Manufacturing Handbook of Best Practices 2011 Part 15 docx

For the individual-X chart shown in Figure 15.1, the control limits are calculated as follows: 15.115.2 The letter x with the bar over it is read “x bar.” The bar notation indicates thea

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is context Even knowing that it’s the measurement in inches for a key characteristic,

we still want more: Is this representative of the other parts? How does this comparewith what we’ve made in the past? Context allows us to process the data intoinformation

Descriptive data are commonly presented as point estimates We see pointestimates in many aspects of our personal and business life: newspapers report theunemployment rate, magazines poll readers’ responses, quality departments reportscrap rate Each of these examples, and countless others, provide us with an estimate

of the state of a population through a sample Yet these point estimates often lackcontext Is the reported reader response a good indicator of the general population?

Is the response changing from what it has been in the past?

Statistics help us to answer these questions In this chapter, we explore sometools for providing an appropriate context for data

15.1.1 H ISTOGRAMS

A histogram is a graphical tool used to visualize data It is a bar chart, where eachbar represents the number of observations falling within a range of data values Anexample is shown in Figure 15.1

An advantage of the histogram is that the process location is clearly identifiable

In Figure 15.1, the central tendency of the data is about 0.4 The variation is alsoclearly distinguishable: we expect most of the data to fall between 0.1 and 1.0 Wecan also see if the data are bounded or have symmetry

If your data are from a symmetrical distribution, such as the bell-shaped normaldistribution, the data will be evenly distributed about a center If the data are notevenly distributed about the center of the histogram, it is skewed If the data appearskewed, you should understand the cause of this behavior Some processes willnaturally have a skewed distribution, and may also be bounded, such as the concen-tricity data in Figure 15.1 Concentricity has a natural lower bound at zero, because

no measurements can be negative The majority of the data is just above zero, sothere is a sharp demarcation at the zero point representing a bound

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320 The Manufacturing Handbook of Best Practices

If double or multiple peaks occur, look for the possibility that the data are comingfrom multiple sources, such as different suppliers or machine adjustments

One problem that novice practitioners tend to overlook is that the histogramprovides only part of the picture A histogram of a given shape may be produced

by many different processes, although the only difference in the data is their order

So the histogram that looks like it fits our needs could have come from data showingrandom variation about the average, or from data clearly trending toward an unde-sirable condition Because the histogram does not consider the sequence of thepoints, we lack this information Statistical process control (SPC) provides thiscontext

15.2 OVERVIEW OF SPC

Statistical process control is a method of detecting changes to a process Unlikemore general enumerative statistical tools, such as hypothesis testing, which allowconclusions to be drawn on the past behavior of static populations, SPC is an

behavior, using its past behavior as a model

Applications of SPC in business are as varied as business itself, includingmanufacturing, chemical processes, banking, healthcare, and general service SPCmay be applied to any time-ordered data, when the observations are statisticallyindependent Methods addressing dependent data are discussed under 15.5.1, Auto-correlation

The tool of SPC is the statistical control chart, or more simply, the control chart.The control chart was developed in the 1920s by Walter Shewhart while he wasworking for Bell Laboratories Shewhart defined statistical control as follows:

A phenomenon is said to be in statistical control when, through the use of past experience, we can predict how the phenomenon will vary in the future.

FIGURE 15.1 Example histogram for non-normal data Concentricity Best-fit curve: Johnson Sb; K–S test: 0.999 Kac K of fit is not significant; specified lower bound = 0.000.

