INTEGRATING RANKED SET SAMPLING IN MEMORY CONTROL CHARTS: A STUDY - - PowerPoint PPT Presentation

INTEGRATING RANKED SET SAMPLING IN MEMORY CONTROL CHARTS: A STUDY ON PEPSI COLA

Mu’azu Ramat Abujiya Presented at the CMS 2019 and MMEI 2019 University of Technology, Chemnitz, Germany March 27 - 30, 2019

1

Outline

Introduction Methodology

Performance Evaluation

Application Conclusions

2

Introduction

• The industrial revolution that brought along highly

advanced machineries has made quality

• f

manufactured products a thing of concern.

• To ensure conformity to certain quality criteria, a

statistical quality control chart is often used to establish a criterion for determining when the process is in-control or out-of-control.

3

Center Line (CL) Upper Control Limit (UCL) Lower Control Limit (LCL)

Introduction

• The Shewhart control charts is the most widely used

procedure for monitoring a process but they are only sensitive to large shifts.

• The primary objective of the control chart is to

quickly detect the formation of assignable causes so that investigation of the process and corrective measure may be taken before many nonconforming units are manufactured.

4

Introduction

• To monitor small and moderate changes in a process,

memory control charts such as the exponentially weighted moving average (EWMA) and the cumulative sum (CUSUM) were introduced.

• Today, both the schemes are gaining their share of

practical uses as a result of recent technological advancements in computers and technical expertise.

5

• Combined applications of the Shewhart, EWMA and

the CUSUM charts have also been suggested to detect changes that may escapes the individual charts.

Introduction

• Most of these charts, however, are based on the

assumption that samples are drawn from a process using the traditional simple random sampling (SRS).

• But since sampling techniques are very essential in all

statistical applications, developing control charts using well structured sampling technique such as ranked set sampling (RSS) can improve estimation process and reduce manufacturing cost.

6

Introduction

• RSS is a data collection method that is more effective

than random sampling in practical problems where the actual measurements of quality characteristics may be costly, destructive or time-consuming but could be ranked by visual inspection or some inexpensive method without actual measurements.

• Why RSS ……. ?

RSS utilizes extra information from specific units in the population to guide its search for a truly representative sample data.

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RSS Methodology

• RSS procedure: Involves drawing n random samples, each of

subgroup size n from a population and rank the units within each set with respect to a variable of interest. Then n measurements are obtained by take the smallest observation from the 1st set…....

• Define 𝑌 𝑗:𝑜 𝑘, 𝑗 = 1, 2, … , 𝑜 and 𝑘 = 1, 2, … , 𝑛 to be the 𝑗th
• rder statistic for RSS of size 𝑜 in the 𝑘th cycle from a process.

The RSS estimator for the mean and variance are respectively, given by

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ത 𝑌𝑆𝑇𝑇 = Τ 1 𝑜𝑛 σ𝑘=1

𝑛

σ𝑗=1

𝑜

𝑌 𝑗:𝑜 𝑘

RSS Methodology

9

• 𝑇𝑆𝑇𝑇

2

=

1 𝑜𝑛−1+𝑤𝑜 σ𝑘=1 𝑛

σ𝑗=1

𝑜

𝑌 𝑗:𝑜 𝑘 − ത 𝑌𝑆𝑇𝑇

2.

In this study, several individual and combined control charts for location and dispersion were developed for effective monitoring of wide range of changes in a process to ensure high performance accuracy, and adequate handling of uncertainties associated with manufacturing process. Most often, these uncertainties results from shifts in mean, standard deviation or both.

