Statistical Process Controlcourses.ieor.berkeley.edu/ieor130/2_Statistical Process...Introduction • Quality control is about controlling manufacturing or service operations such

Statistical Process ControlProf. Robert Leachman

IEOR 130Fall, 2020

Introduction

• Quality control is about controlling manufacturing or service operations such that the output of those operations conforms to specifications of acceptable quality.

• Statistical process control (SPC), also known as statistical quality control (SQC), dates back to the early 1930s and is primarily the invention of one man. The chief developer was Walter Shewhart, a scientist employed by Bell Telephone Laboratories.

• Control charts (discussed below) are sometimes termed Shewhart Control Charts in recognition of his contributions.

Introduction (cont.)

• W. Edwards Deming, the man credited with exporting statistical quality control methodology to the Japanese and popularizing it, was an assistant to Shewhart.

• Deming stressed in his teachings that understanding the statistical variation of manufacturing processes is a precursor to designing an effective quality control system. That is, one needs to quantitatively characterize process variation in order to know how to produce products that conform to specifications.

Introduction (cont.)

• Briefly, a control chart is a graphical method for detecting if the underlying distribution of variation of some measurable characteristic of the product seems to have undergone a shift. Such a shift likely reflects a subtle drift or change to the desired manufacturing process that needs to be corrected in order to maintain good quality output.

• Control charts are very practical and easy to use, yet they are grounded in rigorous statistical theory. In short, they are a fine example of excellent industrial engineering.

History

• Despite their great value, initial use of control charts in the late 1930s and early 1940s was mostly confined to Western Electric factories making telephony equipment. (Western Electric was a manufacturing subsidiary of AT&T.) Evidently, the notion of using formal statistics to manage manufacturing was too much to accept for many American manufacturing managers at the time.

• Following World War II, Japanese industry was decimated and in urgent need of rebuilding. It may be hard to imagine today, but in the 1950s, Japanese products had a low-quality reputation in America.

• W. Edwards Deming went to Japan in the 1950s, and his SPC teachings were quickly embraced by Japanese management.

History (cont.)• Through the 1960s, 1970s and 1980s, many Japanese-made products were

improved dramatically and eventually surpassed competing American-made products in terms of quality, cost and consumer favor.

• Many American industries lost substantial domestic market share or were driven completely out of business.

• This led to a “quality revolution” in US industries during the 1980s and 1990s featuring widespread implementation and acceptance of SPC and other quality management initiatives. Important additions were made to quality control theory and practice, especially Motorola’s Six Sigma controllability methodology (to be discussed).

• It is ironic that a brilliant American invention was not accepted by American industries until threatened by Japanese competition making good use of that invention.

Control Charts

• Control charts provide a simple graphical means of monitoring a process in real time. Today, they have gained wide acceptance in industry – you would be hard-pressed to find a volume manufacturing plant producing technologically advanced products anywhere in the world that is not using SPC extensively.

• A control chart maps the output of a production process over time and signals when a change in the probability distribution generating observations seems to have occurred. To construct a control chart one uses information about the probability distribution of process variation and fundamental results from probability theory.

Central Limit Theorem• Most types of control charts are based on the Central Limit Theorem of

statistics. Roughly speaking, the central limit theorem says that the distribution of a sum of independent and identically distributed (IID) random variables approaches the normal distribution as the number of terms in the sum increases.

• Generally, the distribution of the sum converges very quickly to a normal distribution. For example, consider a random variable with a uniform distribution on the interval (0, 1). See Figure 1.

Figure 1 Probability density of a uniform variate on (0, 1)

Central Limit Theorem• Now assume the three random variables X1, X2, and X3 are independent,

each of which has the uniform distribution on the interval (0, 1). Consider the random variable W = X1 + X2 + X3. If one plots the distribution of W, it tracks remarkably close to a normal distribution with the same mean and variance. See Figure 2. If we were to continue to add independent random variables, the agreement would be even closer.

Figure 2 Density of the sum of three uniform random variables

Central Limit Theorem (cont.)• If (X1, X2, … , Xn) is a random sample, the sample mean is defined as

• The Central Limit Theorem tells us that, regardless of the distribution of Xi, the sample mean will have a normal distribution, provided the variables are IID.