8.0 6.4 4.8 3.2 1.6 0.0 0.100 0.300 0.500 0.700 0.900 1.100 1.300 1.500

0.00 6.67 13.33 20.00 26.67 33.33 High

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Statistical Process Control 321

15.2.1 C ONTROL C HART P ROPERTIES

Control charts take many forms, depending on the process that is being analyzedand the data available from that process All control charts have the followingproperties:

• The x-axis is sequential, usually a unit denoting the evolution of time

• The y-axis is the statistic that is being charted for each point in time.Examples of plotted statistics include an observation, an average of two

or more observations, the median of two or more observations, a count

of items that meet a criteria of interest, or the percentage of items meeting

a criteria of interest

• Limits are defined for the statistic that is being plotted These control

provid-ing an indication of the bounds of expected behavior for the plottedstatistic They are never determined using customer specifications or goals

An example of a control chart is shown in Figure 15.2 In this example, the cycletime for processing an order is plotted on an individual-X control chart, the top chartshown in the figure The cycle time is observed for a randomly selected order eachday and plotted on the control chart For example, the cycle time for the third order

is about 25

In Figure 15.2, the centerline (PCL, for process center line) of the individual-Xchart is the average of the observations (18.6 days) It provides an indication of theprocess location Most of the observations will fall somewhere close to this averagevalue, so it is our best guess for future observations, as long as the observations arestatistically independent of one another

We notice from Figure 15.2 that the cycle time process has variation That is,the observations are different from one another The third observation at 25 days is

FIGURE 15.2 Example of individual-X/moving control charts (shown with histogram).

K-S: 0.929 Cpk: 1.16 Cp: (N/A) AVERAGE(m): 18.6 PROCESS SIGMA: 4.7 HIGH SPEC: 35.0

% HIGH: 0.0265%

UCL : 32.8 LCL: 4.4

37.5 32.5 27.5 22.5 17.5 12.5 7.5 2.5

0 6 12 18

LCL=0.0 RBAR=5.3

UCL=17.4 LCL=4.4 PCL=18.6 UCL=32.8

3

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clearly different from the second observation at 17 days Does this mean that theprocess is changing over time?

The individual-X chart has two other horizontal lines, known as control limits.The upper control limit (UCL) is shown in Figure 15.2 as a line at 32.8 days; thelower control limit (LCL) is drawn at 4.4 days The control limits indicate thepredicted boundary of the cycle time In other words, we don’t expect the cycle time

to be longer than about 33 days or shorter than about 4 days

For the individual-X chart shown in Figure 15.1, the control limits are calculated

as follows:

(15.1)(15.2)

The letter x with the bar over it is read “x bar.” The bar notation indicates theaverage of the parameter, so in this case, the average of the x, where x is anobservation The parameter σx (read as “sigma of x”) refers to the process standarddeviation (or process sigma) of the observations, which in this case is calculatedusing the bottom control chart in Figure 15.2, the moving range chart

The moving range chart uses the absolute value of the difference (i.e., range)between neighboring observations to estimate the short-term variation For example,the first plotted point on the moving range chart is the absolute value of the differencebetween the second observation and the first observation In this case, the firstobservation is 27 and the second is 17, so the first plotted value on the moving rangechart is 10 (27 – 17)

The line labeled RBAR on the moving range chart represents the average movingrange, calculated by simply taking the average of the plotted points on the movingrange chart The moving range chart also has control limits, indicating the expectedbounds on the moving range statistic The lower control limit on the moving rangechart in this example is zero The upper control limit is shown in Figure 15.2 as 17.4.The moving range chart’s control limits are calculated as

(15.3)

(15.4)Process sigma, the process standard deviation, is calculated as

(15.5)

For a moving range chart, the parameters d3 and d2 are 0.853 and 1.128, respectively

UCL x = + 3σx x LCL x= − 3σx x

UCL= + 3R d3σx

LCL=MAX( ,0R−3d3σx)

σx

R d

=2

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15.2.2 G ENERAL I NTERPRETATION OF C ONTROL C HARTS

The control limits on the individual-X chart help us to answer the question posed

in the section above Since all the observations fall within the control limits, theanswer is, “No, the process has not changed,” even though the observations areclearly different

We see variation in all processes, provided we have adequate measurementequipment to detect the variation The control limits represent the amount of variation

we expect to see in the plotted statistic, based on our observations of the process inthe past The fluctuation of the points between the control limits is due to the variationthat is intrinsic (built in) to the process We say that this variation is due to common

the process Although we don’t know what these causes are, their effect on theprocess is consistent over time

