Statistical Process Control 5 Part 6 pot

Figure 7.4 Median chart for herbicide batch moisture content ( ) Other types of control charts for variables 163 The advantage of using sample medians over sample means is that the former are very easy to find, particularly for odd sample sizes where the method of circling the individual item values on a chart is used. No arithmetic is involved. The main disadvantage, however, is that the median does not take account of the extent of the extreme values – the highest and lowest. Thus, the medians of the two samples below are identical, even though the spread of results is obviously different. The sample means take account of this difference and provide a better measure of the central tendency. Sample No. Item values Median Mean 1 134, 134, 135, 139, 143 135 137 2 120, 123, 135, 136, 136 135 130 This failure of the median to give weight to the extreme values can be an advantage in situations where ‘outliers’ – item measurements with unusually high or low values – are to be treated with suspicion. A technique similar to the median chart is the chart for mid-range. The middle of the range of a sample may be determined by calculating the average of the highest and lowest values. The mid-range ( ~ M) of the sample of five, 553, 555, 561, 554, 551, is: Highest Lowest  561 + 551 2 = 556 The central-line on the mid-range control chart is the median of the sample mid-ranges ~ M R . The estimate of process spread is again given by the median of sample ranges and the control chart limits are calculated in a similar fashion to those for the median chart. Hence, Action Lines at ~ M R ± A 4 ~ R Warning Lines at ~ M R ± 2/3 A 4 ~ R. Certain quality characteristics exhibit variation which derives from more than one source. For example, if cylindrical rods are being formed, their diameters may vary from piece to piece and along the length of each rod, due to taper. Alternatively, the variation in diameters may be due in part to the ovality within each rod. Such multiple variation may be represented on the multi-vari chart. 164 Other types of control charts for variables In the multi-vari chart, the specification tolerances are used as control limits. Sample sizes of three to five are commonly used and the results are plotted in the form of vertical lines joining the highest and lowest values in the sample, thereby representing the sample range. An example of such a chart used in the control of a heat treatment process is shown in Figure 7.5a. The Figure 7.5 Multi-vari charts Other types of control charts for variables 165 longer the lines, the more variation exists within the sample. The chart shows dramatically the effect of an adjustment, or elimination or reduction of one major cause of variation. The technique may be used to show within piece or batch, piece to piece, or batch to batch variation. Detection of trends or drift is also possible. Figure 7.5b illustrates all these applications in the measurement of piston diameters. The first part of the chart shows that the variation within each piston is very similar and relatively high. The middle section shows piece to piece variation to be high but a relatively small variation within each piston. The last section of the chart is clearly showing a trend of increasing diameter, with little variation within each piece. One application of the multi-vari chart in the mechanical engineering, automotive and process industries is for trouble-shooting of variation caused by the position of equipment or tooling used in the production of similar parts, for example a multi-spindle automatic lathe, parts fitted to the same mandrel, multi-impression moulds or dies, parts held in string-milling fixtures. Use of multi-vari charts for parts produced from particular, identifiable spindles or positions can lead to the detection of the cause of faulty components and parts. Figure 7.5c shows how this can be applied to the control of ovality on an eight-spindle automatic lathe. 