• Suppose that a random variable Z has the standard normal distribution (i.e., mean 0 and variance 1). Then, according to the table of the unit normal distribution (see Table A-1 in the SPC notes),

.11∑=

=n

iiX

nX

{ } .9974.033 =≤≤− ZP

Central Limit Theorem (cont.)• In other words, the likelihood of obtaining a value of Z either larger than 3

or less than -3 is 0.0026, or roughly 3 chances in 1,000. If such a value of Zis encountered, it is more likely that the IID assumption has been violated, i.e., there has been a drift or shift of the process that needs to be corrected.

• This is the basis of the so-called three-sigma limits that have become the de facto standard in SPC.

Back to Control Charts• Consider the average �𝑋𝑋 of n samples of a process variable. The Central

Limit Theorem tells us it should be (approximately) normally distributed. Suppose the mean of each sample is µ and the standard deviation is σ. The mean of �𝑋𝑋 is expressed as

• The variance of �𝑋𝑋 is derived as follows:

• Now

.)(111111111

µµµ ====

=

= ∑∑∑∑

====

nnn

EXn

XEn

Xn

EXEn

i

n

ii

n

ii

n

ii

.111

21

=

= ∑∑

==

n

ii

n

ii XVar

nX

nVarXVar

( ) ,2

1σnXVarnXVar i

n

ii ==∑

=

Control Charts (cont.)• Therefore,

𝑉𝑉𝑉𝑉𝑉𝑉 �𝑋𝑋 = 𝜎𝜎2

𝑛𝑛.

• Hence the standard deviation of �𝑋𝑋 is

• Therefore, the standardized variate

has (approximately) the normal distribution with mean zero and unit variance.

.nσ

n

XZσµ−

=

Control Charts (cont.)• It follows that

or equivalently,

• That is, the likelihood of observing a value of �𝑋𝑋 either larger than 𝜇𝜇 + 3𝜎𝜎𝑛𝑛

or

less than 𝜇𝜇 − 3𝜎𝜎𝑛𝑛

is 0.0026. Such an event is sufficiently rare that if it were to occur, it is more likely to have been caused by a shift in the population mean µ, than to have been the result of chance.

• This is the basis of the theory of control charts.

,9974.033 =

≤−

≤−

n

XPσµ

.9974.033=

+≤≤−n

Xn

P σµσµ

Control Charts (cont.)• A manufacturing process is said to be in statistical control if a stable system

of chance causes is operating. That is, the underlying probability distribution generating observations of the process variable is not changing with time.

• When the observed value of the sample mean of a group of observations falls outside the appropriate three-sigma limits, it is likely that there has been a change in the probability distribution generating observations. When an observed value falls outside these limits, it is customary to say the process is out-of-control, i.e., it is out of statistical control.

Control Charts for Continuous Variables• For a continuous variable describing the quality of the process, control

charts may be set up to track both the mean of the variable over fixed-size samples (the �𝑋𝑋-chart) and its range (maximum minus minimum) over fixed-size samples (the R-chart).

• An out-of-control signal on the �𝑋𝑋-chart indicates that the process mean has shifted; an out-of-control signal on the R-chart indicates that the process variance has changed. Either out-of-control signal should trigger a halt of the process.

• There should be an investigation to ascertain if and why the process is no longer in statistical control. (An alternative explanation is that the observations were not measured correctly.)

• The investigation culminates in corrective action to restore the process to statistical control, whereupon manufacturing is resumed. In this way, quality losses can be kept to a minimum.

�𝑋𝑋 and 𝑅𝑅 Charts• An �𝑋𝑋-chart requires that collection of data on the process variable be

broken down into subgroups of fixed size. The most common size of subgroups in industrial practice is n = 5. The subgroup size n ought to be at least four in order to have an accurate application of the Central Limit Theorem.

• To construct an �𝑋𝑋-chart, it is necessary to estimate the sample mean and the sample variance of the process variable. This could be done using standard statistical estimates from an initial population of N measurements (ideally, N much larger than n) of the variable:

�𝑋𝑋 =1𝑁𝑁�𝑖𝑖=1

𝑁𝑁

𝑋𝑋𝑖𝑖 , 𝑠𝑠2=1

𝑁𝑁 − 1�𝑖𝑖=1

𝑁𝑁

𝑋𝑋𝑖𝑖 − �𝑋𝑋 2 .