Recall that the control limits are based on process sigma, which for the ual-X chart is calculated based on the moving range statistic We can say that processsigma, and the resulting control limits, are determined by estimating the short-termvariation in the process If the process is stable, or in control, then we would expectwhat we observe now to be about the same as what we’ll observe in the future Inother words, the short-term variation should be a good predictor for the longer-termvariation if the process is stable

individ-Points outside the control limits are attributed to a special cause Although wemay not be able to immediately identify the special cause in process terms (forexample, cycle time increased due to staff shortages), we have statistical evidencethat the process has changed This process change can occur in two ways

• A change in process location, also known as a process shift For example,the average cycle time may have changed from 19 days to 12 days Processshifts may result in process improvement (for example, cycle time reduc-tion) or process degradation (for example, an increased cycle time) Rec-ognizing this as a process change, rather than just random variation of astable process, allows us to learn about the process dynamics, and toreduce variation and maintain improvements

• A change in process variation The variation in the process may also increase

or decrease Generally, a reduction in variation is considered a processimprovement, because the process is then easier to predict and manage.Control charts are generally used in pairs One chart, usually drawn as the bottom

of the two charts, is used to estimate the variation in the process In Figure 15.2,the moving range statistic was used to estimate the process variation, and becausethe chart has no points outside the control limits, the variation is in control.Conversely, if the moving range chart were not in control, the implication would

be that the process variation is not stable (i.e., it varies over time), so a single estimatefor variation would not be meaningful Inasmuch as the individual-X chart’s controllimits are based on this estimate of the variation, the control limits for the individual-Xchart should be ignored if the moving range chart is out of control We must remove

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the special cause that led to the instability in process variation before we can furtheranalyze the process Once the special causes have been identified in process terms,the control limits may be recalculated, excluding the data affected by the specialcauses

15.2.3 D EFINING C ONTROL L IMITS

To define the control limits we need an ample history of the process to set the level

of common-cause variation There are two issues here

• To distinguish between special causes and common causes, you must haveenough subgroups to define the common-cause operating level of yourprocess This implies that all types of common causes must be included

in the data For example, if we observed the process over one shift, usingone operator and a single batch of material from one supplier, we wouldnot be observing all elements of common cause variation that are likely

to be characteristic of the process If we defined control limits under theselimited conditions, then we would likely see special causes arising due tothe natural variation in one or more of these factors

• Statistically, we need to observe a sufficient number of data observationsbefore we can calculate reliable estimates of the variation and, to a lesserdegree, the average In addition, the statistical constants used to definecontrol chart limits (such as d2) are actually variables, and they approachconstants only when the number of subgroups is large For a subgroupsize of 5, for instance, the d2 value approaches a constant at about 25subgroups (Duncan, 1986) When a limited number of subgroups areavailable, short-run techniques may be useful These are covered later inthis chapter

15.2.4 B ENEFITS OF C ONTROL C HARTS

Control charts provide benefits in a number of ways Control limits represent the

lower control limits defines the variation that is expected from the process statistic.This is the variation due to common causes: causes common to all the processobservations We don’t concern ourselves with the differences between the obser-vations themselves If we want to reduce this level of variation, we need to redefinethe process, or make fundamental changes to the design of the process Demingdemonstrated this principle with his red bead experiment, which he regularly con-ducted during his seminars In this experiment, he used a bucket of beads or marbles.Most of the beads were white, but a small percentage (about 10%) of red beadswere thoroughly mixed with the white beads Students volunteered to be processworkers, who would dip a sample paddle into the bucket and produce a day’s

“production” of 50 beads for the “White Bead Company.” Another student wouldvolunteer to be an inspector The inspector counted the number of white beads ineach operator’s daily production The white beads represented usable output that

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could be sold to White Bead Company’s customers, and the red beads were scrap.These results were then reported to a manager, who would invariably chastiseoperators for a high number of red beads If the operator’s production improved onthe next sample, he or she was rewarded; if the production of white beads wentdown, more chastising