7.4 Moving mean, moving range, and exponentially weighted moving average (EWMA) charts As we have seen in Chapter 6, assessing changes in the average value and the scatter of grouped results – reflections of the centring of the process and the spread – is often used to understand process variation due to common causes and detect special causes. This applies to all processes, including batch, continuous and commercial. When only one result is available at the conclusion of a batch process or when an isolated estimate is obtained of an important measure on an infrequent basis, however, one cannot simply ignore the result until more data are available with which to form a group. Equally it is impractical to contemplate taking, say, four samples instead of one and repeating the analysis several times in order to form a group – the costs of doing this would be prohibitive in many cases, and statistically this would be different to the grouping of less frequently available data. An important technique for handling data which are difficult or time- consuming to obtain and, therefore, not available in sufficient numbers to enable the use of conventional mean and range charts is the moving mean and moving range chart. In the chemical industry, for example, the nature of certain production processes and/or analytical methods entails long time 166 Other types of control charts for variables intervals between consecutive results. We have already seen in this chapter that plotting of individual results offers one method of control, but this may be relatively insensitive to changes in process average and changes in the spread of the process can be difficult to detect. On the other hand, waiting for several results in order to plot conventional mean and range charts may allow many tonnes of material to be produced outside specification before one point can be plotted. In a polymerization process, one of the important process control measures is the unreacted monomer. Individual results are usually obtained once every 24 hours, often with a delay for analysis of the samples. Typical data from such a process appear in Table 7.1. If the individual or run chart of these data (Figure 7.6) was being used alone for control during this period, the conclusions may include: Table 7.1 Data on per cent of unreacted monomer at an intermediate stage in a polymerization process Date Daily value Date Daily value April 1 0.29 25 0.16 2 0.18 26 0.22 3 0.16 27 0.23 28 0.18 4 0.24 29 0.33 5 0.21 30 0.21 6 0.22 May 1 0.19 7 0.18 8 0.22 2 0.21 9 0.15 3 0.19 10 0.19 4 0.15 5 0.18 11 0.21 6 0.25 12 0.19 7 0.19 13 0.22 8 0.15 14 0.20 15 0.25 9 0.23 16 0.31 10 0.16 17 0.21 11 0.13 12 0.17 18 0.05 13 0.18 19 0.23 14 0.17 20 0.23 15 0.22 21 0.25 22 0.16 16 0.15 23 0.35 17 0.14 24 0.26 Other types of control charts for variables 167 April 16 – warning and perhaps a repeat sample April 20 – action signal – do something April 23 – action signal – do something April 29 – warning and perhaps a repeat sample From about 30 April a gradual decline in the values is being observed. When using the individuals chart in this way, there is a danger that decisions may be based on the last result obtained. But it is not realistic to wait for another three days, or to wait for a repeat of the analysis three times and then group data in order to make a valid decision, based on the examination of a mean and range chart. The alternative of moving mean and moving range charts uses the data differently and is generally preferred for the following reasons: ᭹ By grouping data together, we will not be reacting to individual results and over-control is less likely. ᭹ In using the moving mean and range technique we shall be making more meaningful use of the latest piece of data – two plots, one each on two different charts telling us different things, will be made from each individual result. ᭹ There will be a calming effect on the process. The calculation of the moving means and moving ranges (n = 4) for the polymerization data is shown in Table 7.2. For each successive group of four, the earliest result is discarded and replaced by the latest. In this way it is Figure 7.6 Daily values of unreacted monomer 168 Other types of control charts for variables Table 7.2 Moving means and moving ranges for data in unreacted monomer (Table 7.1) Date Daily value 4-day moving total 4-day moving mean 4-day moving range Combination for conventional mean and range control charts April 1 0.29 2 0.18 3 0.16 4 0.24 0.87 0.218 0.13 A 5 0.21 0.79 0.198 0.08 B 6 0.22 0.83 0.208 0.08 C 7 0.18 0.85 0.213 0.06 D 8 0.22 0.83 0.208 0.04 A 9 0.15 0.77 0.193 0.07 B 10 0.19 0.74 0.185 0.07 C 11 0.21 0.77 0.193 0.07 D 12 0.19 0.74 0.185 0.06 A 13 0.22 0.81 0.203 0.03 B 14 0.20 0.82 0.205 0.03 C 15 0.25 0.86 0.215 0.06 D 16 0.31 0.98 0.245 0.11 A 17 0.21 