Use the data in Table 10E.2 to set up appropriate short-run x-bar and R charts,

Question

Use the data in Table 10E.2 to set up appropriate short-run x-bar and R charts, assuming that the standard deviations of the measured characteristic for each part type are not the same. The normal dimensions for each part are TA=100, TB=200, and TC=2000. Sample no. Part Type M1 M2 M3 M3 1 A 120 95 100 110 2 A 115 123 99 102 3 A 116 105 114 108 4 A 120 116 100 96 5 A 112 100 98 107 6 A 98 110 116 105 7 B 230 210 190 216 8 B 225 198 236 190 9 B 218 230 199 195 10 B 210 225 200 215 11 B 190 218 212 225 12 C 2150 2230 1900 1925 13 C 2200 2116 2000 1950 14 C 1900 2000 2115 1990 15 C 1968 2250 2160 2100 16 C 2500 2225 2475 2390 17 C 2000 1900 2230 1960 18 C 1960 1980 2100 2150 19 C 2320 2150 1900 1940 20 C 2162 1950 2050 2125

Explanation / Answer

Nominal chart, target chart. There are several different types of short run charts. The most basic are the nominal short run chart, and the target short run chart. In these charts, the measurements for each part are transformed by subtracting a part-specific constant. These constants can either be the nominal values for the respective parts (nominal short run chart), or they can be target values computed from the (historical) means for each part (Target X-bar and R chart). For example, the diameters of piston bores for different engine blocks produced in a factory can only be meaningfully compared (for determining the consistency of bore sizes) if the mean differences between bore diameters for different sized engines are first removed. The nominal or target short run chart makes such comparisons possible. Note that for the nominal or target chart it is assumed that the variability across parts is identical, so that control limits based on a common estimate of the process sigma are applicable. Standardized short run chart. If the variability of the process for different parts cannot be assumed to be identical, then a further transformation is necessary before the sample means for different parts can be plotted in the same chart. Specifically, in the standardized short run chart the plot points are further transformed by dividing the deviations of sample means from part means (or nominal or target values for parts) by part-specific constants that are proportional to the variability for the respective parts. For example, for the short run X-bar and R chart, the plot points (that are shown in the X-bar chart) are computed by first subtracting from each sample mean a part specific constant (e.g., the respective part mean, or nominal value for the respective part), and then dividing the difference by another constant, for example, by the average range for the respective chart. These transformations will result in comparable scales for the sample means for different parts. For attribute control charts (C, U, Np, or P charts), the estimate of the variability of the process (proportion, rate, etc.) is a function of the process average (average proportion, rate, etc.; for example, the standard deviation of a proportion p is equal to the square root of p*(1- p)/n). Hence, only standardized short run charts are available for attributes. For example, in the short run P chart, the plot points are computed by first subtracting from the respective sample p values the average part p's, and then dividing by the standard deviation of the average p's. When the samples plotted in the control chart are not of equal size, then the control limits around the center line (target specification) cannot be represented by a straight line. For example, to return to the formula Sigma/Square Root(n) presented earlier for computing control limits for the X-bar chart, it is obvious that unequal n's will lead to different control limits for different sample sizes. There are three ways of dealing with this situation. Average sample size. If you want to maintain the straight-line control limits (e.g., to make the chart easier to read and easier to use in presentations), then you can compute the average n per sample across all samples, and establish the control limits based on the average sample size. This procedure is not "exact," however, as long as the sample sizes are reasonably similar to each other, this procedure is quite adequate. Variable control limits. Alternatively, you may compute different control limits for each sample, based on the respective sample sizes. This procedure will lead to variable control limits, and result in step-chart like control lines in the plot. This procedure ensures that the correct control limits are computed for each sample. However, you lose the simplicity of straight-line control limits. Stabilized (normalized) chart. The best of two worlds (straight line control limits that are accurate) can be accomplished by standardizing the quantity to be controlled (mean, proportion, etc.) according to units of sigma. The control limits can then be expressed in straight lines, while the location of the sample points in the plot depend not only on the characteristic to be controlled, but also on the respective sample n's. The disadvantage of this procedure is that the values on the vertical (Y) axis in the control chart are in terms of sigma rather than the original units of measurement, and therefore, those numbers cannot be taken at face value.

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