A control chart of the typical white bead output is shown in Figure 15.3 It’sobvious from the figure that there was variation in the process observations: eachdip into the bucket yielded a different number of white beads Has the processchanged? No! No one has changed the bucket, yet the number of white beads is

11 red beads in each sample of 50 beads

seen, process variation is quite natural Once we accept that every process exhibitssome level of variation, we then wonder how much variation is natural for thisprocess If a particular observation seems large, is it unnaturally large, or should anobservation of this magnitude be expected? The control limits remove the subjec-tivity from this decision, and define this level of natural process variation

In the absence of control limits, we assume that an arbitrarily large variation isdue to a shift in the process In our zeal to reduce variation, we adjust the process

to return it to its prior state For example, we sample the circled area in the leftmostdistribution in Figure 15.4 from a process that (unbeknownst to us) is in control Wefeel this value is excessively large, so assume the process must have shifted Weadjust the process by the amount of deviation between the observed value and theinitial process average The process is now at the level shown in the center distri-bution in Figure 15.4 We sample from this distribution and observe several valuesnear the initial average, and then sample a value such as is the circled area in thecenter distribution in the figure We adjust the process upward by the deviationbetween the new value and the initial mean, resulting in the rightmost distributionshown in the figure As we continue this process, we can see that we actually increase

the total process variation, which is exactly the opposite of our desired effect.Responding to these arbitrary observation levels as if they were special causes

is known as tampering This is also called “responding to a false alarm,” since a

FIGURE 15.3 Example, control chart for Deming’s red bead experiment Sample size = 50.

12 10 8 6 4 2 0 2

4 6 8 10121416182022242628303234363840

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false alarm is when we think that the process has shifted when it really hasn’t.Deming’s funnel experiment demonstrates this principle In practice, tamperingoccurs when we attempt to control the process to limits that are narrower than thenatural control limits defined by common cause variation Some causes of this:

• We try to control the process to specifications, or goals These limits aredefined externally to the process, rather than being based on the statistics

of the process

• Rather than using the suggested control limits defined at ±3 standarddeviations from the centerline, we use limits that are tighter (or narrower)than these, based on the faulty notion that this will improve the perfor-mance chart Using limits defined at ±2 standard deviations from thecenterline produces narrower control limits than the ±3 standard deviationlimits, so it would appear that the ±2 sigma limits are better at detectingshifts Assuming normality, the chance of being outside of a ±3 standarddeviation control limit is 0.27% if the process has not shifted On average,

a false alarm is encountered with these limits once every 370 subgroups( = 1/0.0027) Using ±2 standard deviation control limits, the chance ofbeing outside the limits when the process has not shifted is 4.6%, corre-sponding to false alarms every 22 subgroups! If we respond to these falsealarms, we tamper and increase variation

are collected and analyzed for a process, it seems almost second nature to assumethat we can understand the causes of this variation In Deming’s red bead experiment,the manager would congratulate operators when their dips in the bucket resulted in

a relatively low number of red beads, and chastise them if they submitted a highnumber of red beads This should seem absurd, because the operator had no controlover the number of red beads in each random sample Yet, this same experiment

FIGURE 15.4 Tampering increases process variation.

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happens daily in real business environments In the cycle time example shown above,suppose the order-processing supervisor, being unfamiliar with statistical processcontrol, expected all orders to be processed at a quick pace, say 15 days It seemedthe process could deliver at this rate, because it had processed orders at or belowthis many times in the past If this was the supervisor’s expectation, then he or shemay look for a special cause (“This order must be different from the others”) thatdoesn’t exist Instead, he or she should be redesigning the system (i.e., changing thefundamental nature of the bucket)

real-time basis, control charts result in process stability In the absence of a controlchart, a common reaction is to respond to process variation with process adjustments

