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Introduction

- Chapters 4 through 6 focused on Shewhart control

charts. - Major disadvantage of Shewhart control charts is

that it only uses the information about the

process contained in the last plotted point. - Two effective alternatives to the Shewhart

control charts are the cumulative sum (CUSUM)

control chart and the exponentially weighted

moving average (EWMA) control chart. Especially

useful when small shifts are desired to be

detected.

Consider the data in table 8-1, Column(a). The

first 20 of these observations were drawn at

random from normal distribution with µ10 and

s1 UCL µ3s 13 CL µ 10 LCL µ-3s 7

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8-1.2 The Tabular or Algorithmic Cusum

for Monitoring the Process Mean

- Let xi be the ith observation on the process
- If the process is in control then
- Assume ? is known or can be estimated.
- Accumulate derivations from the target ?0 above

the target with one statistic, C - Accumulate derivations from the target ?0 below

the target with another statistic, C - C and C-- are one-sided upper and lower cusums,

respectively.

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8-1.2 The Tabular or Algorithmic Cusum

for Monitoring the Process Mean

- Selecting the reference value, K
- K is often chosen halfway between the target ?0

and the out-of-control value of the mean ?1 that

we are interested in detecting quickly. - Shift is expressed in standard deviation units as

?1 ?0??, then K is

8-1.2 The Tabular or Algorithmic Cusum

for Monitoring the Process Mean

- The statistics are computed as follows
- The Tabular Cusum
- starting values are
- K is the reference value (or allowance or slack

value) - If either statistic exceed a decision interval

H, the process is considered to be out of

control. Often taken as a H 5?

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8-1.2 The Tabular or Algorithmic Cusum for

monitoring the process Mean

- Example 8-1
- The cusum control chart indicates the process is

out of control. (Table 8/2 shows that upper side

cusum at period 29 is . Since this

is the first period at which ).

Tabular cusum also indicates when the shift

probably occured. The counter N records the

number of consecutive periods since the

rose above the calue of zero. Since N 7 at

period 29, we would conclude that the process was

last in control at period 29-722. So the shift

is likely to occur between periods 22 and 23. - The next step is to search for an assignable

cause, take corrective action required, and

reinitialize the cusum at zero. - If an adjustment has to be made to the process,

may be helpful to estimate the process mean

following the shift.

Cusum Status Chart (Figure 8-3a)

MINITAB Version of Cusum Status Chart

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Recommendations for Cusum Design

The tabular cusum is designed by shoosing values

for the reference value K and the decision

interval H. It is usually recommended that these

paramters be selected to provide good ARL

performance. Consider the two sided ARLs shown

below where Hhs Kks

Note that a 1 s shift would be detected in either

8.38 samples or 10.4 samples with k0.5 and h5.

By comparison a shewart CC for individual

measurements would require 43.96

Generally, we want to choose k relative tro the

size of the shift we want to detect that is,

k1/2d, where d is the size of the shift in s

units. This approach comes very close to

minimizing the ARL1 value for detecting a shift

of size d for fixed ARL0

- Several techniques can be used to calculate the

ARL of a cusum. Woodal and adams recommended the

ARL approximation given by Siegmond because of

its simplicity. - For a one sided cusum with parameters h and k,

siegmunds approxiamation is - for ? ? 0, where ? d- k fot the upper one

sided cusum , ? d- k for the lower one

sided cusum , bh1.166, and d (µ1-µ0)/s.

If ? 0, one can use ARLb2. The quantity d

represents the shift in the mean, in the units of

s, for which the ARL is to be calculated.

Therefore, if d 0, we would calculate ARL0

from equation 8.6, whereas if d ? 0, wwe sould

calculate the value of ARL1 corresponding to a

shift of size d.

- To obtain ARL of the two sided cusum from the

ARLs of the two one sided statistics - Say ARL

and ARL- - use - To illustrate, consider the two sided cusum with

k1/2 and h5. To find ARL0 we sould first

calculate the ARL0 values for the two-sided

schemes. Set d 0, then ? d- k 0 ½ -

½, b h 1.166 51.166 6.166 and from

equation 8.6 - By symmetry, we have , and

so from equation 8.7 the in control ARL for the

two-sided cusum is - This is very close to the true ARL0 value of 465

shown in table 8.3

The Standardized Cusum

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Improving CusumResponsiveness for large shifts

- Cusum control chart is not as effective in

detecting large shifts in the process mean as the

Shewhart chart. - An alternative is to use a combined

cusum-Shewhart procedure for on-line control. - The combined cusum-Shewhart procedure can improve

cusum responsiveness to large shifts.

