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256 Part IV: Quality Assuranceselection of a subgroup or sample is perhaps the most important item in setting upa control chart. The next question in the selection of a sample is to determine thesample size. Factors that are usually taken into consideration for determiningthe sample size and the frequency of the samples are the average run length (ARL)and the operating characteristic curve, which is also known as the OC curve.AVERAGE RUN LENGTHA run is a number of successive items possessing the same characteristics. Forexample, number of successive conforming or successive nonconforming forms arun. An average run length (ARL) is the average number of points plotted, that is,the number of subgroups inspected before a point falls outside the control limits,indicating that the process is out of control.In Shewhart control charts the ARL can be determined by using the formulaARL = 1 p(19.1)where p is the probability that any point falls outside the control limits. It is quitecommon to use ARL as a benchmark to check the performance of a control chart.As an illustration, consider a process quality characteristic that is normallydistributed. Then for an X – control chart with three-sigma control limits, the probabilitythat a point will fall outside the control limits when the process is stable isp = 0.0027, which is the probability that a normal random variable deviates fromthe mean m by at least three s. Thus, the average run length for the X – control chartwhen the process is stable isPart IV.B1ARL 0= =370.0.0027In other words, when the process is stable we should expect that, on the average,one out-of-control signal or false alarm will occur once in every 370 samples. TheARL can also be used to determine how often a false alarm will occur, simply bymultiplying the ARL 0 by the time t between the samples. Thus, for example, ifsamples are taken every 30 minutes, then a false alarm will occur on the averageonce every 185 hours. On the other hand, ARL can be used in the same manner tofind how long it will take before a given shift in the process mean is detected. Weillustrate this concept with Example 19.2 using an X – control chart.In practice, the decision of how large and how frequently the samples shouldbe taken is based on the cost of taking samples and how quickly we would like todetect the shift. Large samples taken more frequently would certainly give betterprotection against shifts, since it will take less time to detect any given shift.Thus, for instance, in Example 19.2, if the samples are taken every half an hourinstead of every hour then it will take only one hour instead of taking two hoursto detect the shift of 1.5s. Similarly, it can easily be shown that if larger samplesare taken, the shifts can be detected more quickly. This means if large samples aretaken more frequently, shifts in the process mean will be detected faster, and theprocess will be producing fewer nonconforming units. Thus, when calculating

Chapter 19: B. Statistical Process Control 257EXAMPLE 19.2Suppose a process quality characteristic that is normally distributed is plotted in aShewhart X – control chart with three-sigma control limits. Suppose that the processmean m 0 experiences an upward shift of 1.5s. Determine how long, on the average, itwill take to detect this shift if samples of size four are taken every hour.Solution:Since the process mean has experienced an upward shift of 1.5s the new process meanwill be m 0 + 1.5s. Furthermore, since the sample size is four, the upper control limit inthis case is alsosm0 + 3s x= m0 + 3 = m0+ 15 . s.4In other words, the center line of the control chart will coincide with the upper controllimit. Thus, the probability p that a point will fall beyond the control limits isp= P( z 6)+ P( z 0) 0. 00000+ 0. 5 = 0. 5.Therefore, the ARL is given by1ARL = =05 .2.Thus, it will take, on the average, two hours to detect a shift of 1.5s in the processmean.the cost of taking samples one must take into account how much money willbe saved by detecting shifts more quickly and consequently producing fewernonconforming units.OPERATING CHARACTERISTIC CURVEThe operating characteristic curve is a graph characterizing the relationship betweenthe probabilities of type II error (b ) and the process shifts. The operating characteristiccurves are usually known as OC curves.A set of OC curves are shown in Figure 19.7 for the x – chart with three-sigmalimits, for different sample sizes n. The units of scale on the horizontal axis are inprocess standard deviation s.By carefully observing the OC curves in Figure 19.7, we see that:1. For a given sample size n and a, where a is the probability of apoint exceeding the control limits when the process is stable, a largershift corresponds to a smaller probability, b.2. With a larger sample size there is a smaller probability (b ) for a givenprocess shift.For a detailed discussion of the construction of an operating characteristic curve,refer to Gupta and Walker (2007a).Part IV.B

Chapter 19: B. Statistical Process Control 257

EXAMPLE 19.2

Suppose a process quality characteristic that is normally distributed is plotted in a

Shewhart X – control chart with three-sigma control limits. Suppose that the process

mean m 0 experiences an upward shift of 1.5s. Determine how long, on the average, it

will take to detect this shift if samples of size four are taken every hour.

Solution:

Since the process mean has experienced an upward shift of 1.5s the new process mean

will be m 0 + 1.5s. Furthermore, since the sample size is four, the upper control limit in

this case is also

s

m0 + 3s x

= m0 + 3 = m0

+ 15 . s.

4

In other words, the center line of the control chart will coincide with the upper control

limit. Thus, the probability p that a point will fall beyond the control limits is

p= P( z 6)+ P( z 0)

0. 00000+ 0. 5 = 0. 5.

Therefore, the ARL is given by

1

ARL = =

05 .

2.

Thus, it will take, on the average, two hours to detect a shift of 1.5s in the process

mean.

the cost of taking samples one must take into account how much money will

be saved by detecting shifts more quickly and consequently producing fewer

nonconforming units.

OPERATING CHARACTERISTIC CURVE

The operating characteristic curve is a graph characterizing the relationship between

the probabilities of type II error (b ) and the process shifts. The operating characteristic

curves are usually known as OC curves.

A set of OC curves are shown in Figure 19.7 for the x – chart with three-sigma

limits, for different sample sizes n. The units of scale on the horizontal axis are in

process standard deviation s.

By carefully observing the OC curves in Figure 19.7, we see that:

1. For a given sample size n and a, where a is the probability of a

point exceeding the control limits when the process is stable, a larger

shift corresponds to a smaller probability, b.

2. With a larger sample size there is a smaller probability (b ) for a given

process shift.

For a detailed discussion of the construction of an operating characteristic curve,

refer to Gupta and Walker (2007a).

Part IV.B

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