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Control Charts – Production and Operations Management
Control Tendency and Dispersion Variations in any process can be described in general in terms of two parameters:
1. Central tendency, and
2. Dispersion.
The former has to do with accuracy and the latter with the precision. To understand these two concepts, let us consider the following example.
Example Suppose I weigh 150 pounds (lb). There are two machines and they show the following readings for my weight:
Machine No. 1 Machine No. 2
140, 151, 159 lb 139, 139.5 140 lb
Machine No. 1 gives an average of 150 lb and therefore, this is an accurate machine. Its central tendency does not show deviation, but it has a lot of dispersion’. Machine No. 2 is precise, but it is not accurate because, its central tendency is 139.5 lb which is far removed from the actual weight, whereas, its dispersion is quite low, ranging from 139 to 140.
The above example shows that in controlling errors one has to control not only the central tendency, but also the dispersion. Both controls are necessary.
CONTROL CHARTS
We keep a check on both the above-mentioned aspects by a constant monitoring device; which is graphical; the graphs which are used for such monitoring are called control charts. Process control relies mostly on such graphical or visual representations, and monitoring thereby.
If we were to monitor the process by measuring the characteristics of output from the process, the continuous measurement will interfere considerably with the manufacturing activities. We need to measure, but intermittently.
Normal Distribution
This has also another advantage by virtue of what is known as the Central Limit Theorem in statistics. The theorem states that in means of samples tend to follow a simple statistical distribution, viz. Normal distribution.
Statistical Relationships for Sampling
Based on the sample-size, there is a definite relation between the standard deviation of the population and the standard deviation of the sample means.
X bar chart
Now to keep a watch on the ‘central tendency’ we have to fix limits which are called ‘control limits’ for the values of x. Any time the sample mean exceeds the control limits, we say that the process has gone out of control—that is, there are certain ‘assignable causes’ which should be looked into immediately. It may not be that every time the control limits are exceeded the process has really gone out of control. Therefore we would like to limit the number of times we look for assignable causes for the variations. Depending upon the precision that is involved, we would be setting up the control limits on the + and the – side of the mean of the sample means.
Usually + 3(7 limits are established. Since means of the samples are distributed normally, the 3(7 limits will mean that when the process is under control we will investigate into the assignable causes 3 out of 1000 times. Incidentally, when we look for the assignable cause when none exists, it is called the
Type 1 error; and when we are not looking for assignable causes, when these causes do exist, it is called Type 2 error. We would like to find a proper balance between the Type-1 and the Type-2 errors, and based on this particular balance we can fix the + and – control limits on the central tendency’ (mean of the sample means). As we said, in many cases, we fix these limits at ± 3 cr level. The control chart for the ‘central tendency’ is called the x-chart and is presented in Fig. 9.2.
R-Chart
The chart in Fig. 9.2 is for keeping a control on the ‘central tendency’. Now we look for the control for the ‘dispersion’. The standard deviation as well as the Range will give an indication of the Dispersion. The range is the difference between the maximum value and the minimum value of the observations in a sample. We use the sample’s range for controlling the ‘dispersion’ of the population as it is simpler to use.
If the narrowing down of the dispersion is a permanent feature, it should be incorporated by having a revised R-chart. Moreover, we should ask whether this desirable result is due to some uneconomical methods which might have been employed, or whether it is due simply to the inspection error. These aspects need to be investigated, and hence the need for having a lower control limit even in R-charts.
Stable Values
But, to construct the we need to have information regarding the stable values of the mean of the sample means and the mean of the sample ranges. Unless these charts are constructed from a process which is statistically under control, they cannot be used as reference charts for control.
Cusum Charts
The serve the purpose of locating or indicating the large sudden changes in the process, whereas, the Cusum Charts serve the purpose of noticing small step-by-step changes in the process under consideration. The theory of the Cusum Charts is as follows:
These S’s are the cumulative sum of the deviations of the sample means from the process mean (u). If the process is stable, then the positive and the negative deviations should almost cancel each other out and therefore a plot of the S’s versus the sample number will be an almost horizontal line; whereas, if there is a gradual change in the process, it will be noticed in either gradually increasing or decreasing values of S”s as you go along in time.
p-Charts or Fraction Defective Charts
Suppose we take about 25 samples. We find the average fraction defective, p, This is the central line, We assume normal distribution for the fraction defective data of these 25 samples. Note that though this is a typical case for the application of the Binomial distribution, we can approximate the distribution to a Normal distribution when np > 10. If this requirement is met, then based on the normal distribution assumption, we set the limits for the upper and lower control:
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