• However, it is not recommended that one use the sample standard deviation s as an estimator of σ when constructing an �𝑋𝑋-chart. For sto be an accurate estimator of σ, it is necessary that the underlying mean of the sample population be constant. Because the purpose of an �𝑋𝑋-chart is to determine whether a shift in the mean has occurred, we should not assume a priori that the mean is constant when estimating σ.

• This suggests that one should monitor the process variance and demonstrate that it is in statistical control, whereupon a reliable estimate for σ can be obtained.

�𝑋𝑋 and 𝑅𝑅 Charts

�𝑋𝑋 and 𝑅𝑅 Charts (cont.)• While process variance could be monitored by examining the sample

variances of the subgroup observations, an alternative method for estimating the sample variation that remains accurate when the mean shifts uses the data range. (The range of a sample is defined as the difference between the maximum and minimum value in the sample.)

• Even if the process mean shifts, the range will be stable as long as the process variation is stable.

• The ranges of the subgroups are much easier to compute than standard deviations and they provide equivalent information.

• The theory underlying the R-chart is that when the sample mean has a normal distribution, there is a fixed ratio between the range of the sample and the standard deviation of the sample. This ratio depends on the subgroup size.

�𝑋𝑋 and 𝑅𝑅 Charts (cont.)• Shewhart tabulated these ratios. If �𝑅𝑅 is the average of the ranges of many

subgroups of size n, then

where d2, which depends on n, is tabulated in Table A-5 in the notes.• The purpose of the R-chart is to determine if the process variation is stable.

Upper and lower control limits on the subgroup range may be established that correspond to 3-standard deviation variation in R. They are expressed as

LCL = 𝑑𝑑3 �𝑅𝑅 ,UCL = 𝑑𝑑4 �𝑅𝑅 .

The values of the constants d3 and d4 are tabulated in Table A-6 as a function of n. The values given for these constants assume three-sigma limits for the range of the process.

,ˆ2d

R=σ

�𝑋𝑋 and 𝑅𝑅 Charts (cont.)• Typically, the R-chart is set up first in order to demonstrate that the process

variance is stable and therefore obtain a reliable estimate of σ. • If the range is found to be in statistical control across a reasonably long series

of samples, then the formula

is used to estimate the standard deviation of the process variable for use in the �𝑋𝑋- chart.• Given reliable estimators for the mean and standard deviation of the process,

the �𝑋𝑋 control chart is then constructed in the following way: Lines are drawn for the upper and lower control limits at 𝜇𝜇 ± 3𝜎𝜎/ 𝑛𝑛. (Note that the “three-sigma” limits are drawn at 3 standard deviations of �𝑋𝑋, not at 3 standard deviations of the process variable X.)

,ˆ2d

R=σ

�𝑋𝑋 and 𝑅𝑅 Charts (cont.)• The mean of each observed subgroup is graphed on the �𝑋𝑋- chart and the

range of the group is graphed on the R - chart. The process is said to be out-of-control if an observation falls outside of the control limits on either chart.

• In early days, SPC charts were manually maintained. Nowadays, entry of measurements and maintenance of control charts typically is automated using computers linked to metrology equipment.

• An out-of-control signal triggers an “OCAP” (out-of-control action procedure). Typically, the first step of an OCAP is to check the metrology equipment to ensure it is properly calibrated and repeat the measurement. If the out-of-control reading is confirmed, this triggers further actions, e.g., placing the lot on hold, inhibiting the machine, e-mailing or paging the responsible engineer, performing inspections and tests of the machine.

Additional Control Rules• The three-sigma limits are not the only possible control rules. One could also

watch for other events that have a low probability of occurrence assuming the probability distribution is stationary and that samples are uncorrelated. For example, the probability of two samples in a row lying outside two standard deviations above or below the mean is given by

• This event has sufficiently low probability that, if it occurs, it is more likely due to a shift in the process mean than due to chance. Thus another control rule that could be used is to declare the process out of control if two measurements in a row occur outside of 𝜇𝜇 ± 2𝜎𝜎/ 𝑛𝑛.

