in the process is changing in a regular pattern and therefore the process may be unstable. Hugging the centreline. If the data tends to hug the centreline, defined as 15 or more consecutive points less than 1 standard deviation from the centreline, an abnormality is indicated. This condition is usually as a result of data corruption, poor or inadequate sampling. Hugging the control limits. An abnormality is said to exist if 2 out of 3, 3 out of 7 or 4 out of 10 consecutive points fall between 2 and 3 standard deviations. Even if the points do not all fall on the same side of the centreline, this condition could be an indication that the process is unstable. The usual cause of this type of condition is unnecessary or overadjustment of the process. Shewharts original tests for control charts have been reproduced in Figure In these diagrams, three zones have been defined above and below the mean, these are: Zone C - one standard deviation either side of the mean; Zone B - between one and two standard deviations away from the mean, one each side of the mean; Zone A - between two and three standard deviations out from the mean. Page 42

43 Test 1. One point beyond zone A A B C C B A Test 3. Six points in a row steadily increasing or decreasing A B C C B A Test 5. Two out of three point in a row in zone A or beyond. A B C C B A Test 7. Fifteen point in a row in zone C (above & below the centreline) A B C C B A Test 2. Nine points in a row in Zone C or beyond A B C C B A Test 4. Fourteen points in a row alternating up and down A B C C B A Test 6. Four out of five points in a row in zone B or beyond. A B C C B A Test 8. Eight points in a row on both sides of centreline with none in zone C. A B C C B A Figure 5.14 The Shewhart tests Page 43

44 Process Capability Process capability is a measure of the ability of a process to meet or exceed the customer specifications for that process. Process capability is measured differently, depending on whether or not the process is centred on the mean of the customer specification. The term C p is used where the process is centred, C pk if it is not centred. For these measures to make any sense, the process must first be in statistical control and approximately normal. Calculating C p When the process is centred on the mean of the customer requirement, process capability is measured by C p. Where C p is defined as follows: C p = Specification width Process width In practice, the customer requirements are defined in terms of a range from the lower specification limit to the upper specification limit. LSL and USL respectively. The process width is the 6σ range defined between the mean plus three standard deviations to the mean minus three standard deviations. Process capability is therefore calculated by the formula: C p = USL - LSL 6 σ C p is interpreted as follows: A process with C p = 1 exactly matches the customer s specification limits; A process with C p > 1 exceeds the customer s specification; A process with C p < 1 fails to meet the customer s specification. Page 44

45 Calculating C pk If a process is not centred on the mean of the customer requirement, process capability is measured by C pk. Where C pk is defined as follows: C pk = Minimum of (USL - X, X - LSL ) 3σ 3σ _ Where X is the mean of the process. The C pk calculation overcomes the problem of the process not being centred by calculating the capability for each half of the process and then taking the minimum. Note: the upper one sided capability index is often referred to as C pu and the lower one sided capability index as C pl. ie. _ C pu = USL - X 3 σ _ C pl = X - LSL 3 σ Page 45

46 Box Plots What are Box Plots? A boxplot is a graphical way of displaying information about the spread and location of data. A boxplot focuses attention on certain features of the data without having to plot all the values. In particular, box plots are good at highlighting extreme points which are not typical of the rest of the sample and skewness in the data. Box plots provide a quick way of assessing data for which the team may be considering developing a control chart. When to use Box Plots Box plots can be used in the same types of situations where you would use a histogram or dot plot, to establish the location and spread of data. At the other end of the analytical scale, box plots may be used as an alternative to a control chart for one-off analytical exercises. One of the main advantages of box plots, are that we do not have to plot each individual data point. If we can come to some conclusion will a box plot, further and lengthier analysis will not be necessary. The other main advantage of a box plot, is that is can be used to determine whether or not there is a difference between two sets of data. This makes them extremely useful in situations where we seek to compare two or more sets of data. More so than histograms, dot plots and control charts - where apparent trends may be due to sampling errors or not statistically significant to indicate a real difference. Examples of this would include situations where we wish to establish whether data from to different sources are the same for the purpose of further analysis, or situations where we have implemented some improvements and we wish to establish whether the changes have made a difference or not by comparing before and after performance. Page 46