As discussed above, this tampering results in an unstable process that has increasedvariation Personnel using a control chart to monitor the process in real time (as theprocess produces the observations) are trained to react with process adjustmentsonly when the control chart signals a process shift with an out-of-control point Theresulting process is stable, allowing its future capability to be estimated In fact, thefuture performance of processes may be estimated only if the process is stable (seealso, process capability later in this chapter)

15.3 CHOOSING A CONTROL CHART

Many control charts are available for our use One differentiator between controlcharts is the type of data to be analyzed:

Attribute data: also known as “count” data Typically, we will count the number

of times we observe some condition (usually something we don’t like, such as adefect or an error) in a given sample from the process

Variables data: also known as measurement data Variables data are continuous

in nature, generally capable of being measured to enough resolution to provide atleast ten unique values for the process being analyzed

Attribute data have less resolution than variables data, because we count only

if something occurs, rather than take a measurement to see how close we are to thecondition For example, attribute data for a manufacturing process might include thenumber of items in which the diameter exceeds the specification, whereas variablesdata for the same process might be the measurement of that part’s diameter.Attribute data generally provide us with less information than variables data wouldfor the same process Attribute data would generally not allow us to predict if the process

is trending toward an undesirable state, because it is already in this condition As aresult, variables data are considered more useful for defect prevention

15.3.1 A TTRIBUTE C ONTROL C HARTS

There are several attribute control charts, each designed for slightly different uses:

• NP chart — for monitoring the number of times a condition occurs,relative to a constant sample size NP charts are used for binomial data,

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which exist when each sample can either have this condition of interest,

or not have this condition For example, if the condition is “the product

is defective,” then each sample unit either is defective or not defective

In the NP chart, the value that is plotted is the observed number of units

that meet the condition in the sample For example, if we sample 50 items,

and 4 are defective, we plot the value 4 for this sample The NP chart

requires a constant sample size, inasmuch as we cannot directly compare

4 observations from 50 units with 5 observations from 150 units

Figure 15.3 provided an example of an NP chart

• P chart — for monitoring the percentage of samples having the condition,

relative to either a fixed or varying sample size Use the P chart for the

same data types and examples as the NP chart The value plotted is a

percentage, so we can use it for varying sample sizes When the samples

vary by more than 20% or so, it’s common to see the control limits vary

as well

• C chart — for monitoring the number of times a condition occurs, relative

to a constant sample size, when each sample can have more than one

instance of the condition C charts are used for Poisson data For example,

if the condition is a surface scratch, then each sample unit can have 0, 1,

2, 3 … etc., defects The value plotted is the observed number of defects

in the sample For example, if we sample 50 items and 65 scratches are

detected, we plot the value 65 for this sample The C chart requires a

constant sample size

• U chart — for monitoring the percentage of samples having the condition,

relative to either a fixed or varying sample size, when each sample can

have more than one instance of the condition Use the U chart for the

same data types and examples as the C chart The value that is plotted is

a percentage, so we can use it for varying sample sizes When the samples

vary by more than 20% or so, it’s common to see the control limits vary

as well An example of a U chart is shown in Figure 15.5

FIGURE 15.5 U control chart, number of cracks per injection molding piece.

0.30 0.25 0.20 0.15 0.10 0.05 0.00

08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00

UCL

PCL=0.105

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15.3.2 V ARIABLES C ONTROL C HARTS

Several variables charts are also available for use The first selection is generally

the subgroup size The subgroup size is the number of observations, taken in close

proximity of time, used to estimate the short-term variation In the cycle-time

example at the beginning of the chapter, the subgroup size was equal to one, since

only one observation was used for each plotted point

Sometimes we choose to collect data in larger subgroups because a single

observation provides only limited information about the process at that time By

increasing the subgroup size, we obtain a better estimate of both the process location