The Fast Initial Response or headstart feature

- These procedures were introduced to increase

sensitivity of the cusum control chart upon

start-up. - The fast initial response (FIR) or headstart sets

the starting values equal to some

nonzero value, typically H/2. - Setting the starting values to H/2 is called a 50

percent headstart.

The Fast Initial Response (FIR) Cusum

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More on Cusums

- Cusums are often used to determine if a process

has shifted off a specified target because it is

easy to calculate the required adjustment - One-sided cusums are often useful
- Cusums can also be used to monitor variability
- Cusums are available for other sample statistics

(ranges, standard deviations, counts,

proportions) - Rational subgroups and cusums

One-Sided Cusums

- There are practical situations where a single

one-sided cusum is useful. - If a shift in only one direction is of interest

then a one-sided cusum would be applicable.

Rational Subgroups

- Shewhart chart performance is improved with

rational subgrouping - Cusum is not necessarily improved with rational

subgrouping - Only if there is significant economy of scale or

some other reason for taking larger samples

should rational subgrouping be considered with

the cusum

A Cusum for Monitoring process Variability

- Let
- The standardized value of xi is
- A new standardized quantity (Hawkins (1981)

(1993)) is given by - Hawkins suggest that the ?i are sensitive to

variance changes rather than mean changes.

A Cusum for Monitoring process Variability

- ?I N(0, 1), two one-sided standardized scale

cusums are - The Scale Cusum
- where
- if either statistic exceeds h, the process is

considered out of control.

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The V-Mask Procedure

- The V-mask procedure is an alternative to the

tabular cusum. - It is often strongly advised not to use the

V-mask procedure for several reasons. - The V-mask is a two-sided scheme it is not very

useful for one-sided process monitoring problems. - The headstart feature, which is very useful in

practice, cannot be implemented with the V-mask. - It is sometimes difficult to determine how far

backwards the arms of the V-mask should extend,

thereby making interpretation difficult for the

practitioner. - Ambiguity associated with with ? and ?

The Cusum V-Mask

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EWMA control chart is also a good alternative to

the shewart control chart when we are interested

in detecting small shifts. The performance of the

EWMA control chart is aproximately equivalent to

that of the cumulative sum control chart. EWMA is

defined as

Starting value is the process target, so that

Sometimes the average of prelimnary data is used

as the starting value of the EWMA so that z0

. If the observations xi are independent random

variables with variance s2, then the variance of

zi is

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Design of an EWMA Control Chart

- The design parameters of the chart are L and ?.
- The parameters can be chosen to give desired ARL

performance. - In general, 0.05 ? ? ? 0.25 works well in

practice. - L 3 works reasonably well (especially with the

larger value of ?. - L between 2.6 and 2.8 is useful when ? ? 0.1
- Similar to the cusum, the EWMA performs well

against small shifts but does not react to large

shifts as quickly as the Shewhart chart. - EWMA is often superior to the cusum for larger

shifts particularly if ? 0.1

Design of the EWMA

Hunter has also stucked the EWMA and suggested

choosing ? so that the weight given to current

and previous observations watches as closely as

possible the weight given to these observations

by a shewart chart with the western Electric

rules. Then results in a recommended value of

?0.4. If ?3.054, then table 8.10 indicates that

this chart would have ARL0500 and for detecting

a shift of 1s in the process mean the ARL114.3

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Robustness of the EWMA to Non-normality

- As discussed in Chapter 5, the individuals

control chart is sensitive to non-normality. - A properly designed EWMA is less sensitive to the

normality assumption.

Robustness of EWMA toNon-normal Process Data

The ARL0 if the shewart individuals chart and

several EWMA control charts for these non-normal

distributions are given in table 8.11 and 8.12.

Two aspects of the information in these tables

are very striking. First, even moderately

non-normal distributions have the effect of

greately reducing the in control ARL of the

shewart individual chart. This will dramatically

increase the rate of false alarms. Second, on

EWMA with ?0.05 or ?0.05 and an approximately

chosen control limit will perform very well

against both normal and non-normal distributions.

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Extensions of the EWMA

- Fast initial response feature
- Monitoring variability
- Monitoring count data
- The EWMA as a predictor of process level

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Moving average control chart is more effective

than the shewart chart in detecting small process

shifts. However, it is generally not as effective

against small shifts as either the cusum or the

EWMA. The moving average cc is considered by some

to be simpler to implement than the cusum.

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