RSS based Control Charts

Combined Shewhart-EWMA ത 𝑌 charts:

• CS-EWMA control limit for the Shewhart component

𝑉𝐷𝑀𝑆𝑇𝑇−𝑇 = 𝜈0 + 𝑙 Var ത 𝑌𝑆𝑇𝑇 𝑀𝐷𝑀𝑆𝑇𝑇−𝑇 = 𝜈0 − 𝑙 Var ത 𝑌𝑆𝑇𝑇

• CS-EWMA control limit for the EWMA component

𝑉𝐷𝑀𝑆𝑇𝑇−𝐹 = 𝜈0 + 𝑀

𝜇Var ത 𝑌𝑆𝑇𝑇 2−𝜇

1 − 1 − 𝜇 2𝑗 𝑀𝐷𝑀𝑆𝑇𝑇−𝐹 = 𝜈0 − 𝑀

𝜇Var ത 𝑌𝑆𝑇𝑇 2−𝜇

1 − 1 − 𝜇 2𝑗

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RSS based Control Charts

• The RSS based CS-EWMA control chart gives an out-
• f-control signal when either

where

• The design parameters k and L are chosen based on

the choice of the smoothing constant λ to satisfy the required process in-control needs.

ത 𝑌𝑆𝑇𝑇 𝑘 < 𝑀𝐷𝑀𝑇ℎ𝑓𝑥 or ത 𝑌𝑆𝑇𝑇𝑘 > 𝑉𝐷𝑀𝑇ℎ𝑓𝑥 EWMA𝑆𝑇𝑇𝑘 < 𝑀𝐷𝑀𝐹𝑋𝑁𝐵 or EWMA𝑆𝑇𝑇𝑘 > 𝑉𝐷𝑀𝐹𝑋𝑁𝐵

EWMA𝑆𝑇𝑇 𝑘 = 𝜇 ത 𝑌𝑆𝑇𝑇𝑘 + 1 − 𝜇 EWMA𝑆𝑇𝑇 𝑘−1 and/or

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RSS based Control Charts

Combined Shewhart-CUSUM 𝑇 charts:

• The CUSUM component for the CS-CUSUM chart using RSS

is based on the statistic

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𝐷𝑉𝑇𝑉𝑁𝑆𝑇𝑇 𝑘

+

= max 0, 𝐷𝑉𝑇𝑉𝑁𝑆𝑇𝑇 𝑘−1

+

+ Τ 𝑇𝑆𝑇𝑇 𝑘 𝜏0 − 𝑙+ 𝐷𝑉𝑇𝑉𝑁𝑆𝑇𝑇 𝑘

−

= max 0, 𝐷𝑉𝑇𝑉𝑁𝑆𝑇𝑇 𝑘−1

−

− Τ 𝑇𝑆𝑇𝑇 𝑘 𝜏0 + 𝑙−

where 𝑙+ = Τ 𝑑4

∗ 1 + 𝜐

2; 𝜐 ≥ 1 and 𝑙− = Τ 𝑑4

∗ 1 + 𝜐

2 ; 𝜐 ≤ 1

thus, the CS-CUSUM 𝑇 chart gives an out-of-control signal when 𝐷𝑉𝑇𝑉𝑁𝑆𝑇𝑇 𝑘

+

> 𝐼

• r

Τ 𝑇𝑆𝑇𝑇 𝑘 𝜏0 > 𝑉𝐷𝑀S 𝐷𝑉𝑇𝑉𝑁𝑆𝑇𝑇 𝑘

−

> 𝐼

• r

Τ 𝑇𝑆𝑇𝑇 𝑘 𝜏0 < 𝑀𝐷𝑀𝑇

Performance Evaluation

• Average Ratio of ARLs (ARARL)
• Average Extra Quadratic Loss (AEQL)
• Performance Comparison Index (PCI)

ARARL = 1 𝜀max − 𝜀min න

𝜀min 𝜀max

ARL 𝜀 ARLbest 𝜀 d𝜀 AEQL = 1 𝜀max − 𝜀min න

𝜀min 𝜀max

𝜀2ARL 𝜀 d𝜀 PCI = Τ AEQL AEQLbest

• Average Run Length (ARL)

13

Performance Evaluation

where δ = Τ 𝑜 𝜈out − 𝜈0 𝜏0, is the process mean shift ARL δ is the ARL value of a control chart at 𝜀; ARLbest δ and AEQLbest are generated by the best performing charts.