{ } .0021.0)0228.0*2(}2{2or 2 222==>=−<+> ZPnXnXP σµσµ

Additional Control Rules (cont.)• As another example, consider the probability of eight measurements in a row

occurring on the same side of the mean. The probability of this event is(0.5)8 = 0.004 .

• Again, this event has sufficiently low probability that, if it occurs, it is more likely due to a shift in the process mean than due to chance. Thus another control rule that could be used is to declare the process out of control if eight measurements in a row occur on the same side of the process mean.

• Another kind of event suggesting a shift in the process mean is a series of strictly increasing or strictly decreasing samples.

• A popular set of control rules is known as the Western Electric control rules. These rules include the ones discussed above as well as several others. Except for the standard three-sigma control rules, all of these rules involve comparing a series of samples. The benefit of using these additional rules is that they may detect an out-of-control occurrence before the standard three-sigma rule does. The disadvantage is the additional computation and data storage required.

Control Charts for Binary Variables: The p-Chart

• �𝑋𝑋 and R charts are valuable tools for process control when the process result may be characterized using a single real variable. This is appropriate when there is a single quality dimension of interest such as length, width, thickness, resistivity, and so on.

• In two circumstances, continuous-variable control charts are not appropriate: (1) when one’s concern is whether an item has a particular attribute or set of attributes, and (2) there are too many different quality variables. In case (2) it might not be practical or cost-effective to maintain separate control charts for each variable. Either the item has the desired attributes or it does not.

• When using control charts for attributes, each sample value is either a 1 or a 0. A 1 means that the item is not acceptable, and a 0 means that it is.

The p-Chart (cont.)

• Let n be the size of the sampling subgroup, and define the random variable X as the total number of defectives in the subgroup. Because X counts the number of defectives in a fixed sample size, the underlying distribution of Xis binomial with parameters n and p.

• Interpret p as the proportion of defectives produced and n as the number of items sampled in each group. A p-chart is used to determine if there is a significant shift in the true value of p.

• Although one could construct p charts based on the exact binomial distribution, it is more common to use a normal approximation. Also, as our interest is in estimating the value of p, we track the random variable X / n, whose expectation is p, rather than X itself.

The p-Chart (cont.)• For a binomial distribution, we have

• For large n, the Central Limit Theorem tell us that X / n is approximately normally distributed with parameters µ = p and 𝜎𝜎 = 𝑝𝑝(1 − 𝑝𝑝)/𝑛𝑛. Using a normal approximation, the traditional three-sigma limits are

( ) ,/ pnXE =

./)1()/( nppnXVar −=

,)1(3n

pppUCL −+=

.)1(3,0Max

−

−=n

pppLCL

The p-Chart (cont.)• The estimate for p, the true proportion of defectives in the population, is �̅�𝑝,

the average fraction of defectives observed over some reasonable time period.

• The process is said to be in control as long as the observed fraction defective for each subgroup remains within the upper and lower control limits in a chart as above for which p is set to be �̅�𝑝.

• If n is too small to apply the Central Limit Theorem, then cumulative Binomial distribution probabilities must be used to establish appropriate control limits (i.e., control limits such that roughly 99% of observations from a stable distribution fall within the control limits). Cumulative Binomial probabilities are provided in Table A-2.

Attribute with Varying Subgroup Sizes: The Z-Chart• In some circumstances, the number of items produced per unit time may

be varying, making it impractical to use a fixed subgroup size. Suppose there is 100 percent inspection. We can modify the p-chart analysis to accommodate a varying subgroup size as follows. Consider the standardized variate Z:

• Z is approximately standard normal independent of n. The lower and upper control limits would be set at -3 and +3 respectively to obtain three-sigma limits, and the control chart would monitor successive values of the standardized variate Z. The resulting chart is sometimes called a “Z-chart.”

.)1(

npp

ppZ−−

=

Control Charts for Countable Defects: The c-Chart• In some cases, one is concerned with the number of defects of a particular

type in an item or a collection of items. An item is acceptable if the number of defects is not too large.

• The number of non-working pixels on a liquid crystal display is a good example: if only a few are not working, the eye will not be able to detect them and the display has good quality, but if too many are failed, the display will not be acceptable. (In fact, every flat-panel display includes some non-functional pixels.)