and the short-term variation at that time

Control charts available for variables data include

• Individual-X/moving range chart (a.k.a individuals chart, I chart, IMR

chart) Limited to subgroup size equal to one An example was provided

in the previous sections, with the calculations used to develop the chart

(Equations 15.1 through 15.5) Those calculations are valid for many

applications, as long as the distribution of the observations is not severely

non-normal The chart has been shown to be fairly robust to departures

from normality, but data that are severely bounded can cause irrational

control limits Figure 15.6a shows cycle-time data on an individual-X/

moving range chart using the standard calculations The lower control

limits are calculated as a negative number, which clearly cannot exist for

cycle-time data in the real world Figure 15.6b provides the same data on

an individual-X/moving range chart that uses a fitted curve to calculate

control limits with the same detection ability as a normal distribution’s

±3 sigma limits These revised control limits allow us to detect process

shifts (in this case, improvements to the process) that would go undetected

using the standard calculations Other techniques for dealing with

non-normality include data transformations, such as the Box-Cox transformation

• X-bar chart Used for subgroup size two and larger The plotted statistic

is the average of the observations in the subgroup The average value has

been shown to be insensitive to departures from normality, even for a

subgroup size as small as three or five, so the control limits need not be

adjusted for non-normal process distributions

X-bar control limits are calculated as follows:

(15.6)

(15.7)

The letter x with the two bars over it is read “x double bar.” Because the

bar notation indicates the average of the parameter, x double bar is the

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average of the subgroup averages The process sigma σx (read as “sigma

of x”) is calculated using either the range chart or the sigma chart The

range and sigma charts, like the moving range chart described earlier, are

used to estimate, and detect instability in, the process variation

• Range chart Plots the range of observations (i.e., largest minus the

small-est observation) within the subgroup Because it attempts to small-estimate the

variation within the subgroup using only two of the observations in the

subgroup (the smallest and largest), the estimate is not as precise as the

sigma statistic described below The range chart should not be used for

subgroup sizes larger than ten because of its poor performance Its

pop-ularity is due largely to its ease of use before computers Its control limits

are calculated as in Equations 15.3 through 15.5, where the parameters

d3 and d2 are found in reference tables, such as in Montgomery and Runger

FIGURE 15.6 Individual X/moving range charts.

g

Group range: Selected (1-30)

Auto drop: OFF

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

UCL=1.723 PCL=0.564

LCL=0.594 UCL=1.424

RBAR=0.436 LCL=0.000

Group range: Selected (1-30)

Auto drop: OFF

LCL=0.000 RBAR=0.436

UCL=1.424 LPCL=0.078 PCL=0.483 UPCL=1.747

1 3 5 7 9

10 12 14 16 18 20

0.3

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• Sigma chart Plots the sample standard deviation of observations within

the subgroup, where x-barj is the average of the j th subgroup, and n is

the subgroup size:

(15.8)

The sigma chart is always more accurate than the range chart The sigma

chart’s control limits are calculated as follows:

(15.9)

(15.10)Process sigma, the process standard deviation, is calculated as:

(15.11)

• Other charts The EWMA (exponentially weighted moving average) chart

and the CuSum (cumulative sum) chart each have unique properties that

make them preferable for particular situations Both charts are robust to

departures from normality, so they can be used for the bounded process

of Figure 15.6 Another valuable characteristic is their increased

sensitiv-ity to small process shifts, as an alternative to increasing the sample size

Although the plotted statistics are inconvenient to calculate by hand, the

use of computer software to generate the charts allows ease of use

com-parable to any of the other charts

15.3.3 S ELECTING THE S UBGROUP S IZE

Control charts rely upon rational subgroups to estimate the short-term variation in

the process This short-term variation is then used to predict the longer-term variation

defined by the control limits

A rational subgroup is simply “a sample in which all of the items are produced

under conditions in which only random effects are responsible for the observed

variation” (Nelson, 1988) As such, a rational subgroup has the following properties:

• The observations composing the subgroup are independent Two

obser-vations are independent if neither observation influences, or results from,

the other When observations are dependent on one another, we say the

process has autocorrelation, or serial correlation (these terms mean the

same thing) Autocorrelation is covered later in this chapter

UCL S = +S 3σx 1−c42

LCL S=MAX( ,0 S−3σx 1−c42)

σx

S c

=4

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