UCL LCL

σ0 µout µ0

14

Performance Comparison

15

Applications

16

• Here,

we present practical examples to illustrate the application of the RSS based memory control charts using real dataset on fill volume of a soft drink bottle.

• The
• riginal

dataset was

• btained from a production

line

• f

the Pepsi-Cola production company, Al- Khobar, Saudi Arabia. We used re-sampling approach to obtain new dataset.

Applications

17

MRSS based CS-CUSUM ത 𝑌 control chart using real dataset. Classical CS-CUSUM ത 𝑌 control chart using real data.

1 2 3 4 5 6 7 8 9 1 5 9 13 17 21 25 29 33 37 41 45 49 53 Cumulative sum Observation number Z_srs C+_srs Shewhart control limit CUSUM control limit 1 2 3 4 5 6 7 8 9 10 1 5 9 13 17 21 25 29 33 37 41 45 49 53 Cumulative sum Observation number Z_mrss C+_mrss Shewhart control limit CUSUM control limit

Applications

18

MRSS based CS-EWMA ത 𝑌 control chart using real dataset. Classical CS-EWMA ത 𝑌 control chart using real dataset.

5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 6.1 6.2 6.3 6.4 6.5 6.6 1 4 7 10 13 16 19 22 25 28 31 34 Ploting statistics Observation number X bar Z+ srs SCL ECL

Indroduction of shift

5.4 5.5 5.6 5.7 5.8 5.9 6.0 6.1 6.2 6.3 6.4 6.5 1 4 7 10 13 16 19 22 25 28 31 34 Ploting statistics Observation number X bar Z+dmrss SCL ECL

Indroduction of shift

Applications

19

ERSS based CS-CUSUM 𝑆 control chart using real dataset. Classical CS-CUSUM 𝑆 control chart using real dataset.

2 4 6 8 10 12 14 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 Ploting statistics Observation number

Classical Shewhart-R Classical CUSUM-R Shewhart limit CUSUM limit

Mean line for Shewhart R plot Indroduction of shift

2 4 6 8 10 12 14 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 Ploting statistics Observation number

Shewhart-R DERSS CUSUM-R DERSS Shewhart limit CUSUM limit

Mean line for Shewhart R plot Indroduction of shift

Conclusion

• In this work, the RL properties of the charts were

evaluated and found to be substantially more effective in the detection of different sizes of shifts than most

• f the existing schemes without increasing the false

alarm rate.

• We demonstrated the practicability for the application
• f the proposed charts using Pepsi-Cola data sets. It is

found that apart from the detection effectiveness of the charts, the design procedures are easy to follow and implement.

20

Setup of RSS Scheme

x x x x x x x x x x x x x x x x x x x x x x x x x

Site

`Sets

Ranked Sets Randomly allocate to sets Rank within sets x11 x12 x13 x14 x15 1 2 3 4 5 x21 x22 x23 x24 x25 x31 x32 x33 x34 x35 x41 x42 x43 x44 x45 x51 x52 x53 x54 x55 x(11) x(12) x(13) x(14) x(15) 1 2 3 4 5 x(21) x(22) x(23) x(24) x(25) x(31) x(32) x(33) x(34) x(35) x(41) x(42) x(43) x(44) x(45) x(51) x(52) x(53) x(54) x(55)

Setup of RSS Scheme

Measure only diagonal samples

x(11) x(12) x(13) x(14) x(15) 1 2 3 4 5 x(21) x(22) x(23) x(24) x(25) x(31) x(32) x(33) x(34) x(35) x(41) x(42) x(43) x(44) x(45) x(51) x(52) x(53) x(54) x(55) Ranked Sets

XRSS = ( x(11) + x(22) + x(33) + x(44) + x(55) ) / 5