• Other examples are the number of knots in a board of lumber, the number of defects per yard of cloth, etc.

The c-Chart (cont.)• The c-chart is based on the observation that if the defects are occurring

completely at random, then the probability distribution of the number of defects per unit of production has the Poisson distribution. If c represents the true mean number of defects in a unit of production, then the likelihood that there are k defects in a unit is

• In using a control chart for the number of defects, the sample size must be the same at each inspection. One estimates the value of c from baseline data by computing the sample mean of the observed number of defects per unit of production.

....,2,1,0for !

} unit onein defects ofNumber { ===−

kkcekP

kc

The c-Chart (cont.)• When 𝑐𝑐 ≥ 20, the normal distribution provides a reasonable approximation

to the Poisson. Because the mean and the variance of the Poisson are both equal to c, it follows that for large c,

is approximately standard normal. Using traditional three-sigma limits, the upper and lower control limits for the c-chart are

• One develops and uses a c-chart in the same way as �𝑋𝑋, R and p charts.• In the case that c < 20, cumulative Poisson distribution probabilities must be

used to establish appropriate control limits (i.e., control limits such that roughly 99% of observations from a stable distribution fall within the control limits). Cumulative Poisson probabilities are provided in Table A-3.

ccXZ −

=

,3LCL cc −=

.3 UCL cc +=

Theoretical Design of an �𝑋𝑋-Chart• In practice, �𝑋𝑋 control charts are usually set up with standard 3-sigma limits

and a sample size of 5. But from an operations research perspective, one might wonder if these are the best values for the parameters.

• The choice of control limits and sample size impacts three different kinds of system costs: (1) the cost of the performing the inspections, (2) the cost of investigating the process when an out-of-control signal is received but the process is actually in statistical control (a so-called Type 1 error), and (3) the cost of operating the process out-of-control when no out-of-control signal has been received yet ( a so-called Type 2 error).

• We therefore may view the selection of parameters as a stochastic optimization problem, seeking to minimize expected system operating costs per unit time.

Theoretical Design of an �𝑋𝑋-Chart (cont.)• The trade-off with respect to control limits is clear: for tight control limits,

we will have more Type 1 errors but fewer Type 2 errors, whereas for loose control limits, we will have less Type 1 errors but more Type 2 errors.

• There also is a trade-off with respect to the sample size parameter: For large sample sizes, we will have a better approximation to the normal distribution and thus we would expect fewer Type 1 and Type 2 errors, but we would have to perform more inspection measurements.

• In general, stochastic optimization problems are difficult to solve, and this problem is no exception, even though it is somewhat simplified.

• The model here does not include the sampling interval as a decision variable. In many cases the sampling interval is determined from considerations other than cost. There may be convenient or natural times to sample based on the nature of the process, the items being produced, or personnel constraints.

Theoretical Design of an �𝑋𝑋-Chart (cont.)• The three costs we do consider are defined as follows:1. Sampling cost. We assume exactly n items are sampled each period. (A “period” in this context might refer to a duration such as a production shift or to a production run quantity such as a manufacturing lot.) In many cases, sampling consumes workers’ time, so that personnel costs are incurred. There also may be costs associated with the equipment required for sampling. In some cases, sampling may require destructive testing, adding the cost of the item itself. We will assume that for each item sampled, there is a cost of a1. It follows that the sampling cost incurred each period is a1n.

Theoretical Design of an �𝑋𝑋-Chart (cont.)2. Search cost. When an out-of-control condition is signaled, the presumption is that there is an assignable cause for the condition. The search for an assignable cause generally will require that the process be shut down. When an out-of-control signal occurs, there are two possibilities: either the process is truly out of control or the signal is a false alarm. In either case, we will assume that there is a cost a2 incurred each time a search is required for an assignable cause of the out-of-control condition. The search cost could include the costs of shutting down the facility, engineering time required to identify the cause of the signal, time required to determine if the out-of-control signal was a false alarm, and the cost of testing and possibly adjusting the equipment. Note that the search cost is probably a random variable; it might not be possible to predict the degree of effort required to search for an assignable cause of the out-of-control signal. When this is the case, interpret a2 as the expected search cost.