0.97 0.243 0.11 B 18 0.05 0.82 0.205 0.26 C 19 0.23 0.80 0.200 0.26 D 20 0.23 0.72 0.180 0.18 A 21 0.25 0.76 0.190 0.20 B 22 0.16 0.87 0.218 0.09 C 23 0.35 0.99 0.248 0.19 D 24 0.26 1.02 0.255 0.19 A 25 0.16 0.93 0.233 0.19 B 26 0.22 0.99 0.248 0.19 C 27 0.23 0.87 0.218 0.10 D 28 0.18 0.79 0.198 0.07 A 29 0.33 0.96 0.240 0.15 B 30 0.21 0.95 0.238 0.15 C May 1 0.19 0.91 0.228 0.15 D 2 0.21 0.94 0.235 0.14 A 3 0.19 0.80 0.200 0.02 B 4 0.15 0.74 0.185 0.06 C 5 0.18 0.73 0.183 0.06 D 6 0.25 0.77 0.193 0.10 A 7 0.19 0.77 0.193 0.10 B 8 0.15 0.77 0.193 0.10 C 9 0.23 0.82 0.205 0.10 D 10 0.16 0.73 0.183 0.08 A 11 0.13 0.67 0.168 0.10 B 12 0.17 0.69 0.173 0.10 C 13 0.18 0.64 0.160 0.05 D 14 0.17 0.65 0.163 0.05 A 15 0.22 0.74 0.185 0.05 B 16 0.15 0.72 0.180 0.07 C 17 0.14 0.68 0.170 0.08 D Other types of control charts for variables 169 possible to obtain and plot a ‘mean’ and ‘range’ every time an individual result is obtained – in this case every 24 hours. These have been plotted on charts in Figure 7.7. The purist statistician would require that these points be plotted at the mid- point, thus the moving mean for the first four results should be placed on the chart at 2 April. In practice, however, the point is usually plotted at the last result time, in this case 4 April. In this way the moving average and moving range charts indicate the current situation, rather than being behind time. An earlier stage in controlling the polymerization process would have been to analyse the data available from an earlier period, say during February and March, to find the process mean and the mean range, and to establish the mean and range chart limits for the moving mean and range charts. The process was found to be in statistical control during February and March and capable of meeting the requirements of producing a product with less than 0.35 per cent monomer impurity. These observations had a process mean of 0.22 per cent and, with groups of n = 4, a mean range of 0.079 per cent. So the control chart limits, which are the same for both conventional and moving mean and range charts, would have been calculated before starting to plot the moving mean and range data onto charts. The calculations are shown below: Moving mean and mean chart limits n=4A 2 = 0.73 X = 0.22 from the results from table R = 0.079 · for February/March 2/3A 2 = 0.49 · (Appendix B) Figure 7.7 Four-day moving mean and moving range charts (unreacted monomer) 170 Other types of control charts for variables UAL = X +A 2 R = 0.22 + (0.73 ϫ 0.079) = 0.2777 UWL = X + 2/3 A 2 R = 0.22 + (0.49 ϫ 0.079) = 0.2587 LWL = X – 2/3 A 2 R = 0.22 – (0.49 ϫ 0.079) = 0.1813 LAL = X – A 2 R = 0.22 – (0.73 ϫ 0.079) = 0.1623 Moving range and range chart limits D 1 .001 = 2.57 from table (Appendix C) D 1 .025 = 1.93 · UAL = D 1 .001 R = 2.57 ϫ 0.079 = 0.2030 UWL = D 1 .025 R = 1.93 ϫ 0.079 = 0.1525 The moving mean chart has a smoothing effect on the results compared with the individual plot. This enables trends and changes to be observed more readily. The larger the sample size the greater the smoothing effect. So a sample size of six would smooth even more the curves of Figure 7.7. A disadvantage of increasing sample size, however, is the lag in following any trend – the greater the size of the grouping, the greater the lag. This is shown quite clearly in Figure 7.8 in which sales data have been plotted using moving means of three and nine individual results. With such data the technique may be used as an effective forecasting method. In the polymerization example one new piece of data becomes available each day and, if moving mean and moving range charts were being used, the result would be reviewed day by day. An examination of Figure 7.7 shows that: Figure 7.8 Sales figures and moving average charts . 142 150 150 149. 25 13 5. 38 21 1 46 1 56 148 160 152 .50 14 6. 61 22 152 147 158 154 152 . 75 11 4 .57 23 143 1 56 151 151 150 . 25 13 5. 38 24 151 152 157 149 152 . 25 8 3.40 25 154 140 157 151 150 .50 17. 19 8. 76 3 1 45 139 143 152 144. 75 13 5. 44 4 154 1 46 152 148 150 .00 8 3. 65 5 157 153 155 157 155 .50 4 1.91 6 157 150 1 45 147 149. 75 12 5. 25 7 149 144 137 155 1 46. 25 18 7 .63 8 141 147 149 155 148.00. 149 150 . 75 16 6.70 15 150 1 46 148 157 150 . 25 11 4.79 16 147 144 148 149 147.00 5 2. 16 17 155 150 153 148 151 .50 7 3.11 18 157 148 149 153 151 . 75 9 4.11 19 153 155 149 151 152 .00 6 2 .58 20 155 142