Statistical control of
multiple-stream processes
— a literature review
Eugenio K. Epprecht
PUC-Rio
Rio de Janeiro, Brazil
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multiple-stream processes
processes in which a same type of item is
manufactured in several streams of output
in parallel
 continuous processes in which several
measures are taken at a cross section of
the product

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examples
Source: Lanning et al. (2002)
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MSP
literature on this topic: scarce
 first literature review on this topic (to my
knowledge)
 Focus: Industrial applications

◦ Main points/works (stressing essential
differences between approaches)
◦ Opportunities for research, challenges and
◦ Other fields for application
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Classic SPC scheme for MSP:
group-charts (Boyd, 1950)
recommended in textbooks and guides:
Burr (1976);
Pyzdek (1992),
Wise and Fair (1998),
Montgomery (until 3rd edition, 1997)
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Group-charts (Boyd, 1950)
Source: Pyzdek (1992)
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Boyd (1950) reported :
“the group chart for X-bar and R was developed by
the British during World War II and was described in
‘A First Guide to Quality Control for Engineers’, a
British Ministry of Supply publication compiled by Dr.
E. H. Sealy and issued in 1943.”
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Between 1943 and 1995:

Ott and Snee (JQT, 1973)

Nelson’s runs scheme (JQT, 1986)
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Assumptions
(validity conditions for application)
1.
Streams with same distribution;
Or it is possible to adjust them in the same mean value, and they have
roughly the same variance
No serial correlation;
3. No cross correlation between streams.
2.
Nelson admits the possibility of the existence of crosscorrelation, but does not tackle this case: he says that if
the streams are 100% correlated, one chart will suffice;
and considers then the case where the streams present
no correlation (or negligible)
Assumptions not realistic in many practical
situations
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Other limitations
Overall false-alarm rate (multiplicity issue)

Scarcely considered (if ever) in early works
Limited sensitivity, due to:


Necessarily wide limits
Very few observations (often only one) per stream
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A reasonable model for typical MSPs:
two components of variation
value of the quality variable at stream i in time t:
xi(t) = b(t) + ei(t)
where
b (“mean level”): common component
- may be random or have some dynamics;
e : individual component of stream i
ei(t) ~ N(0, s2) i.i.d. over t and i, and independent from b
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Limitations of the classic group
chart
Large variance requires large limits
Reduced sensitivity to shifts in one stream if
V(b) is relevant with respect to V(e)
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Limitation of the classic group chart: example
(Mortell & Runger, 1995)
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Limitations of the classic group
chart
converse situation:
when V(e) is large with respect to V(b):
the GCC will have little sensitivity to causes
that affect all streams
— at least less sensitivity than a chart on the
average of the measurements across all streams
(since this one will have tighter limits than the
GCC)
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Recent schemes - using the model
xi(t) = b(t) + ei(t)

Mortell & Runger (1995)

Runger, Alt & Montgomery (1996)

Epprecht & Barbosa (2008)
Monitoring statistics:
For b(t): the average of all streams in time t
Chart: according to the dynamics of b(t)
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Schemes using the model xi(t) = b(t) + ei(t)
Monitoring statistics for ei(t):

Mortell & Runger (1995):
the range between streams (Rt-chart)

Runger, Alt & Montgomery (1996):
the variance between streams (S2-chart)

Epprecht & Barbosa (2008):
the differences between each stream and
the average over streams (residuals GCC)
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Other approaches / particular
problems:

Amin et al. (1999): MaxMin EWMA chart

Wludyka and Jacobs (2002): group chart
and runs scheme for MS binomial processes

Lanning et al. (2008): MSP “where it is
possible to monitor only a fraction of the
total streams at a given time”: VSSI chart
for the average of all streams (shifts in a
single stream or in a very few streams were
rare and/or not relevant)
(largest and smallest observations in each sample)
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Other approaches / particular
problems (cont.)

Bothe (2008): capability index (average Cpk)

Liu et al. (2008): multiple gauges in parallel

Xiang and Tsung (2008): multiple-stream

Mei (2010): sum (over all streams) of CUSUM
model to monitor a multi-stage process
statistics of the individual streams.
(The individual CUSUM statistics are based on the logarithm
of a likelihood ratio statistic).
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Other approaches / particular
problems (cont.)

Epprecht and Barros (2013): real filling process
where the stream means differed, wandered, changed from
day to day, were very difficult to adjust, and production
runs were too short to enable good estimation of the
parameters of the individual streams

Jirasettapong and Rojanarowan (2011):
selection of the appropriate control charts for monitoring
a MSP
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Perspectives: other applications, open issues,
challenges and opportunities for research
Other applications



Similarity between MSPs and the “large number of multiple
streams of data” used in health-related surveillance,
for example “data available over time on a number of
different subregions, hospitals or physicians” (Woodall,
2000)
... underlying assumptions of the models and methods for
industrial MSPs “may be too restrictive for these methods
to be applied directly to health-related monitoring.”
“...it can be a challenge to control the rate of false alarms
and yet retain power to detect meaningful outbreaks.”
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Perspectives: other applications, open issues,
challenges and opportunities for research (cont.)
Open issues and challenges

Some processes may have hundreds of streams  how to
control the false-alarm rate while keeping enough detection
power??

Real multiple-stream processes can be very ill-behaved.
Example: I have seen a plant with six 20-stream filling processes in
which the stream levels had different means and variances and could
not be adjusted separately (one single pump and 20 hoses). For many
real cases with particular twists like this one, it happens that no
previous solution in the literature is applicable.
Developing methods for such specific real cases is a need.
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Perspectives: other applications, open issues,
challenges and opportunities for research (cont.)
Characterizing the process in order to select the
appropriate monitoring scheme (or adapt one, or
develop a new one), according to:
dynamic behaviour of the process over time, degree of
cross-correlation between streams, ratio between the
variabilities of the individual streams and of the common
component, type and size of shifts that are likely and/or
relevant to detect, ease or difficulty to adjust all streams in
the same target, process capability, on the number of
streams, feasibility of taking samples of more than one
observation per stream at each sampling time (or even the
feasibility of taking one observation of every stream at each
sampling time!), length of the production runs, etc...
This may not be trivial for the average practitioner in industry.
 A methodological guide for such an analysis can be a
very useful contribution.
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Perspectives: other applications, open issues,
challenges and opportunities for research (cont.)
Phase I analysis
 etc., etc...

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