Theoretical Design of an �𝑋𝑋-Chart (cont.)

3. Operating out of control. The third and final cost we consider is the cost of operating the process after it has gone out of control.

There is a greater likelihood that defective items are produced if the process is out of control. If defectives are discovered during inspection, they would either be scrapped or reworked.

An even more serious consequence is that the defective item becomes part of a larger subassembly which must be disassembled or scrapped.

Finally, defective items can make their way into the marketplace, resulting in possible costs of warranty claims, liability suits, and overall customer dissatisfaction.

Assume that there is a cost a3 each period that the process is operated in an out-of-control condition.

Theoretical Design of an �𝑋𝑋-Chart (cont.)

Assume that the process mean is µ and that the process standard deviation is σ. A sufficient history of observations is assumed to exist so that µ and σcan be estimated accurately.

We also assume that an out-of-control condition means that the underlying mean undergoes a shift from µ to µ + δσ or to µ – δσ. Hence, out of control in this case means that the process mean shifts by δ standard deviations.Define a cycle as the time interval from the start of production just after an adjustment to detection and elimination of the next out-of-control condition. A cycle consists of two parts. Define T as the number of periods that the process remains in control directly following an adjustment and S as the number of periods the process remains out of control until a detection is made. A cycle is the sum T + S. Note that both T and S are random variables, so the length of each cycle is a random variable as well.

Theoretical Design of an �𝑋𝑋-Chart (cont.)The �𝑋𝑋-chart is assumed to be constructed using the following control limits:

Heretofore we assumed k = 3, but this may not be optimal. The goal of the analysis of this section is to determine the economically optimal values for both k and n.

The method of analysis is to determine an expression for the expected total cost incurred in one cycle and an expression for the expected length of each cycle. That is,

After determining an expression for the expected cost per unit time, we will find the optimal values of n and k that minimize this cost rate.

, UCLn

kσµ += . LCLn

kσµ −=

.}cycle ofLength {

}cycleper Cost {}unit timeper Cost {EEE =

Theoretical Design of an �𝑋𝑋-Chart (cont.)Assume that T, the number of periods that the system remains in control following an adjustment, is a discrete random variable having a geometric distribution. That is,

The geometric model arises as follows. Suppose that in a given period the process is in control. Then π is the conditional probability that the process will shift out of control in the next period. The geometric distribution is the discrete analog of the exponential distribution. Like the exponential distribution, the geometric distribution has the memoryless property. In applying the memoryless property, we are assuming that the production process exhibits no aging or decay. That is, the process is just as likely to shift out of control right after an assignable cause is found as it is many periods later. This assumption is reasonable when process shifts are due to random causes or when the process is recalibrated on an ongoing basis.

....,3,2,1,0for )1(}{ =−== ttTP tππ

Theoretical Design of an �𝑋𝑋-Chart (cont.)An out-of-control signal is indicated when

Let α denote the probability of a Type 1 error. (A Type 1 error occurs when an out-of-control signal is observed but the process actually is in statistical control.) We can express α as

or, equivalently, as

where Φ denotes the cumulative standard normal distribution function (tabulated in Table A-4).

.n

kX σµ >−

=>−= µσµα )(XEn

kXP

,)(2}{)( kkZPXEkn

XP −Φ=>=

=>−

= µσ

µα

Theoretical Design of an �𝑋𝑋-Chart (cont.)Let β denote the probability of a Type 2 error. (A Type 2 error occurs when the process is out of control, but this is not detected by the control chart.) Here we assume that an out-of-control condition means that the process mean has shifted to µ + δσ or to µ – δσ. Suppose that we condition on the event that the mean has shifted from µ to µ + δσ. The probability that the shift is not detected after observing a sample of n observations is

+=≤−= δσµσµβ )(XEn

kXP

+=≤−≤−= δσµσµσ )(XEn

kXn

kP

+=−≤

−−≤−−= δσµδ

σδσµδ )(XEnk

n

XnkP }{ nkZnkP δδ −≤≤−−=

.)()( nknk δδ −−Φ−−Φ=

Theoretical Design of an �𝑋𝑋-Chart (cont.)If we had conditioned on 𝐸𝐸 �𝑋𝑋 = 𝜇𝜇 − 𝛿𝛿𝜎𝜎, we would have obtained