47 How to construct a Box Plot Unlike histograms and dot plots, box plots required us to have an understanding of some common terms and concepts associated with analysing and characterising data. In particular, we will need to know how to calculate: The median The mean The lower quartile The upper quartile These terms and others associated with analysing and characterising data are presented and discussed in Appendix 1: "Understanding Variation and Data". Refer to this Appendix if you need a refresher on these concepts. 1. Define the problem to be investigated Example: A bank decides to investigate customer queue waiting times. 2. Collect data. The average waiting times for a random sample of customers joining a queue in front of a teller position are collected at five randomly selected times during the day, for a week. The average waiting time is calculated at each point. The data is given in Table 6.1. Table 6.1 Customer Queuing Times in Minutes (Averages of samples taken 5 times per day) Mon Tues Wed Thurs Fri Sample Sample Sample Sample Sample Page 47

48 3. Rearrange the data in ascending order To calculate the median, upper and lower quartiles, requires the data to be in order from the smallest value to the largest (ascending order) irrespective of what order the values occurred in when the measurements were taken. When you have only a few data points, it is relatively easy to re-order the data by examining it and rewriting it in the appropriate order. When there are many values, it is some what more difficult to do this by inspection. A tally chart is a useful tool to use to assist with the reordering of the data. To develop your tally chart, use the following procedure. i) Determine the spread of the data. From Table 6.1, the shortest waiting time is 1 minute (minimum data value), the longest is 12 minutes (maximum data value). ii) Set the spread of your tally chart. Set the minimum and maximum values for your tally chart as equal to the minimum and maximum values of your data. From Table 6.1, we will need a tally chart spanning the range from 1 (the minimum data value) to 12 (the maximum data value). iii) Set the class boundaries. The class boundaries need to be set to each of the discrete data intervals that are naturally occurring in the data. iv) Fill in the Tally Chart from your data. Analyse the data given in Table 6.1 with the tally chart. The completed Tally chart for the data given in Table 6.1 is shown as Table 6.2. Table 6.2 Tally chart for data in Table 6.1 Waiting Time Number of Occurrences Number of Occurrences (Minutes) (Tally) 1 I 1 2 II 2 3 II 2 4 III 3 5 IIII 5 6 II 2 7 II 2 8 III 3 9 II 2 10 I 1 12 II 2 Total 25 Page 48

49 The table tells us how many times each measured value occurred in the collected data. Using this information we can rewrite the data collected in ascending order, as is shown in Table 6.3. This table shows that we have 25 data points, the lowest being 1 and the highest being 12. With reference to the tally chart (Table 6.2), we note that the value 5 minutes was recorded on 5 occasions during the week. This is shown in Table 6.3 as a block of 5 consecutive 5 s. Similarly, the tally chart shows that the value 8 minutes was observed on three occasions. This appears on Table 6.3 as a block of three 8 s. Table 6.3. Table 6.1 data, reordered in ascending order Position Measured Value Lower quartile (middle of the lower half) Median (the middle value) Upper quartile (middle of upper half) Page 49

50 4. Calculate the value required to draw the Box Plot. The key values we need to calculate in order to draw a box plot are the: Median; Upper Quartile; Lower Quartile; Interquartile Range; Mean. The mean, upper and lower quartiles can be determined from Table 6.3 by observations. i) Determine the Median The median is the middle value in the data. We have 25 data values, the middle value occupies the 13th position. ie. it has 12 values either side of it. The 13th data point, the median value is 5 minutes. ii) Determine the Lower Quartile The lower quartile is the middle value of the lower half of the data. The usual convention is to include the overall median as the upper limit of the bottom half of the data. The middle value of the lower half of the data is the 7th value. ie. It has 6 data points below it, 1 to 6 and 6 data points above it, 8 to 13. The data value corresponding to the 7th point is 4 minutes. Therefore, the lower quartile is 4 minutes. iii) Determine the Upper Quartile The upper quartile is the middle value of the upper half of the data. The usual convention is to include the overall median as the lower limit of the upper half of the data. From Table 6.3. we can see that the middle value of the top half of the data is the 19th data point which corresponds to the value of 8 minutes. Therefore, the upper quartile is 8 minutes. Page 50