Statistical control of multiple-stream processes — a literature review Eugenio K. Epprecht PUC-Rio Rio de Janeiro, Brazil ISQC Syney1

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 ISQC Syney2

examples examples ISQC Syney3 Source: Lanning et al. (2002)

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 ISQC Syney4

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) ISQC Syney5

Group-charts (Boyd, 1950) Source: Pyzdek (1992) ISQC Syney6

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 ” ISQC Syney7

Between 1943 and 1995: Ott and Snee (JQT, 1973) Nelson’s runs scheme (JQT, 1986) ISQC Syney8

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 2. No serial correlation; 3. No cross correlation between streams. Nelson admits the possibility of the existence of cross- correlation, 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 practi cal situations ISQC Syney9

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 ISQC Syney10

A reasonable model for typical MSPs: two components of variation value of the quality variable at stream i in time t: x i (t) = b(t) + e i (t) where b (“mean level”): common component - may be random or have some dynamics ; e : individual component of stream i e i (t) ~ N(0,  2 ) i.i.d. over t and i, and independent from b ISQC Syney11

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) ISQC Syney12

Limitation of the classic group chart: example (Mortell & Runger, 1995) ISQC Syney13

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) ISQC Syney14

Recent schemes - using the model 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) ISQC Syney15 x i (t) = b(t) + e i (t)

Schemes using the model Schemes using the model Monitoring statistics for e i (t): Mortell & Runger (1995): the range between streams (R t -chart) Runger, Alt & Montgomery (1996): the variance between streams (S 2 -chart) Epprecht & Barbosa (2008): the differences between each stream and the average over streams (residuals GCC) ISQC Syney16 x i (t) = b(t) + e i (t)

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Other approaches / particular problems: Amin et al. (1999): MaxMin EWMA chart (largest and smallest observations in each sample) 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) ISQC Syney18

Other approaches / particular problems (cont.) Bothe (2008): capability index (average C pk ) Liu et al. (2008): multiple gauges in parallel Xiang and Tsung (2008): multiple-stream model to monitor a multi-stage process Mei (2010): sum (over all streams) of CUSUM statistics of the individual streams. (The individual CUSUM statistics are based on the logarithm of a likelihood ratio statistic). ISQC Syney19

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 ISQC Syney20

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.” ISQC Syney21

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. ISQC Syney22

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. ISQC Syney23

Perspectives: other applications, open issues, challenges and opportunities for research (cont.) Phase I analysis etc., etc... ISQC Syney24