By the symmetry of the normal distribution (specifically, that Φ(t) = 1 – Φ(-t) for any t), it is easy to show that these two expressions for β are the same.Consider the random variable T. We assumed that T is a geometric random variable taking values 0, 1, 2, … . Then

.)()( nknk δδβ +−Φ−+Φ=

∑∞

=

−=0

)1()(t

t tTE ππ ∑∞

=

−−−−−=0

1)1()1(t

tt πππ t

t)1()1(

0π

πππ −

∂∂

−−= ∑∞

=

∑∞

=

−∂∂

−−=0

)1()1(t

tππ

ππ

∂∂

−−=ππ

ππ 1)1( .1ππ−

=

Theoretical Design of an �𝑋𝑋-Chart (cont.)Now consider S. The random variable S is the number of periods that the process remains out of control after a shift occurs. The probability that a shift is not detected while the process is out of control is exactly β. It follows that S is also a geometric random variable, except that it assumes only the values 1, 2, 3, … . That is,

The expected value of S is given by. ... 3, 2, 1, for )1(}{ 1 =−== − ssSP sββ

s

ss

s sSE ββ

βββ ∑∑∞

=

∞

=

−

∂∂

−=−=11

1 )1()1()(

−

∂∂

−=∂∂

−= ∑∑∞

=

∞

=

1)1()1(01 s

s

s

s ββ

βββ

ββββ

β−

=

−

−∂∂

−=1

111

1)1(

Theoretical Design of an �𝑋𝑋-Chart (cont.)The expected length of a cycle is therefore

Now consider the expected sampling cost per cycle. In each period there are n items sampled. As there are, on average, E(C) periods per cycle, the expected sampling cost per cycle is therefore a1nE(C).

Now consider the expected search cost. The process is shut down each time an out-of-control signal is observed. One or more of these signals per cycle could be a false alarm.

Suppose there are exactly M false alarms in a cycle. The random variable Mhas a binomial distribution with probability of “success” (i.e., a false alarm) equal to α for a total of T trials. It follows that E(M) = αE(T).

.1

11)()()(βπ

π−

+−

=+= SETECE

Theoretical Design of an �𝑋𝑋-Chart (cont.)The expected number of searches per cycle is exactly 1 + E(M), as the final search is assumed to discover and correct the assignable cause. Hence the total search cost per cycle is

We also assume that there is a cost a3 for each period that the process is operated in an out-of-control condition. The process is out of control for exactly S periods. Hence, the expected out-of-control cost per cycle is a3 E(S) = a3 / (1-β).Collecting terms, the expected cost per cycle is

[ ] [ ] .)1(1)(1 22 ππαα −+=+ aTEa

[ ] .)1()1(1)( 321 βππα −+−++ aaCnEa

Theoretical Design of an �𝑋𝑋-Chart (cont.)Dividing by the expected length of a cycle, E(C), results in the following expression for the expected cost per unit time:

where α = 2Φ(-k) and 𝛽𝛽 = Φ 𝑘𝑘 − 𝛿𝛿 𝑛𝑛 −Φ(−𝑘𝑘 − 𝛿𝛿 𝑛𝑛), π is the probability of a real process shift in each period, and δ is the degree of such a shift (in units of σ). The optimization problem is to find the values for n and k that minimize G(n,k).

This is a difficult optimization problem because α and β require evaluation of the cumulative normal distribution function.

,

111

)1()1(1

),(

32

1

βππ

βππα

−+

−−

+

−+

+=

aa

naknG

Theoretical Design of an �𝑋𝑋-Chart (cont.)An approximation of the standard normal cumulative distribution function due to Herron is as follows:

This is accurate to within 0.5 percent for 0 < z < 3.

A solution strategy is to compute G(n,k) for various values of n and k over some practical range and then select the best values.

This operations research analysis is due to Baker (1971). As mentioned earlier, economic optimization of control charts is rare in industrial practice and the topic is primarily one of theoretical interest. More useful concepts for deciding sampling rates and the level of control effort are process capability and process performance indices, which we discuss next.

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